Scalar curvature rigidity of domains in a 3-dimensional warped product

Xiaoxiang Chai Department of Mathematics, POSTECH, Pohang, Gyeongbuk, South Korea xxchai@kias.re.kr, xxchai@postech.ac.kr and Gaoming Wang Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China gmwang@tsinghua.edu.cn

Abstract.

A warped product with a spherical factor and a logarithmically concave warping function satisfies a scalar curvature rigidity of the Llarull type. We develop a scalar curvature rigidity of the Llarull type for a general class of domains in a three dimensional spherical warped product. In the presence of rotational symmetry, we identify this class of domains as those satisfying a boundary condition analogous to the logarithmic concavity of the warping function.

Key words and phrases:

Scalar curvature, mean curvature, stable capillary surface, prescribed mean curvature, rigidity, foliation.

1991 Mathematics Subject Classification:

53C24, 49Q20

1. Introduction

Llarull proved a scalar curvature rigidity theorem for the standard $n$ -spheres.

Theorem 1.1 ([Lla98]).

Let $g$ be a smooth metric on the n-sphere with the metric comparison $g\geqslant\bar{g}$ and the scalar curvature comparison $R_{g}\geqslant n(n-1)$ . Then $g=\bar{g}$ .

An interesting feature of this scalar curvature rigidity for spheres comparing to that of torus [SY79a], the Euclidean space [SY79b] and the hyperbolic space [MO89] is the requirement of a metric comparison $g\geqslant\bar{g}$ . A counterexample without the requirement $g\geqslant\bar{g}$ was given in [BM11] (for half $n$ -sphere fixing the geometry of the boundary), but $g\geqslant\bar{g}$ can be weakened, see [Lis10]. Llarull also showed that the condition $g\geqslant\bar{g}$ can be formulated more generally as the existence of a distance non-increasing map $F:(M,g)\to(\mathbb{S}^{n},\bar{g})$ of non-zero degree.

Recently, there were efforts in extending Llarull’s theorem to a spherical warped product

(1.1)

(\bar{M}^{n},\bar{g}):=([t_{-},t_{+}]\times S^{n-1},\mathrm{d}t^{2}+\psi(t)^{2}g_{\mathbb{S}^{n-1}})\text{ with }t_{-}<t_{+},\text{ }(\log\psi)^{\prime\prime}<0,

by spinors [CZ24], [BBHW24], [WX23b], by $\mu$ -bubbles [Gro21], [HLS23] and by spacetime harmonic functions [HKKZ].

We are interested in the Llarull type theorems of domains in the spherical warped product (1.2). Although the form (1.2) can also be considered as a domain in a larger spherical warped product, our focus will be on such domains with boundaries that are not necessarily given by $t$ -level sets. Previously, this direction has been explored by Lott [Lot21], Wang-Xie [WX23a] and Chai-Wan [CW24], which all involved spinors. Gromov first suggested the use of stable capillary minimal surface in studying the scalar curvature rigidity of Euclidean balls (see Section 5.8.1 of [Gro21]; Spin-Extremality of Doubly Punctured Balls) and Li [Li20] in three-dimensional Euclidean dihedral rigidity. In this article, we make use of the stable capillary surfaces with prescribed (varying) contact angle and prescribed mean curvature, or in the terminology of Gromov [Gro21], (part of) the boundary of a stable capillary $\mu$ -bubble. This is also a further development of the previous work [CW23] and the recent work of Ko-Yao [KY24] in the Euclidean case.

We consider three dimensions, fix a metric $g_{S^{2}}$ of positive Gauss curvature on the $2$ -sphere $S^{2}$ , and the following

(1.2)

(\bar{M}^{3},\bar{g}):=([t_{-},t_{+}]\times S^{2},\mathrm{d}t^{2}+\psi(t)^{2}g_{{S}^{2}})\text{ with }t_{-}<t_{+},\text{ }(\log\psi)^{\prime\prime}<0.

We reserve $g_{\mathbb{S}^{2}}$ for the standard round metric on the 2-sphere $S^{2}$ . We use $t_{x}$ to indicate the $t$ coordinate and $p_{x}$ to denote the $S^{2}$ coordinate of a point $x\in M\subset[t_{-},t_{+}]\times S^{2}$ . Let $K(p)$ be the Gauss curvature at $p\in(S^{2},g_{S^{2}})$ .

Let $P_{\pm}=\{t_{\pm}\}\times{S}^{2}$ , we always assume that $M$ lies between $P_{\pm}$ such that $P_{\pm}\cap\partial M$ is non-empty. If $P_{\pm}\cap\partial M$ contains only a point, we set this point to be $p_{\pm}$ . Let $\partial_{s}M=\partial M\backslash(P_{+}\cup P_{-})$ and $\bar{X}$ be the unit outward normal of $\partial_{s}M$ in $M$ with respect to $\bar{g}$ . We fix

(1.3)

\bar{h}(t)=2\psi^{\prime}(t)/\psi(t)

and $\bar{\gamma}$ to be the dihedral angles formed by $\partial_{s}M$ and $\Sigma_{t}=(\{t\}\times{S}^{2})\cap M$ that is given by

(1.4)

\cos\bar{\gamma}=\bar{g}(\bar{X},\partial_{t}).

We call the $t$ -suplevel set $\Omega_{t}^{+}$ and $t$ -sublevel set $\Omega_{t}^{-}$ . We also need to fix some more conventions for the direction of the unit normal, the sign of the mean curvatures and the dihedral angles. Let $\Sigma$ be a surface with boundary on $\partial_{s}M$ and separates $P_{+}\cap\partial M$ and $P_{-}\cap\partial M$ , we always fix the direction of the unit normal $N$ of $\Sigma$ to be the direction which points inside of the region bounded by $\Sigma,$ $P_{+}\cap\partial M$ and $\partial_{s}M$ . The mean curvature is then the trace of the second fundamental form $\nabla N$ . We fix $\gamma_{\Sigma}$ to be the contact angle formed by $\Sigma$ and $\partial_{s}M$ , that is, $\cos\gamma_{\Sigma}=\langle X,N\rangle$ . For the mean curvature of $\partial_{s}M$ , it is always computed with respect to the outward unit normal. The geometric quantity on $(M,\bar{g})$ comes with a bar unless otherwise specified (see Figure 2.1).

As is well known, the warped product metric (1.2) is conformal to a direct product metric. Indeed, let $s=\int^{t}\tfrac{1}{\psi(\tau)}\mathrm{d}\tau$ , then $\mathrm{d}s=\tfrac{1}{\psi(t)}\mathrm{d}t$ and

(1.5)

\mathrm{d}t^{2}+\psi(t)^{2}g_{{S}^{2}}=\psi(t)^{2}\mathrm{d}s^{2}+\psi(t)^{2}g_{{S}^{2}}=\psi(t)^{2}(\mathrm{d}s^{2}+g_{{S}^{2}})

where $t=t(s)$ is implicitly given by $s=\int^{t}\tfrac{1}{\psi(\tau)}\mathrm{d}\tau$ .

Now we state our scalar curvature rigidity result. (For a quick feel, we refer to Theorem 1.4.)

Theorem 1.2.

Assume that $\bar{g}$ is a Riemannian metric on $\bar{M}^{3}$ as in (1.2). Assume that $M$ is a domain in $\bar{M}$ such that $\partial_{s}M$ is convex with respect to the conformally related metric $\mathrm{d}s^{2}+g_{S^{2}}$ where $s$ is given in (1.5) and $g$ is a smooth metric on $M$ . If $g$ satisfies the metric comparison

(1.6)

g\geqslant\bar{g},

and the scalar curvature comparison

(1.7)

R_{g}\geqslant R_{\bar{g}},

and the mean curvature comparison

(1.8)

H_{\partial_{s}M}\geqslant\bar{H}_{\partial_{s}M}

along $\partial_{s}M$ , and further that $P_{\pm}\cap M$ and $\psi(t)$ satisfy either of the following conditions:

(1)

$\psi(t_{\pm})>0$ , $P_{\pm}\cap\partial M$ only contains a point where $\partial M$ is smooth under both metrics $g$ and $\bar{g}$ ;
(2)

$\psi(t_{\pm})>0$ , $P_{\pm}\cap\partial M$ only contains a point where $\partial M$ where the tangent cone of $(M,g)$ at $p_{\pm}$ is Euclidean, and the tangent cone of $(M,\bar{g})$ at $p_{\pm}$ is a round (solid) circular cone and that $(|t-t_{\pm}|^{-1}\Sigma_{t},|t-t_{\pm}|^{-2}\bar{g})$ converges to its axial section as $t\to t_{\pm}$ ;
(3)

$\psi(t)=a|t-t_{\pm}|+o(|t-t_{\pm}|^{2})$ as $t\to t_{\pm}$ where $a>0$ is a constant such that the tangent cone of $(M,\bar{g})$ at $p_{\pm}$ is of non-negative Ricci curvature and $(M,g)$ admits a metric tangent cone at $p_{\pm}$ ;
(4)

$\psi(t)>0$ on $[t_{-},t_{+}]$ , $P_{\pm}\cap\partial M$ is a disk, the mean curvatures $\pm H_{P_{\pm}\cap\partial M}\geqslant\pm\bar{h}_{P_{\pm}\cap\partial M}=\pm\bar{h}(t_{\pm})$ and the dihedral angles $\pm\gamma_{P_{+}\cap\partial M}\geqslant\pm\bar{\gamma}|_{P_{\pm}\cap\partial M}$ formed by $\partial_{s}M$ and $P_{\pm}\cap\partial M$ ,

then $g=\bar{g}$ .

Remark 1.3.

The mean curvature comparisons can be reformulated as $H_{\partial M}\geqslant\bar{H}_{\partial M}$ on $\partial M$ if all mean curvatures are computed with respect to the outward unit normal. For convergence of sequences of Riemannian manifolds and notions of tangent cones at a point of a Riemannian manifold, we refer to [BBI01, Chapter 8]. We believe that the extra condition on the tangent cone of $(M,\bar{g})$ at $p_{\pm}$ can be dropped in item (2).

We use a special case of $(M,\bar{g})$ to illustrate the convexity of $\partial_{s}M$ with respect to the metric $\mathrm{d}s^{2}+g_{S^{2}}$ given in (1.5). Assume that $g_{S^{2}}$ is just the standard round metric, that is,

(1.9)

g_{S^{2}}=g_{\mathbb{S}^{2}}=\mathrm{d}r^{2}+\sin^{2}r\mathrm{d}\theta^{2},\text{ }r\in[0,\pi],\text{ }\theta\in\mathbb{S}^{1},

using the polar coordinates and $M$ is given by

M=\{(t,r,\theta):\text{ }t\in[t_{-},t_{+}],\text{ }r\leqslant\rho(t)<\tfrac{\pi}{2},\text{ }\theta\in\mathbb{S}^{1}\}

for some positive function $\rho(t)$ on $[t_{-},t_{+}]$ . In this case, the prescribed contact angle $\bar{\gamma}$ only depends on $t$ . It is easy to check that the convexity of $\partial_{s}M$ with respect to $\mathrm{d}s^{2}+g_{\mathbb{S}^{2}}$ is equivalent to $\tfrac{\mathrm{d}\bar{\gamma}}{\mathrm{d}s}<0$ . Also, we find that

(1.10)

\tfrac{\mathrm{d}\bar{\gamma}}{\mathrm{d}t}<0

using (1.5) or that the angles are conformally invariant. The mean curvature of a $t$ -level set is given by

\bar{h}(t):=2\psi^{-1}\tfrac{\mathrm{d}\psi}{\mathrm{d}t}=2\tfrac{\mathrm{d}(\log\psi)}{\mathrm{d}t}.

The logarithmic concavity of $\psi$ is equivalent to the more geometric statement that the mean curvature of the $t$ -level set is monotonically decreasing as $t$ increases. The condition $\tfrac{\mathrm{d}\bar{\gamma}}{\mathrm{d}t}<0$ can be viewed as a boundary analog of the logarithmic concavity. Geometrically, (1.10) says that the dihedral angles (1.4) formed by $\Sigma_{t}$ and $\partial_{s}M$ monotonically decreases along the $\partial_{t}$ direction with respect to the metric $\bar{g}$ . This condition answers a question raised by Gromov at the end of [Gro21, Section 5.8.1]. See also Chai-Wan [CW24, Theorem 1.1].

We can have various scalar curvature rigidity assuming different combinations of geometric structures at $P_{\pm}$ . The following is a simple corollary of Theorem 1.2 where both $P_{\pm}\cap\partial M$ take the structure as specified in item (1) of Theorem 1.2.

Theorem 1.4.

Let $M$ be a smooth domain in $[t_{-},t_{+}]\times{S}^{2}$ with the metric $\mathrm{d}t^{2}+\psi(t)^{2}g_{S^{2}}$ where $(\log\psi)^{\prime\prime}<0$ , $\psi(t)>0$ on $[t_{-},t_{+}]$ . Assume that $\partial M$ is convex with respect to the conformally related metric $\mathrm{d}s^{2}+g_{S^{2}}$ where $s$ is given in (1.5) and $g$ is another metric on $M$ which satisfies the comparisons of:

(1)

the scalar curvatures $R_{g}\geqslant R_{\bar{g}}$ ,
(2)

the mean curvatures $H_{g}\geqslant H_{\bar{g}}$ of the boundary $\partial M$ in $M$ ,
(3)

and the metrics $g\geqslant\bar{g}$ ,

then $g=\bar{g}$ .

Our approach toward items (1)-(3) is by construction of surfaces of prescribed mean curvature and prescribed contact angles near $t=t_{-}$ which serves as barriers, a concept which now introduce. The existence of barriers enables us to find a stable capillary $\mu$ -bubble, if fact, $P_{\pm}\cap\partial M$ are natural barriers if we are in item (4) of Theorem 1.2.

Definition 1.5.

We say that a surface $\Sigma_{+}$ ( $\Sigma_{-}$ ) whose boundary separates $\partial(P_{+}\cap\partial M)$ and $\partial(P_{-}\cap\partial M)$ is an upper (lower) barrier if $H_{\Sigma_{+}}\geqslant\bar{h}|_{\Sigma_{+}}$ ( $H_{\Sigma_{-}}\leqslant\bar{h}|_{\Sigma_{-}}$ ) and $\gamma_{\Sigma_{+}}\geqslant\bar{\gamma}|_{\partial\Sigma_{+}\cap\partial M}$ ( $\gamma_{\Sigma_{-}}\leqslant\bar{\gamma}|_{\partial\Sigma_{-}\cap\partial M}$ ) along $\partial\Sigma_{+}$ ( $\partial\Sigma_{-}$ ). We call $\Sigma_{+}$ and $\Sigma_{-}$ are a set of barriers if $\Sigma_{+}$ and $\Sigma_{-}$ are respectively an upper barrier and a lower barrier and $\Sigma_{+}$ is closer to $P_{+}$ than $\Sigma_{-}$ .

Our construction of barriers near $t=t_{\pm}$ are purely local, and in this way, our proof of items (1)-(3) can be reduced to the last item of Theorem 1.2, that is, the case with a set of barriers. In particular, if both $P_{\pm}\cap\partial M$ take the structure as specified in item (3) of Theorem 1.2, we obtain a genuine generalization of Theorem 1.1, since in the case of round metric, $\psi(t)=\sin t$ , $t\in[0,\pi]$ is allowed to take zero values at $t=0$ and $t=\pi$ . Hu-Liu-Shi [HLS23] (see also Gromov [Gro21]) used a $\mu$ -bubble approach for Theorem 1.1. As indicated earlier, we use the capillary version of the $\mu$ -bubble approach. However, our method differs from theirs in a technical manner when handling

\psi(t)=\sin t=t+o(|t|)

near $t=0$ . (Similarly, near $t=\pi$ .) They constructed a family of perturbations on the function $2\psi^{\prime}/\psi$ while our strategy is to perform a careful tangent cone analysis near $t=0$ or $t=\pi$ . As a result of this new strategy, we are able to generalize the Llarull Theorem 1.1 to the case where the background metric $\bar{g}$ are equipped with antipodal conical points.

Theorem 1.6.

Let $n=3$ and $(\bar{M},\bar{g})$ be a three dimensional warped product given in (1.1) such that

\psi(t_{\pm})=a_{\pm}|t-t_{\pm}|+o(|t-t_{\pm}|),\text{ }0<a_{\pm}\leqslant 1,

If $g$ is another smooth metric on $\bar{M}$ with possible cone singularity at only $t=t_{\pm}$ which satisfies the comparisons of metrics $g\geqslant\bar{g}$ and scalar curvatures $R_{g}\geqslant R_{\bar{g}}$ , then $g=\bar{g}$ .

Theorem 1.6 directly follows from the proof of item (3) of Theorem 1.2 with only slight changes and we omit its proof. See Remark 3.13. Note that the condition $0<a_{\pm}\leqslant 1$ ensures that the Ricci curvature of the tangent cone with respect to $\bar{g}$ at $t=t_{\pm}$ is non-negative. The scalar curvature rigidity of the Llarull type for $a_{\pm}>1$ is an interesting question. One could also compare Theorem 1.6 with [CLZ24] where conical singularities with respect to the metric $g$ are allowed at multiple points on $S^{n}$ .

Some of the essential difficulties of items (1) and (2) are already present in [CW23, Theorem 1.2 (2) and (3)]. In light of this, we only give a sketch for their proof in Section 4 , and refer relevant details to [CW23].

It is possible that the inequalities in $(\log\psi)^{\prime\prime}<0$ and the convexity of $\partial_{s}M$ can be weakened in some cases. For instance, we can consider a convex disk $M$ in $(-\infty,\infty)\times S^{2}$ with the direct product metric $\mathrm{d}t^{2}+g_{S^{2}}$ . In this case, $\log\psi$ vanishes. The Llarull type rigidity Theorem 1.4 is still valid for this $M$ .

Now one could naturally ask what are other shapes of point singularities, in particular, asymptotics of $\psi$ near $t=t_{\pm}$ , such that Theorem 1.2 remain valid. However, it is a quite intricate matter to which we do not have an answer at the moment. It is also desirable to find a proof for higher dimensional analogs of our results using the stable capillary $\mu$ -bubbles. This seems to be a promising direction to investigate being aware of the recent works [CWXZ24, WWZ24].

The article is organized as follows:

In Section 2, we introduce basics of stable capillary $\mu$ -bubble and we use it to show item (4) of Theorem 1.2.

In Section 3, we use the tangent cone analysis at $t=t_{-}$ to construct barriers and reduce item (3) of Theorem 1.2 to item (4).

In Section 4, we revisit our constructions in [CW23] and use the techniques developed there to show items (2) and (1) of Theorem 1.2.

Acknowledgments X. Chai was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. RS-2024-00337418) and an NRF grant No. 2022R1C1C1013511. G. Wang was supported by the China Postdoctoral Science Foundation (No. 2024M751604).

2. Stable capillary $\mu$ -bubble

In this section, we introduce the functional (2.1) whose minimiser is a stable capillary $\mu$ -bubble. We introduce the barrier condition which combining with a maximum principle ensures the existence of a regular minimiser to (2.1). By a rigidity analysis on the second variation of (2.1), we conclude the proof of item (4) of Theorem 1.2.

2.1. Notations

We set up some notations. Let $E\subset M$ be be a set such that $\partial E\cap M$ is a regular surface with boundary which we name it $\Sigma$ . We set

•

$N$ , unit normal vector of $\Sigma$ pointing inside $E$ ;
•

$\nu$ , unit normal vector of $\partial\Sigma$ in $\Sigma$ pointing outside of $\Sigma$ ;
•

$\eta$ , unit normal vector of $\partial\Sigma$ in $\partial M$ pointing outside of $\partial E\cap\partial M$ ;
•

$X$ : unit normal vectors of $\partial M$ in $M$ pointing outside of $M$ ;
•

$\gamma$ : the contact angle formed by $\Sigma$ and $\partial M$ and the magnitude of the angle is given by $\cos\gamma=\langle X,N\rangle$ ,
•

$\langle Y,Z\rangle=g(Y,Z)$ , the inner product of vectors $Y$ and $Z$ with respect to the metric $g$ ;
•

$\langle Y,Z\rangle_{\bar{g}}=\bar{g}(Y,Z)$ , the inner product of vectors $Y$ and $Z$ with respect to the metric $\bar{g}$ .

See Figure 2.1. We use a bar on every quantity to denote that the quantity is computed with respect to the metric $\bar{g}$ given in (1.2).

Refer to caption — Figure 2.1. Notations.

2.2. Functional and first variation

We fix $\bar{h}=2\psi^{\prime}/\psi$ and $\bar{\gamma}$ to be given by (1.4). We define the functional

(2.1)

I(E)=\mathcal{H}^{2}(\partial^{\ast}E\cap\operatorname{int}M)-\int_{E}\bar{h}-\int_{\partial^{\ast}E\cap\partial M}\cos\bar{\gamma},

where $\partial^{\ast}E$ denotes the reduced boundary of $E$ and the variational problem

(2.2)

\mathcal{I}=\inf\{I(E):\text{ }E\in\mathcal{E}\},

where $\mathcal{E}$ is the collection of contractible open subsets $E^{\prime}$ such that $P_{+}\subset E^{\prime}$ . Let $\Sigma$ be a surface with boundary $\partial\Sigma$ such that $\partial\Sigma$ separates $P_{\pm}$ . Then $\Sigma$ separates $M$ into two components and the component closer to $P_{+}$ is just $E$ . We reformulate the functional (2.1) in terms of $\Sigma$ . We define

(2.3)

F(\Sigma)=I(E)=|\Sigma|-\int_{E}\bar{h}-\int_{\partial E\cap\partial M}\cos\bar{\gamma}.

Let $\phi_{t}$ be a family of immersions $\phi_{t}:\Sigma\to M$ such that $\phi_{t}(\partial\Sigma)\subset\partial M$ and $\phi_{0}(\Sigma)=\Sigma$ . Let $\Sigma_{t}=\phi_{t}(\Sigma)$ and $E_{t}$ be the corresponding component separated by $\Sigma_{t}$ . Let $Y$ be the vector field $\tfrac{\partial\phi_{t}}{\partial t}$ . Define $\mathcal{A}(t)=F(\Sigma_{t})$ and $f=\langle Y,N\rangle$ , then by the first variation

(2.4)

\mathcal{A}^{\prime}(0)=\int_{\Sigma}f(H-\bar{h})+\int_{\partial\Sigma}\langle Y,\nu-\eta\cos\bar{\gamma}\rangle.

We know that if $\Sigma$ is regular, then it is of mean curvature $\bar{h}$ and meets $\partial M$ at a prescribed angle $\bar{\gamma}$ . And $E$ is called a capillary $\mu$ -bubble. The second variation at such $\Sigma$ is

(2.5)

\mathcal{A}^{\prime\prime}(0)=Q(f,f):=-\int_{\Sigma}(f\Delta f+(|A|^{2}+\operatorname{Ric}(N)+\partial_{N}\bar{h})f^{2})+\int_{\partial\Sigma}f(\tfrac{\partial f}{\partial\nu}-qf).

where $f\in C^{\infty}(\Sigma)$ and

(2.6)

q:=\tfrac{1}{\sin\bar{\gamma}}A_{\partial M}(\eta,\eta)-\cot\bar{\gamma}A(\nu,\nu)+\tfrac{1}{\sin^{2}\bar{\gamma}}\partial_{\eta}\cos\bar{\gamma}.

We define two operators

(2.7)

L=-\Delta-(|A|^{2}+\operatorname{Ric}(N)+\partial_{N}\bar{h})\text{ in }\Sigma,

and

B=\tfrac{\partial}{\partial\nu}-q\text{ on }\partial\Sigma.

The surface $\Sigma$ is called stable if

(2.8)

Q(f,f)\geqslant 0

for all $f\in C^{\infty}(M)$ . The second variation (2.5) is closely related to the variation of $H-\bar{h}$ and $\cos\gamma-\cos\bar{\gamma}$ . Indeed, let $f=\langle Y,N\rangle$ , we have that the first variation of $H-\bar{h}$ is

(2.9)		$\displaystyle\nabla_{Y}(H-\bar{h})$	$\displaystyle=Lf+\nabla_{Y^{\top}}(H-\bar{h})$
(2.10)			$\displaystyle=-\Delta f-(\|A\|^{2}+\operatorname{Ric}(N)+\partial_{N}\bar{h})f+\nabla_{Y^{\top}}(H-\bar{h}).$

And the first variation of the angle difference $\langle X,N\rangle-\cos\bar{\gamma}$ is

(2.11)		$\displaystyle\nabla_{Y}(\cos\gamma-\cos\bar{\gamma})=$	$\displaystyle-\sin\bar{\gamma}\tfrac{\partial f}{\partial\nu}$
(2.12)		$\displaystyle+(A_{\partial M}(\eta,\eta)$	$\displaystyle-\cos\bar{\gamma}A(\nu,\nu)+\tfrac{1}{\sin\bar{\gamma}}\partial_{\eta}\cos\bar{\gamma})f+\nabla_{Y^{\top}}(\langle X,N\rangle-\cos\bar{\gamma}).$

For $\Sigma$ , Schoen-Yau [SY79b] rewrote the term $|A|^{2}+\operatorname{Ric}(N)$ as

(2.13)

|A|^{2}+\operatorname{Ric}(N)=\tfrac{1}{2}(R_{g}-2K+|A|^{2}+H^{2})

where $K$ is the Gauss curvature of $\Sigma$ . Along the boundary $\partial\Sigma$ , we have the rewrite (see [RS97, Lemma 3.1] or [Li20, (4.13)])

(2.14)

\tfrac{1}{\sin\bar{\gamma}}A_{\partial M}(\eta,\eta)-\cos\bar{\gamma}A(\nu,\nu)=-H\cot\bar{\gamma}+\tfrac{H_{\partial M}}{\sin\bar{\gamma}}-\kappa

where $\kappa$ is the geodesic curvature of $\partial\Sigma$ in $\Sigma$ .

2.3. Analysis of stability

Starting from now on, we assume that $\Sigma$ is a regular stable capillary $\mu$ -bubble in $(M,g)$ which satisfies the assumptions of item (4) of Theorem 1.2.

Lemma 2.1.

Let $\Sigma$ be a regular stable capillary $\mu$ -bubble, then $\Sigma$ is a $t$ -level set.

Proof.

First, we note that the second variation $\mathcal{A}^{\prime\prime}(0)\geqslant 0$ as in (2.5). First, using Schoen-Yau’s rewrite (2.13) we see that

(2.15)		$\displaystyle\|A\|^{2}+\operatorname{Ric}(N)+\partial_{N}\bar{h}$
(2.16)	$\displaystyle=$	$\displaystyle\tfrac{1}{2}(R-2K+\|A\|^{2}+H^{2})+\partial_{N}\bar{h}$
(2.17)	$\displaystyle=$	$\displaystyle\tfrac{1}{2}(R-2K+\|A^{0}\|^{2}+\tfrac{H^{2}}{2}+H^{2})+\partial_{N}\bar{h}$
(2.18)	$\displaystyle=$	$\displaystyle\tfrac{1}{2}(R+\tfrac{3}{2}\bar{h}^{2}+2\partial_{N}\bar{h})-K+\tfrac{1}{2}\|A^{0}\|^{2},$

where $A^{0}$ is the traceless part of the second fundamental form. Similarly using (2.14), we see

q=-H\cot\bar{\gamma}+\tfrac{H_{\partial M}}{\sin\bar{\gamma}}-\kappa+\tfrac{1}{\sin^{2}\bar{\gamma}}\partial_{\eta}\cos\bar{\gamma}.

We obtain by letting $f\equiv 1$ in the (2.8) (also using (2.5) and (2.6)),

(2.19)	$\displaystyle 2\pi\chi(\Sigma)$	$\displaystyle=\int_{\Sigma}K+\int_{\partial\Sigma}\kappa$
(2.20)		$\displaystyle\geqslant\int_{\Sigma}\left[\tfrac{1}{2}(R+\tfrac{3}{2}\bar{h}^{2}+2\partial_{N}\bar{h})+\tfrac{1}{2}\|A^{0}\|^{2}\right]+\int_{\partial\Sigma}\left(\tfrac{H_{\partial M}}{\sin\bar{\gamma}}-\bar{h}\cot\bar{\gamma}+\tfrac{1}{\sin^{2}\bar{\gamma}}\partial_{\eta}\cos\bar{\gamma}\right)$
(2.21)		$\displaystyle\geqslant\int_{\Sigma}\tfrac{1}{2}\left(R+\tfrac{3}{2}\bar{h}^{2}+2\partial_{N}\bar{h}\right)+\int_{\partial\Sigma}\left(\tfrac{H_{\partial M}}{\sin\bar{\gamma}}-\bar{h}\cot\bar{\gamma}+\tfrac{1}{\sin^{2}\bar{\gamma}}\partial_{\eta}\cos\bar{\gamma}\right)$
(2.22)		$\displaystyle\geqslant\int_{\Sigma}\tfrac{1}{2}\left(R_{\bar{g}}+\tfrac{3}{2}\bar{h}^{2}+2\partial_{N}\bar{h}\right)+\int_{\partial\Sigma}\left(\tfrac{\bar{H}_{\partial M}}{\sin\bar{\gamma}}-\bar{h}\cot\bar{\gamma}+\tfrac{1}{\sin^{2}\bar{\gamma}}\partial_{\eta}\cos\bar{\gamma}\right),$

where in the last line we have incorporated the comparisons $R_{g}\geqslant R_{\bar{g}}$ in $M$ and $H_{\partial M}\geqslant\bar{H}_{\partial M}$ on $\partial M$ .

Now we estimate $R_{\bar{g}}+\tfrac{3}{2}\bar{h}^{2}+2\partial_{N}\bar{h}$ . We have that

\partial_{N}\bar{h}=\bar{g}(N,\nabla^{\bar{g}}\bar{h})\geqslant-|N|_{\bar{g}}|\nabla^{\bar{g}}\bar{h}|_{\bar{g}}=|N|_{\bar{g}}\bar{h}^{\prime},

since $g\geqslant\bar{g}$ , so

1=|N|_{g}\geqslant|N|_{\bar{g}},

and we get

\partial_{N}\bar{h}\geqslant\bar{h}^{\prime}.

R_{\bar{g}}+\tfrac{3}{2}\bar{h}^{2}+2\partial_{N}\bar{h}\geqslant R_{\bar{g}}+\tfrac{3}{2}\bar{h}^{2}+2\bar{h}^{\prime}.

For any point $x\in\Sigma$ , the right hide side is just $\tfrac{2K(p_{x})}{\psi^{2}(t_{x})}$ . Recall that $x=(t_{x},p_{x})$ is the coordinate of $x\in\bar{\Sigma}_{t}$ . This is by a direct calculation of the scalar curvature of the warped product metric (1.2). So

(2.23)

R_{\bar{g}}+\tfrac{3}{2}\bar{h}^{2}+2\partial_{N}\bar{h}\geqslant\tfrac{2K(p_{x})}{\psi^{2}(t_{x})}.

Let $\hat{g}=\mathrm{d}s^{2}+g_{{S}^{2}}$ and it is conformally related to $\bar{g}$ via (1.5). Let $\hat{X}$ be the unit outward normal of $\partial_{s}M$ in $M$ and $\hat{H}_{\partial_{s}M}$ be the mean curvature of $\partial_{s}M$ in $M$ with respect to $\hat{g}$ . Since $\bar{g}$ is conformal to $\hat{g}$ , by a well known formula of conformal change of mean curvature,

(2.24)

\bar{H}_{\partial_{s}M}=\tfrac{1}{\varphi(s)}(\hat{H}_{\partial_{s}M}+2\partial_{\hat{X}}\log\varphi).

Similarly, the mean curvature $\bar{h}$ of $\Sigma_{t}$ in $M$ is

(2.25)

\bar{h}(t)=\tfrac{2}{\varphi(s)^{2}}\varphi^{\prime}(s).

Hence, by (2.24), (2.25), (1.5) and that $\hat{g}(\partial_{s},\hat{X})=\cos\bar{\gamma}$ ,

\tfrac{\bar{H}_{\partial M}}{\sin\bar{\gamma}}-\bar{h}\cot\bar{\gamma}+\tfrac{1}{\sin^{2}\bar{\gamma}}\partial_{\eta}\cos\bar{\gamma}=\tfrac{1}{\psi(t_{x})\sin\bar{\gamma}}(\hat{H}_{\partial_{s}M}-\partial_{\psi\eta}\bar{\gamma}).

Inserting (2.23) and the above in (2.22) yields

2\pi\chi(\Sigma)\geqslant\int_{\Sigma}\tfrac{K(p_{x})}{\psi(t_{x})^{2}}\mathrm{d}\sigma+\int_{\partial\Sigma}\tfrac{1}{\psi(t_{x})\sin\bar{\gamma}}(\hat{H}_{\partial_{s}M}-\partial_{\psi\eta}\bar{\gamma})\mathrm{d}\lambda,

where we have written the area element $\mathrm{d}\sigma$ and line length element $\mathrm{d}\lambda$ explicitly in the metric $g$ . The rest of the proof is deferred to the next Lemma 2.2. ∎

Lemma 2.2.

Assume that $g\geqslant\bar{g}$ . If $\Sigma$ is a surface in $M$ whose boundary $\partial\Sigma$ is a simple smooth curve that separates $\partial(P_{+}\cap\partial M)$ and $\partial(P_{-}\cap\partial M)$ , then

(2.26)

\int_{\Sigma}\tfrac{K(p_{x})}{\psi(t_{x})^{2}}\mathrm{d}\sigma+\int_{\partial\Sigma}\tfrac{1}{\psi(t_{x})\sin\bar{\gamma}}(\hat{H}_{\partial_{s}M}-\partial_{\psi\eta}\bar{\gamma})\mathrm{d}\lambda\geqslant 2\pi,

where equality occurs if and only if $\Sigma$ is a $t$ -level set.

Proof.

It suffices to prove (2.26) for $\psi\equiv 1$ since $g\geqslant\bar{g}=\psi^{2}\hat{g}$ . In this case, $s=t$ , we just use $t$ . We also suppress the subscript $\partial_{s}M$ for clarity in this proof. In addition, we assume that every $t$ -coordinate of $\Sigma$ is strictly less than $t_{+}$ , since we can increase $t_{+}$ a little. Let $e_{1}$ be the unit tangent vector of $\partial\Sigma_{t}$ with respect to $\hat{g}$ and $e_{2}$ be the unit outward normal of $\partial\Sigma_{t}$ in $\partial_{s}M$ with respect to $\hat{g}$ . Let $T$ (resp. $\hat{T}$ ) be the unit tangent vector of $\partial\Sigma$ with respect to $g$ (resp. $\hat{g}$ ). We recall an ingenious inequality of [KY24, Lemma 3.2],

(2.27)

\hat{H}-\nabla_{\eta}\bar{\gamma}\geqslant\langle T,e_{2}\nabla_{1}\bar{\gamma}+(\hat{H}-\nabla_{2}\bar{\gamma})e_{1}\rangle,

where $\langle\cdot,\cdot\rangle$ denotes the inner product with respect to $\hat{g}$ .

Let $\lambda$ be an arc-length parameter of $\partial\Sigma$ with respect to $g$ , then the length element of $\partial\Sigma$ with respect to $\hat{g}$ is given by $|T|_{\hat{g}}\mathrm{d}\lambda=:\mathrm{d}\hat{\lambda}$ and $T=|T|_{\hat{g}}\hat{T}$ . Therefore,

(2.28)		$\displaystyle\int_{\partial\Sigma}\tfrac{1}{\sin\bar{\gamma}}(\hat{H}-\partial_{\eta}\bar{\gamma})\mathrm{d}\lambda$
(2.29)	$\displaystyle\geqslant$	$\displaystyle\int_{\partial\Sigma}\tfrac{1}{\sin\bar{\gamma}}\langle T,\partial_{1}\bar{\gamma}e_{2}+(\hat{H}-\partial_{2}\bar{\gamma})e_{1}\rangle\mathrm{d}\lambda$
(2.30)	$\displaystyle=$	$\displaystyle\int_{\partial\Sigma}\tfrac{1}{\sin\bar{\gamma}}\langle\hat{T},\nabla_{1}\bar{\gamma}e_{2}+(\hat{H}-\nabla_{2}\bar{\gamma})e_{1}\rangle\mathrm{d}\hat{\lambda}.$

Let $\hat{\eta}$ be the unit outward normal of $\partial\Sigma$ in $\partial_{s}M$ with respect to $\hat{g}$ . Recall the notation $\hat{A}=\hat{A}_{\partial_{s}M}$ , and we know that the components of $\hat{A}$ satisfy

(2.31)

\partial_{1}\bar{\gamma}=\hat{A}_{12},\text{ }\partial_{2}\bar{\gamma}=\hat{A}_{22},\text{ }\hat{H}-\partial_{2}\bar{\gamma}=\hat{A}_{11},

in the frame $\{e_{1},e_{2}\}$ . So

\langle\hat{T},\partial_{1}\bar{\gamma}e_{2}+(\hat{H}-\partial_{2}\bar{\gamma})e_{1}\rangle=\langle\hat{T},\hat{A}_{12}e_{2}+\hat{A}_{11}e_{1}\rangle=\hat{A}_{1\hat{T}}.

Since the orthonormal frames $\{\hat{T},\hat{\eta}\}$ and $\{e_{1},e_{2}\}$ have the same orientation, we can set $\hat{T}=a_{1}e_{1}+a_{2}e_{2}$ and so $\hat{\eta}=-a_{2}e_{1}+a_{1}e_{2}$ for some $a_{1}$ and $a_{2}$ which satisfy $a_{1}^{2}+a_{2}^{2}=1$ . It then follows that

(2.32)

\hat{A}_{1\hat{T}}=-(\hat{A}-\hat{H}\hat{g})(e_{2},\hat{\eta})=:-\hat{W}(e_{2},\hat{\eta}),

by considering also that $\hat{H}=\hat{A}_{11}+\hat{A}_{22}$ . Hence

\int_{\partial\Sigma}\tfrac{1}{\sin\bar{\gamma}}(\hat{H}-\partial_{\eta}\bar{\gamma})\mathrm{d}\lambda\geqslant-\int_{\partial\Sigma}\tfrac{1}{\sin\bar{\gamma}}\hat{W}(e_{2},\hat{\eta})\mathrm{d}\hat{\lambda}.

Suppose that $\partial\Sigma(t_{+})$ (we set $\Sigma(t_{+\pm})=\Sigma_{t_{\pm}}$ to avoid double subscripts) and $\partial\Sigma$ enclose a region $S$ in $\partial_{s}M$ . By the divergence theorem,

(2.33)		$\displaystyle-\int_{\partial\Sigma}\tfrac{1}{\sin\bar{\gamma}}\hat{W}(e_{2},\hat{\eta})\mathrm{d}\hat{\lambda}$
(2.34)	$\displaystyle=$	$\displaystyle-\int_{S}\hat{\nabla}^{S}_{i}\left(\hat{W}_{ij}\langle\tfrac{1}{\sin\bar{\gamma}}e_{2},e_{j}\rangle\right)\mathrm{d}\hat{\sigma}-\int_{\partial\Sigma(t_{+})}\tfrac{1}{\sin\bar{\gamma}}(\hat{A}-\hat{H}\hat{g})(e_{2},e_{2})\mathrm{d}\hat{\lambda}$
(2.35)	$\displaystyle=:$	$\displaystyle-\int_{S}\hat{\nabla}^{S}_{i}(\hat{A}_{ij}-\hat{H}\hat{g}_{ij})\langle\tfrac{1}{\sin\bar{\gamma}}e_{2},e_{j}\rangle\mathrm{d}\hat{\sigma}-\int_{S}\hat{W}_{ij}\langle\hat{\nabla}_{i}^{S}(\tfrac{1}{\sin\bar{\gamma}}e_{2}),e_{j}\rangle\mathrm{d}\hat{\sigma}-I_{3}$
(2.36)	$\displaystyle=:$	$\displaystyle-I_{1}-I_{2}-I_{3},$

where $\hat{\nabla}^{S}$ is the induced connection on $S$ with respect to the metric $\hat{g}$ . For $I_{1}$ , we use Gauss-Codazzi equation,

I_{1}=-\int_{S}\operatorname{Ric}_{\hat{g}}(\hat{X},\tfrac{1}{\sin\bar{\gamma}}e_{2})\mathrm{d}\hat{\sigma}=-\int_{S}K(p_{x})\cos\bar{\gamma}\mathrm{d}\hat{\sigma}=-\int_{S}K(p_{x})\langle\partial_{t},\hat{X}\rangle\mathrm{d}\hat{\sigma},

where the fact that $\hat{g}=\mathrm{d}t^{2}+g_{S^{2}}$ was also used in determining the Ricci curvature. For $I_{3}$ , we use (2.32), we see

I_{3}=\int_{\partial\Sigma(t_{+})}\tfrac{1}{\sin\bar{\gamma}}\hat{A}_{11}\mathrm{d}\hat{\lambda}=\int_{\partial\Sigma(t_{+})}\hat{\kappa}(p_{x},t)\mathrm{d}\hat{\lambda},

where $\hat{\kappa}(p,t)$ is the geodesic curvature of $\partial\Sigma_{t}$ in $\Sigma_{t}$ at $(p,t)\in\partial\Sigma_{t}$ . It seems tricky to calculate $\hat{\nabla}_{i}\left(\tfrac{1}{\sin\bar{\gamma}}e_{2}\right)$ in $I_{2}$ directly at a first sight, but we can convert to terms that are easier. By the identity,

\tfrac{1}{\sin\bar{\gamma}}e_{2}=\partial_{t}-\tfrac{\cos\bar{\gamma}}{\sin\bar{\gamma}}\hat{\nu}=\partial_{t}-\tfrac{\cos\bar{\gamma}}{\sin^{2}\bar{\gamma}}\hat{X}+\tfrac{\cos^{2}\bar{\gamma}}{\sin^{2}\bar{\gamma}}\partial_{t}

at $x\in\partial_{s}M$ , we see

(2.37)		$\displaystyle\hat{W}_{ij}\langle\hat{\nabla}_{i}(\tfrac{1}{\sin\bar{\gamma}}e_{2}),e_{j}\rangle$
(2.38)	$\displaystyle=$	$\displaystyle\hat{W}_{ij}\left\langle\hat{\nabla}_{i}\left(\partial_{t}-\tfrac{\cos\bar{\gamma}}{\sin^{2}\bar{\gamma}}\hat{X}+\tfrac{\cos^{2}\bar{\gamma}}{\sin^{2}\bar{\gamma}}\partial_{t}\right),e_{j}\right\rangle$
(2.39)	$\displaystyle=$	$\displaystyle-\tfrac{\cos\bar{\gamma}}{\sin^{2}\bar{\gamma}}\hat{W}_{ij}\hat{A}_{ij}+\hat{W}_{ij}\partial_{i}\left(\tfrac{\cos^{2}\bar{\gamma}}{\sin^{2}\bar{\gamma}}\right)\langle\partial_{t},e_{j}\rangle$
(2.40)	$\displaystyle=$	$\displaystyle-\tfrac{\cos\bar{\gamma}}{\sin^{2}\bar{\gamma}}\hat{W}_{ij}\hat{A}_{ij}-2\hat{W}_{i2}\tfrac{\cos\bar{\gamma}}{\sin^{3}\bar{\gamma}}\partial_{i}\bar{\gamma}\langle\partial_{t},e_{2}\rangle$

at $x$ . It is now a tedious task to check from the definition of $\hat{W}$ , $\langle\partial_{t},e_{2}\rangle=-\sin\bar{\gamma}$ and (2.31) that the above vanishes. Therefore, $I_{2}=0$ and to sum up, we have shown that

(2.41)

\int_{\partial\Sigma}\tfrac{1}{\sin\bar{\gamma}}(\hat{H}-\partial_{\eta}\bar{\gamma})\mathrm{d}\lambda\geqslant-\int_{S}K(p_{x})\langle\partial_{t},\hat{X}\rangle\mathrm{d}\hat{\sigma}+\int_{\partial\Sigma(t_{+})}\hat{\kappa}(p_{x},t)\mathrm{d}\hat{\lambda}.

Now we set the region enclosed by $\Sigma(t_{+})$ , $\Sigma$ and $\partial_{s}M$ to be $\Omega$ . Let $\hat{G}=\operatorname{Ric}_{\hat{g}}-\frac{1}{2}R_{\hat{g}}\hat{g}$ . Using the divergence free property of $\hat{G}$ (twice-contracted Gauss-Codazzi equation), $\partial_{t}$ , and the divergence theorem,

0=\int_{\Omega}\hat{\nabla}_{i}(\hat{G}_{ij}(\partial_{t})_{j})=\int_{\partial\Omega}\hat{G}(\partial_{t},\hat{X})=-\frac{1}{2}\int_{\partial\Omega}R_{\hat{g}}\langle\partial_{t},\hat{X}\rangle,=-\int_{\partial\Omega}K(p_{x})\langle\partial_{t},\hat{X}\rangle,

where $\hat{X}$ now also denotes the unit outward normal of $\partial\Omega$ with respect to $\hat{g}$ and $(\partial_{t})_{j}$ denotes the $j$ -th component of the vector field $\partial_{t}$ . Note that $\partial\Omega=S\cup\Sigma(t_{+})\cup\Sigma$ and so

0=\int_{\Sigma(t_{+})}K(p_{x})\langle\partial_{t},\partial_{t}\rangle-\int_{\Sigma}K(p_{x})\langle\partial_{t},\hat{N}\rangle+\int_{S}K(p_{x})\langle\partial_{t},\hat{X}\rangle,

and it follows from that $K(p_{x})>0$ that

(2.42)

\int_{\Sigma}K(p_{x})\geqslant\int_{\Sigma}K(p_{x})\langle\partial_{t},\hat{N}\rangle=\int_{\Sigma(t_{+})}K(p_{x})+\int_{S}K(p_{x})\langle\partial_{t},\hat{X}\rangle.

Finally, it follows from (2.41) and (2.42) that

\int_{\Sigma}K(p_{x})+\int_{\partial\Sigma}\tfrac{1}{\sin\bar{\gamma}}(\hat{H}-\partial_{\eta}\bar{\gamma})\mathrm{d}\lambda\geqslant\int_{\Sigma(t_{+})}K(p_{x})+\int_{\partial\Sigma(t_{+})}\hat{\kappa}(p_{x},t)\mathrm{d}\hat{\lambda}.

An application of the Gauss-Bonnet theorem on the right hand finishes the proof of (2.26). The equality case is easy to trace. ∎

Remark 2.3.

The convexity of $\partial_{s}M$ in $(M,\mathrm{d}s^{2}+g_{S_{2}})$ with $\mathrm{d}s^{2}+g_{S_{2}}$ given in (1.5) is used in the inequality (2.27).

2.4. Infinitesimally rigid surface

The surface $\Sigma$ is a stable capillary $\mu$ -bubble has more consequences than the mere Lemma 2.1. We can conclude that $\Sigma$ is a so-called infinitesimally rigid surface. See Definition 2.4.

All inequalities are in fact equalities by Lemma 2.2 and tracing the equalities in (2.22), we arrive that

(2.43)

R_{g}=R_{\bar{g}},N=\bar{N},|A^{0}|=0\text{ in }\Sigma

and

(2.44)

H_{\partial M}=\bar{H}_{\partial M}\text{ along }\partial\Sigma.

It then follows from the equality case of Lemma 2.2 that

(2.45)

t_{x}=t_{0}\text{ at all }x\in\bar{\Sigma}

for some constant $t_{0}\in[t_{-},t_{+}]$ . Because $\Sigma$ is stable (equivalently $Q(f,f)\geqslant 0$ ), so the eigenvalue problem

(2.46)

\left\{\begin{array}[]{ll}Lf&=\mu f\text{ in }\Sigma\\ Bf&=0\text{ on }\partial\Sigma\end{array}\right.

has a non-negative first eigenvalue $\mu_{1}\geqslant 0$ .

The analysis now is similar to [FCS80]. Letting $f\equiv 1$ in (2.8), using (2.43), (2.44) and (2.45), we get

(2.47)		$\displaystyle Q(1,1)=$	$\displaystyle\int_{\Sigma}\left[K-\tfrac{1}{2}(R+\tfrac{3}{2}\bar{h}^{2}+2\partial_{N}\bar{h})\right]$
(2.48)			$\displaystyle\quad+\int_{\partial\Sigma}\left[\kappa-(\tfrac{H_{\partial M}}{\sin\bar{\gamma}}-\bar{h}\cot\bar{\gamma}-\tfrac{1}{\sin\bar{\gamma}}\tfrac{\partial\bar{\gamma}}{\partial\eta})\right]=0.$

And so the first eigenvalue $\mu_{1}$ is zero, and the constant 1 is its corresponding eigenfunction.

By (2.43) and (2.18), the stability operator $L$ reduces to

L=-\Delta-\left(\tfrac{K(p_{x})}{\psi(t_{0})^{2}}-K\right);

by considering (2.44) and that $t_{x}=t_{0}$ , the boundary stability operator $B$ reduces to

B=\partial_{\nu}-\left(\frac{\hat{\kappa}(p_{x},t_{0})}{\psi(t_{0})}-\kappa\right).

Putting $f=1$ and $\mu_{1}=0$ in the eigenvalue problem (2.46), we get

(2.49)

K=\tfrac{K(p_{x})}{\psi^{2}(t_{0})}\text{ in }\Sigma,\text{ }\kappa=\tfrac{\hat{\kappa}(p_{x},t_{0})}{\psi(t_{0})}\text{ on }\partial\Sigma.

Now we summarize the properties of $\Sigma$ in the definition of an infinitesimally rigid surface.

Definition 2.4.

We say that $\Sigma$ is infinitesimally rigid if it satisfies (2.43), (2.44), (2.45) and (2.49).

2.5. Capillary foliation of constant $H-\bar{h}$

See for instance the works [Ye91], [BBN10] and [Amb15] on constructing CMC foliations. First, we construct a foliation with prescribed angles $\bar{\gamma}$ whose leaf is of constant $H-\bar{h}$ . Let $\phi(x,t)$ be a local flow of a vector field $Y$ which is tangent to $\partial M$ and transverse to $\Sigma$ and that $\langle Y,N\rangle=1$ .

In the following theorem, we only require that the scalar curvature of $(M,g)$ and the mean curvature of $\partial M$ are bounded below.

Theorem 2.5.

Suppose $(M,g)$ is a three manifold with boundary, if $\Sigma$ is an infinitesimally rigid surface, then there exists $\varepsilon>0$ and a function $w(x,t)$ on $\Sigma\times(-\varepsilon,\varepsilon)$ such that for each $t\in(-\varepsilon,\varepsilon)$ , the surface

(2.50)

\Sigma_{t}=\{\phi(x,w(x,t)):x\in\Sigma\}

is a surface of constant $H-\bar{h}$ intersecting $\partial M$ with prescribed angle $\bar{\gamma}$ . Moreover, for every $x\in\Sigma$ and every $t\in(-\varepsilon,\varepsilon)$ ,

(2.51)

w(x,0)=0,\text{ }\int_{\Sigma}(w(x,t)-t)=0\text{ and }\tfrac{\partial}{\partial t}w(x,t)|_{t=0}=1.

Proof.

Given a function in the Hölder space $C^{2,\alpha}(\Sigma)\cap C^{1,\alpha}(\bar{\Sigma})$ ( $0<\alpha<1$ ), we consider

\Sigma_{u}=\{\phi(x,u(x)):x\in\Sigma\},

which is a properly embedded surface if the norm of $u$ is small enough. We use the subscript $u$ to denote the quantities associated with $\Sigma_{u}$ .

Consider the space

\mathcal{Y}=\left\{u\in C^{2,\alpha}(\Sigma)\cap C^{1,\alpha}(\bar{\Sigma}):\int_{\Sigma}u=0\right\}

and

\mathcal{Z}=\left\{u\in C^{0,\alpha}(\Sigma):\int_{\Sigma}u=0\right\}.

Given small $\delta>0$ and $\varepsilon>0$ , we define the map

\Phi:(-\varepsilon,\varepsilon)\times B(0,\delta)\to\mathcal{Z}\times C^{0,\alpha}(\partial\Sigma)

given by

(2.52)			$\displaystyle\Phi(t,u)$
(2.53)		$\displaystyle=$	$\displaystyle\left((H_{t+u}-\bar{h}_{t+u})-\tfrac{1}{\|\Sigma\|}\int_{\Sigma}(H_{t+u}-\bar{h}_{t+u}),\langle X_{t+u},N_{t+u}\rangle-\cos\bar{\gamma}_{t+u}\right).$

Here, $B(0,\delta)$ is a ball of radius $\delta>0$ centered at the zero function in $\mathcal{Y}$ . For each $v\in\Sigma$ , the map

f:(x,s)\in\Sigma\times(-\varepsilon,\varepsilon)\to\phi(x,sv(x))\in M

gives a variation with

\tfrac{\partial f}{\partial s}|_{s=0}=\tfrac{\partial}{\partial s}\phi(x,sv(x))|_{s=0}=vN.

Since $\Sigma$ is infinitesimally rigid and using also (2.10) and (2.12), we obtain that

D\Phi_{(0,0)}(0,v)=\tfrac{\mathrm{d}}{\mathrm{d}s}\Phi(0,sv)|_{s=0}=\left(-\Delta v+\tfrac{1}{|\Sigma|}\int_{\partial\Sigma}\Delta v,-\sin\bar{\gamma}\tfrac{\partial v}{\partial\nu}\right).

It follows from the elliptic theory for the Laplace operator with Neumann type boundary conditions that $D\Phi(0,0)$ is an isomorphism when restricted to $0\times\mathcal{Y}$ .

Now we apply the implicit function theorem: For some smaller $\varepsilon$ , there exists a function $u(t)\in B(0,\delta)\subset\mathcal{X}$ , $t\in(-\varepsilon,\varepsilon)$ such that $u(0)=0$ and $\Phi(t,u(t))=\Phi(0,0)=(0,0)$ for every $t$ . In other words, the surfaces

\Sigma_{t+u(t)}=\{\phi(x,t+u(t)):x\in\Sigma\}

are of constant $H-\bar{h}$ with prescribed angles $\bar{\gamma}$ .

Let $w(x,t)=t+u(t)(x)$ where $(x,t)\in\Sigma\times(-\varepsilon,\varepsilon)$ . By definition, $w(x,0)=0$ for every $x\in\Sigma$ and $w(\cdot,t)-t=u(t)\in B(0,\delta)\subset\mathcal{X}$ for every $t\in(-\varepsilon,\varepsilon)$ . Observe that the map $s\mapsto\phi(x,w(x,s))$ gives a variation of $\Sigma$ with variational vector field is given by

\tfrac{\partial\phi}{\partial t}\tfrac{\partial w}{\partial s}|_{s=0}=\tfrac{\partial w}{\partial s}|_{s=0}Y.

Since for every $t$ we have that

(2.54)		$\displaystyle 0=$	$\displaystyle\Phi(t,u(t))$
(2.55)		$\displaystyle=$	$\displaystyle\left((H_{w(\cdot,t)}-\bar{h}_{w(\cdot,t)})-\tfrac{1}{\|\Sigma\|}\int_{\Sigma}(H_{w(\cdot,t)}-\bar{h}_{w(\cdot,t)}),\langle X_{t+u},N_{t+u}\rangle-\cos\bar{\gamma}_{t+u}\right),$

by taking the derivative at $t=0$ we conclude that

\langle\tfrac{\partial w}{\partial t}|_{t=0}Y,N\rangle=\tfrac{\partial w}{\partial t}|_{t=0}

satisfies the homogeneous Neumann problem. Therefore, it is constant on $\Sigma$ . Since

\int_{\Sigma}(w(x,t)-t)=\int_{\Sigma}u(x,t)=0

for every $t$ , by taking derivatives at $t=0$ again, we conclude that

\int_{\Sigma}\tfrac{\partial w}{\partial t}|_{t=0}=|\Sigma|.

Hence, $\tfrac{\partial w}{\partial t}|_{t=0}=1$ . Taking $\varepsilon$ small, we see that $\phi(x,w(x,t))$ parameterize a foliation near $\Sigma$ . ∎

Theorem 2.6.

There exists a continuous function $\Psi(t)$ such that

\tfrac{\mathrm{d}}{\mathrm{d}t}\left(\exp(-\int_{0}^{t}\Psi(\tau)\mathrm{d}\tau)(H-\bar{h})\right)\leqslant 0.

Proof.

Let $\psi:\Sigma\times I\to M$ parameterize the foliation, $Y=\tfrac{\partial\psi}{\partial t}$ , $v_{t}=\langle Y,N_{t}\rangle$ . Then

(2.56)

-\tfrac{\mathrm{d}}{\mathrm{d}t}(H-\bar{h})=\Delta_{t}v_{t}+(\operatorname{Ric}(N_{t})+|A_{t}|^{2})v_{t}+v_{t}\nabla_{N_{t}}\bar{h}\text{ in }\Sigma_{t},

and

(2.57)

\tfrac{\partial v_{t}}{\partial\nu_{t}}=[-\cot\bar{\gamma}A_{t}(\nu_{t},\nu_{t})+\tfrac{1}{\sin\bar{\gamma}}A_{\partial M}(\eta_{t},\eta_{t})+\tfrac{1}{\sin^{2}\bar{\gamma}}\nabla_{\eta_{t}}\cos\bar{\gamma}]v_{t}.

By shrinking the interval if needed, we assume that $v_{t}>0$ for $t\in I$ . By multiplying of (2.56) and integrate on $\Sigma_{t}$ , we deduce by integration by parts and applying the Schoen-Yau rewrite (2.13) that

(2.58)		$\displaystyle-(H-\bar{h})^{\prime}\int_{\Sigma_{t}}\tfrac{1}{v_{t}}$
(2.59)	$\displaystyle=$	$\displaystyle\int_{\Sigma_{t}}\tfrac{\Delta_{t}v_{t}}{v_{t}}+(\operatorname{Ric}(N_{t})+\|A_{t}\|^{2}+\nabla_{N_{t}}\bar{h})$
(2.60)	$\displaystyle=$	$\displaystyle\int_{\partial\Sigma_{t}}\tfrac{1}{v_{t}}\tfrac{\partial v_{t}}{\partial\nu_{t}}+\tfrac{1}{2}\int_{\Sigma_{t}}(R_{g}+\|A_{t}\|^{2}+H_{t}^{2}+2\nabla_{N_{t}}\bar{h})-\int_{\Sigma_{t}}K_{\Sigma_{t}}+\int_{\Sigma_{t}}\tfrac{\|\nabla v_{t}\|^{2}}{v_{t}^{2}}.$

Let $\chi=A-\tfrac{1}{2}\bar{h}\sigma$ , we have that

(2.61)		$\displaystyle\|A_{t}\|^{2}$
(2.62)	$\displaystyle=$	$\displaystyle\|\chi+\tfrac{1}{2}\bar{h}\sigma\|^{2}$
(2.63)	$\displaystyle=$	$\displaystyle\|\chi\|^{2}+\langle\chi,\bar{h}\sigma\rangle+\tfrac{1}{2}\bar{h}^{2},$
(2.64)	$\displaystyle=$	$\displaystyle\|\chi^{0}\|^{2}+\tfrac{1}{2}(\operatorname{tr}_{\sigma}\chi)^{2}+\bar{h}\operatorname{tr}_{\sigma}\chi+\tfrac{1}{2}\bar{h}^{2},$

where $\chi^{0}$ is the traceless part of $\chi$ . Also,

H^{2}=(\operatorname{tr}_{\sigma}\chi+\bar{h})^{2}=(\operatorname{tr}_{\sigma}\chi)^{2}+2\operatorname{tr}_{\sigma}\chi\bar{h}+\bar{h}^{2}.

(2.65)		$\displaystyle-(H-\bar{h})^{\prime}\int_{\Sigma_{t}}\tfrac{1}{v_{t}}$
(2.66)	$\displaystyle=$	$\displaystyle\int_{\partial\Sigma_{t}}\tfrac{1}{v_{t}}\tfrac{\partial v_{t}}{\partial\nu_{t}}+\tfrac{1}{2}\int_{\Sigma_{t}}(R_{g}+\|A_{t}\|^{2}+H_{t}^{2}+2\nabla_{N_{t}}\bar{h})-\int_{\Sigma_{t}}K_{\Sigma_{t}}+\int_{\Sigma_{t}}\tfrac{\|\nabla v_{t}\|^{2}}{v_{t}^{2}}$
(2.67)	$\displaystyle=$	$\displaystyle\int_{\partial\Sigma_{t}}\tfrac{1}{v_{t}}\tfrac{\partial v_{t}}{\partial\nu_{t}}+\tfrac{1}{2}\int_{\Sigma_{t}}(R_{g}+\tfrac{3}{2}\bar{h}^{2}+2\nabla_{N_{t}}\bar{h})$
(2.68)		$\displaystyle+\tfrac{1}{2}\int_{\Sigma_{t}}\|\chi^{0}\|^{2}+\tfrac{3}{2}(\operatorname{tr}_{\sigma}\chi)^{2}+3\bar{h}\operatorname{tr}_{\sigma}\chi-\int_{\Sigma_{t}}K_{\Sigma_{t}}+\int_{\Sigma_{t}}\tfrac{\|\nabla v_{t}\|^{2}}{v_{t}^{2}}$
(2.69)	$\displaystyle\geqslant$	$\displaystyle\int_{\partial\Sigma_{t}}\tfrac{1}{v_{t}}\tfrac{\partial v_{t}}{\partial\nu_{t}}+\int_{\Sigma_{t}}\tfrac{K(p_{x})}{\psi^{2}(t_{x})}+\tfrac{3}{2}(H-\bar{h})\int_{\Sigma_{t}}\bar{h}-\int_{\Sigma_{t}}K_{\Sigma_{t}},$

where in the last line we have also used the bound (2.23). Now we use (2.57) and also the rewrite (2.14), we see that

(2.70)		$\displaystyle-(H-\bar{h})^{\prime}\int_{\Sigma_{t}}\tfrac{1}{v_{t}}$
(2.71)	$\displaystyle\geqslant$	$\displaystyle\int_{\partial\Sigma_{t}}[-\cot\bar{\gamma}A_{t}(\nu_{t},\nu_{t})+\tfrac{1}{\sin\bar{\gamma}}A_{\partial M}(\eta_{t},\eta_{t})+\tfrac{1}{\sin^{2}\bar{\gamma}}\nabla_{\eta_{t}}\cos\bar{\gamma}]$
(2.72)		$\displaystyle\quad+\int_{\Sigma_{t}}\tfrac{K(p_{x})}{\psi^{2}(t_{x})}+\tfrac{3}{2}(H-\bar{h})\int_{\Sigma_{t}}\bar{h}-\int_{\Sigma_{t}}K_{\Sigma_{t}}$
(2.73)	$\displaystyle\geqslant$	$\displaystyle\int_{\partial\Sigma_{t}}[-\kappa_{\partial\Sigma_{t}}-H(t)\cot\bar{\gamma}+\tfrac{1}{\sin\bar{\gamma}}H_{\partial M}+\tfrac{1}{\sin^{2}\bar{\gamma}}\nabla_{\eta_{t}}\cos\bar{\gamma}]$
(2.74)		$\displaystyle\quad+\int_{\Sigma_{t}}\tfrac{K(p_{x})}{\psi^{2}(t_{x})}+\tfrac{3}{2}(H-\bar{h})\int_{\Sigma_{t}}\bar{h}-\int_{\Sigma_{t}}K_{\Sigma_{t}}$
(2.75)	$\displaystyle=$	$\displaystyle-\left(\int_{\Sigma_{t}}K_{\Sigma_{t}}+\int_{\partial\Sigma_{t}}\kappa_{\partial\Sigma_{t}}\right)+\left[\int_{\Sigma_{t}}\tfrac{K(p_{x})}{\psi^{2}(t_{x})}+\int_{\partial\Sigma_{t}}\left(\tfrac{1}{\sin\bar{\gamma}}H_{\partial M}-\bar{h}\cot\bar{\gamma}+\tfrac{1}{\sin^{2}\bar{\gamma}}\nabla_{\eta_{t}}\cos\bar{\gamma}\right)\right]$
(2.76)		$\displaystyle\quad+\tfrac{3}{2}(H-\bar{h})\int_{\Sigma_{t}}\bar{h}-(H-\bar{h})\int_{\partial\Sigma_{t}}\cot\bar{\gamma}.$

It follows from Lemma 2.2 and the proof of Lemma 2.1 that the second term in the big bracket is bounded below by $2\pi$ . Using also the Gauss-Bonnet theorem on the first term in the bracket, we see that

(2.77)

-(H-\bar{h})^{\prime}\int_{\Sigma_{t}}\tfrac{1}{v_{t}}\geqslant(H-\bar{h})(\tfrac{3}{2}\int_{\Sigma_{t}}\bar{h}-\int_{\partial\Sigma_{t}}\cot\bar{\gamma}).

Let

(2.78)

\Psi(t)=\left(\int_{\Sigma_{t}}\tfrac{1}{v_{t}}\right)^{-1}(\int_{\partial\Sigma_{t}}\cot\bar{\gamma}-\tfrac{3}{2}\int_{\Sigma_{t}}\bar{h}),

then note that we have assume that $v_{t}>0$ near $t=0$ , so $H-\bar{h}$ satisfies the ordinary differential inequality

(2.79)

(H-\bar{h})^{\prime}-\Psi(t)(H-\bar{h})\leqslant 0.

We see then

\tfrac{\mathrm{d}}{\mathrm{d}t}\left(\exp\left(-\int_{0}^{t}\Psi(\tau)\mathrm{d}\tau\right)(H-\bar{h})\right)\leqslant 0.

So the function $\exp(-\int_{0}^{t}\Psi(\tau)\mathrm{d}\tau)(H-\bar{h})$ is non-increasing. ∎

2.6. From local foliation to rigidity

Let $\Sigma_{t}$ be the constant mean curvature surfaces with prescribed contact angles $\bar{\gamma}$ with $\partial M$ .

Proposition 2.7.

Every $\Sigma_{t}$ constructed in Theorem 2.5 is infinitesimally rigid.

Proof.

Let $\Omega_{t}$ be the component of $M\backslash\Sigma_{t}$ whose closure contains $P_{+}\cap\partial M$ . We abuse the notation and define

F(t)=|\Sigma_{t}|-\int_{\Omega_{t}}\bar{h}-\int_{\partial\Omega_{t}}\cos\bar{\gamma}.

By the first variation formula (2.4),

F(t_{2})-F(t_{1})=\int_{t_{1}}^{t_{2}}\mathrm{d}t\int_{\Sigma_{t}}(H-\bar{h})v_{t}.

By Theorem 2.6,

H-\bar{h}\leqslant 0\text{ if }t\geqslant 0;\text{ }H-\bar{h}\geqslant 0\text{ if }t\leqslant 0,

which in turn implies that

F(t)\leqslant 0\text{ if }t\geqslant 0;\text{ }F(t)\leqslant 0\text{ if }t\leqslant 0.

However, $\Omega_{t}$ is a minimiser to the functional (2.1), hence

F(t)\equiv F(0).

It then follows every $\Sigma_{t}$ is a minimiser, hence infinitesimally rigid. ∎

We can conclude the proof of item (4) of Theorem 1.2.

Proof of item (4) of Theorem 1.2.

We note easily by the assumptions of item (4) of Theorem 1.2 that $\Sigma_{\pm}=P_{\pm}\cap\partial M$ are a set of barriers (see Definition 1.5), by the maximum principle, there exists a minimiser $E$ to (2.2) such that $E$ is either empty or $\partial E\backslash\partial_{s}M$ or lies entirely away from $P_{\pm}$ . Without loss of generality, we assume that $\Sigma=\partial E\cap\operatorname{int}M$ non-empty. By [DPM15], $\Sigma$ is a regular stable surface of prescribed mean curvature $\bar{h}$ and prescribed contact angle $\bar{\gamma}$ . Moreover, the second variation $\mathcal{A}^{\prime\prime}(0)\geqslant 0$ in (2.5) for any smooth family $\Sigma_{s}$ such that $\Sigma_{0}=\Sigma$ .

Let $Y=\tfrac{\mathrm{d}}{\mathrm{d}t}\phi(x,w(x,t))$ where $\phi$ and $w$ are as Theorem 2.5, we show first that $N_{t}$ is conformal. It suffices to show that $Y^{\bot}$ is conformal.

Since every $\Sigma_{t}$ is infinitesimally rigid by Proposition 2.7, from (2.46) and (2.48), we know that $\langle Y,N_{t}\rangle$ is a constant. Let $\partial_{i}$ , $i=1,2$ be vector fields induced by local coordinates on $\Sigma$ , $\partial_{i}$ also extends to a neighborhood of $\Sigma$ via the diffeomorphism $\phi$ . We have $\nabla_{\partial_{i}}\langle Y,N\rangle=0$ . Note that $\Sigma_{t}$ are umbilical with constant mean curvature $\bar{h}$ , so

\nabla_{\partial_{i}}N\equiv\tfrac{1}{2}\bar{h}\partial_{i}

and

(2.80)	$\displaystyle 0$	$\displaystyle=\nabla_{\partial_{i}}\langle Y,N\rangle$
(2.81)		$\displaystyle=\langle\nabla_{\partial_{i}}Y,N\rangle+\langle Y,\nabla_{\partial_{i}}N\rangle$
(2.82)		$\displaystyle=\langle\nabla_{\partial_{i}}Y,N\rangle+\tfrac{1}{2}\bar{h}\langle Y,\partial_{i}\rangle.$

On the other hand,

(2.83)	$\displaystyle 0$	$\displaystyle=\langle\nabla_{\partial_{i}}Y,N\rangle=\langle\nabla_{Y}\partial_{i},N\rangle$
(2.84)		$\displaystyle=Y\langle\partial_{i},N\rangle-\langle\partial_{i},\nabla_{Y}N\rangle$
(2.85)		$\displaystyle=-\langle\partial_{i},\nabla_{Y}N\rangle$
(2.86)		$\displaystyle=-\langle\partial_{i},\nabla_{Y^{\top}}N\rangle-\langle\partial_{i},\nabla_{Y^{\bot}}N\rangle$
(2.87)		$\displaystyle=-\tfrac{1}{2}\bar{h}\langle Y^{\top},\partial_{i}\rangle-\langle\partial_{i},\nabla_{Y^{\bot}}N\rangle.$

Combining the two equations above, we conclude that $\nabla_{Y^{\bot}}N=0$ which implies that $\Sigma$ foliates a warped product under the diffeomorphism $\phi$ (parameterized by $t$ ). Considering that the induced metric on $\Sigma$ agrees with the induced metric from $\bar{g}$ , we conclude that $g=\bar{g}$ . ∎

3. Construction of barriers (I)

In this section, we prove item (3) of Theorem 1.2. Our strategy is to construct a surface $\Sigma_{-}$ ( $\Sigma_{+}$ ) which serves as a lower (upper) barrier, and to use item (4) of Theorem 1.2 to finish the proof. This section is occupied by such a construction of $\Sigma_{-}$ .

3.1. Setting up coordinates and notations

For convenience, we set $t_{-}=0$ . As before, for any $t>0$ , we set $\Sigma_{t}$ to be the $t$ -level set of $t$ and $\Omega_{t}$ to be the $t$ -sublevel set, that is, all points of $M$ which lie below $\Sigma_{t}$ . Since both $(M,g)$ and $(M,\bar{g})$ has cone structures near where $t_{-}=0$ where each cross-section of the cone is a topological disk and it collapses to a point which we denote by $p_{0}$ .

In the following subsections, we construct graphical perturbations $\Sigma_{t,t^{2}u}$ of $\Sigma_{t}$ . Let $\Sigma_{t,t^{2}u}$ be the surface which consists of points $x+t^{2}u(x,t)N_{t}(x)$ where $N_{t}$ is the unit normal of $\Sigma_{t}$ with respect to the metric $g$ at $x\in\Sigma_{t}$ . The boundary $\partial\Sigma_{t,t^{2}u}$ might not lie in $\partial_{s}M$ , we can compensate this by expanding or shrinking $\Sigma_{t,t^{2}u}$ a little, and we still denote the resulting surface $\Sigma_{t,t^{2}u}$ .

We use a $t$ subscript on every geometric quantity on $\Sigma_{t}$ and a $t,t^{2}u$ subscript on every geometric quantity on $\Sigma_{t,t^{2}u}$ . We will explicitly indicate when there was confusion or change.

3.2. Capillary foliation with constant $H-\bar{h}$

We assume that $(M,g)$ and $(M,\bar{g})$ have isometric tangent cones at $p_{0}$ and we construct a foliation of constant $H-\bar{h}$ with prescribed angles $\bar{\gamma}$ near $p_{0}$ . In fact, later in Subsection 3.3, it is shown that this is the only case.

By the first variation formula of the mean curvatures

(3.1)

H_{t,t^{2}u}-H_{t}=-\Delta_{t}u-t^{2}(\operatorname{Ric}(N_{t})+|A_{t}|^{2})u+O(t),

where $\Delta_{t}$ is the Laplacian with respect to the induced rescaled metric $t^{-2}g|_{\Sigma_{t}}$ . Note that $\operatorname{Ric}(N_{t})=O(t^{-1})$ by the fact the tangent cone is $\mathrm{d}t^{2}+a^{2}t^{2}g_{S^{2}}$ . By the Taylor expansion of the function $\bar{h}$ , we see that

(3.2)

\bar{h}_{t,t^{2}u}-\bar{h}_{t}=\bar{h}^{\prime}(t)t^{2}u=t^{2}u\nabla_{N_{t}}\bar{h}+O(t).

(3.3)

(H_{t,t^{2}u}-\bar{h}_{t,t^{2}u})-(H_{t}-\bar{h}_{t})=-\Delta_{t}u-t^{2}(\operatorname{Ric}(N_{t})+|A_{t}|^{2}+\nabla_{N_{t}}\bar{h})u+O(t).

Note that both $H_{t}-\bar{h}_{t}$ and $H_{t,t^{2}u}-\bar{h}_{t,t^{2}u}$ are finite and $|A_{t}|^{2}+\nabla_{N_{t}}\bar{h}=O(t^{-1})$ considering that $(M,g)$ and $(M,\bar{g})$ have isometric tangent cones at $p_{0}$ .

Remark 3.1.

We elaborate a bit more on (3.1) and its $O(t)$ remainder term. Since the metric $g$ is close to $\mathrm{d}t^{2}+\psi(t)^{2}g_{S^{2}}$ when $t\to 0^{+}$ , we calculate the expansions with respect to the rescaled metric $t^{-2}g$ when computing for small $t>0$ . This is similar to [Ye91]. Then we rescale back and we obtain (3.1). The term $O(t)$ involves products of $|A_{t}|$ which is of order $t^{-1}$ with terms of order at most $O(1)$ . That is why the remainder is only of order $O(t)$ instead of $O(t^{2})$ .

Also, the variation of angles give

(3.4)			$\displaystyle t^{-1}[\langle X_{t,t^{2}u},N_{t,t^{2}u}\rangle-\langle X_{t},N_{t}\rangle]$
(3.5)		$\displaystyle=$	$\displaystyle-\sin\gamma\tfrac{\partial u}{\partial\nu_{t}}+t(-\cos\gamma A(t^{-1}\nu_{t},t^{-1}\nu_{t})+A_{\partial M}(\eta_{t}{,}\eta_{t}))u+O(t^{2}),$

where $\nu_{t}$ is the outward unit normal of $\partial\Sigma_{t}$ in $\Sigma_{t}$ with respect to the rescaled induced metric $t^{-2}g|_{\Sigma_{t}}$ (note that $t^{-1}\nu_{t}$ is of unit length with respect to $g$ ). Other geometric quantities are not rescaled. By the variation of the prescribed angle $\bar{\gamma}$ ,

(3.6)

t^{-1}(\cos\bar{\gamma}_{t,t^{2}u}-\cos\bar{\gamma}_{t})=-\tfrac{tu}{\sin\bar{\gamma}}\partial_{\bar{\eta}_{t}}\cos\bar{\gamma}+O(t^{2}).

(3.7)		$\displaystyle t^{-1}[(\langle X_{t,t^{2}u},N_{t,t^{2}u}\rangle-\cos\bar{\gamma}_{t,t^{2}u})-(\langle X_{t},N_{t}\rangle-\cos\bar{\gamma}_{t})]$
(3.8)	$\displaystyle=$	$\displaystyle-\sin\gamma\tfrac{\partial u}{\partial\nu_{t}}$
(3.9)		$\displaystyle\quad+t(-\cos\gamma A(t^{-1}\nu_{t},t^{-1}\nu_{t})+A_{\partial M}(\eta_{t}{,}\eta_{t})+\tfrac{1}{\sin\bar{\gamma}}\partial_{\bar{\eta}_{t}}\cos\bar{\gamma})u+O(t^{2}).$

Remark 3.2.

The term $A(t^{-1}\nu_{t},t^{-1}\nu_{t})=O(t^{-1})$ , however, we observe that $\lim_{t\to 0}\bar{\gamma}_{t}=\pi/2$ , and $A_{\partial M}(\eta_{t},\eta_{t})=O(1)$ since $A_{\partial M}(\bar{\eta}_{t},\bar{\eta}_{t})=O(1)$ . Or we can calculate with respect to the rescaling metric as in Remark 3.1.

Since $g$ and $\bar{g}$ has isometric tangent cone at $p_{0}$ , we see that the limit of the surface $(\Sigma_{t},t^{-2}g|_{\Sigma_{t}})$ as $t\to 0$ is $(\Sigma,a^{2}g_{{S}^{2}})$ where $\Sigma$ is a scaling copy of a geodesic disk of radius $\rho(0)=\lim_{t\to 0}\rho(t)>0$ in the standard 2-sphere. Consider the spaces

\mathcal{Y}=\left\{u\in C^{2,\alpha}(\Sigma)\cap C^{1,\alpha}(\bar{\Sigma}):\int_{\Sigma}u=0\right\}

and

(3.10)

\mathcal{Z}=\left\{u\in C^{0,\alpha}(\Sigma):\int_{D}u=0\right\}.

Given small $\delta>0$ and $\varepsilon>0$ , we define the map

(3.11)

\Phi:(-\varepsilon,\varepsilon)\times B(0,\delta)\to\mathcal{Z}\times C^{1,\alpha}(\partial\Sigma)

given by $\Phi(t,u)=(\Phi_{1}(t,u),\Phi_{2}(t,u))$ where $\Phi_{i}$ , $i=1,2$ are given by

(3.12)		$\displaystyle\Phi_{1}(t,u)=$	$\displaystyle(H_{t,t^{2}u}-\bar{h}_{t,t^{2}u})-\frac{1}{\|\Sigma\|}\int_{\Sigma}(H_{t,t^{2}u}-\bar{h}_{t,t^{2}u}),$
(3.13)		$\displaystyle\Phi_{2}(t,u)=$	$\displaystyle t^{-1}(\langle X_{t,t^{2}u},N_{t,t^{2}u}\rangle-\cos\bar{\gamma}_{t,t^{2}u})$

for $t\neq 0$ . Here $B(0,\delta)\subset\mathcal{Y}$ is an open ball with radius $\delta$ in the $C^{2,\alpha}$ norm and the integration on $\Sigma$ is with respect to the metric $g_{\mathbb{S}^{2}}$ . We extend $\Phi(t,u)$ to $t=0$ by taking limits, that is,

(3.14)

\Phi(0,u)=\lim_{t\to 0}\Psi(t,u).

We have the following proposition.

Proposition 3.3.

For each $t\in[0,\varepsilon)$ with $\varepsilon$ small enough, we can find $u_{t}=u(\cdot,t)\in C^{2,\alpha}(\Sigma)\cap C^{1,\alpha}(\bar{\Sigma})$ such that $\int_{\Sigma}u(\cdot,t)=0$ and

\Phi(t,u_{t})=(0,0).

In particular, each of the surfaces $\Sigma_{t,t^{2}u}$ have constant $\lambda_{t}:=H_{t,t^{2}u}-\bar{h}_{t,t^{2}u}$ and prescribed angles $\gamma_{t,t^{2}u}=\bar{\gamma}_{t,t^{2}u}$ . Moreover, $\lambda_{t}\leqslant 0$ for all small $t\in[0,\varepsilon)$ .

Before proving this proposition, we give a variational lemma.

Lemma 3.4.

Suppose that $(\Omega,\hat{g})$ is a compact manifold with piecewise smooth boundary $\partial\Omega$ and $\Sigma$ is a relatively open, smooth subset of $\partial\Omega$ . Let $g_{s}$ be a smooth family of metrics indexed by $s\in[0,\varepsilon)$ such that $g_{s}\to\hat{g}$ as $s\to 0$ , let $h_{s}=g_{s}-\hat{g}$ . We now omit the subscript on $h_{s}$ . Let $\nu$ be the unit outward normal of $\partial\Omega$ in $(\Omega,g)$ , $H_{g}$ and $A_{g}$ be the mean curvatures and the second fundamental form of $\partial\Omega$ in $(\Omega,g)$ computed with respect to the unit normal pointing outward of $\Omega$ , and $\gamma$ be the dihedral angles formed by $\Sigma$ and $\partial\Omega\backslash\Sigma$ with respect to the metric $g$ . We put a hat at appropriate places for the geometric quantities with respect to $\hat{g}$ .

Then

(3.15)			$\displaystyle 2\left[-\int_{\Sigma}(H_{g}-H_{\hat{g}})+\int_{\partial\Sigma}\tfrac{1}{\sin\gamma_{\hat{g}}}(\cos\gamma_{\hat{g}}-\cos\gamma_{g})\right]$
(3.16)		$\displaystyle=$	$\displaystyle\int_{\Omega}((R_{g}-R_{\hat{g}})+\langle\operatorname{Ric}_{\hat{g}},h\rangle_{\hat{g}})+2\int_{\partial\Omega\backslash\Sigma}(H_{g}-H_{\hat{g}})+\int_{\partial\Omega}\langle h,A_{\hat{g}}\rangle+O(s^{2}).$

Here, we have used $O(s^{2})$ to denote a remainder term comparable to $|h|_{\hat{g}}^{2}+|h|_{\hat{g}}|\hat{\nabla}h|_{\hat{g}}+|\hat{\nabla}h|_{\hat{g}}^{2}$ .

Proof.

From the variational formulas of the scalar curvature and the mean curvature, we have

(3.17)

R_{g}-R_{\hat{g}}=-\langle\operatorname{Ric}_{\hat{g}},h\rangle_{\hat{g}}-\operatorname{div}_{\hat{g}}(\mathrm{d}(\operatorname{tr}_{\hat{g}}h)-\operatorname{div}_{\hat{g}}h)+O(s^{2}),

and

(3.18)

2(H_{g}-H_{\hat{g}})=(\mathrm{d}(\operatorname{tr}_{\hat{g}}h)-\operatorname{div}_{\hat{g}}h)(\hat{\nu})-\operatorname{div}_{\sigma}Y-\langle h,A_{\hat{g}}\rangle_{\sigma}+O(s^{2})

where $Y$ is the tangential component dual to the 1-form $h(\cdot,\hat{\nu})$ . For the explicit form of the remainder terms, refer to [BM11, Proposition 4] and [MP21].

We integrate the variation of the mean curvature (3.18) on the boundary $\partial\Omega$ with respect to the metric $\hat{g}$ , we see

\int_{\partial\Omega}[(\mathrm{d}(\operatorname{tr}_{\hat{g}}h)-\operatorname{div}_{\hat{g}}h)(\hat{\nu})-\operatorname{div}_{\hat{\sigma}}Y-\langle h,A_{\hat{g}}\rangle]=2\int_{\partial\Omega}(H_{g}-H_{\hat{g}})+O(s^{2}).

By the divergence theorem and the variation of the scalar curvature,

\int_{\partial\Omega}(\mathrm{d}(\operatorname{tr}_{\hat{g}}h)-\operatorname{div}_{\hat{g}}h)(\hat{g})=\int_{\Omega}[-(R_{g}-R_{\hat{g}})-\langle\operatorname{Ric}_{\hat{g}},h\rangle_{\hat{g}}]+O(s^{2}).

For the term $\int_{\partial\Omega}\operatorname{div}_{\hat{\sigma}}Y$ , we follow [MP21, (3.18)] and obtain

\int_{\partial\Omega}\operatorname{div}_{\hat{\sigma}}Y=\int_{\Sigma}\operatorname{div}_{\hat{g}}Y+\int_{\partial\Omega\backslash\Sigma}\operatorname{div}_{\hat{\sigma}}Y=2\int_{\partial\Sigma}\tfrac{1}{\sin\hat{\gamma}}(\cos\hat{\gamma}-\cos\gamma)+O(s^{2}).

Collecting all the formulas in the proof proves the lemma. ∎

Lemma 3.5 implies the following by taking the difference of two families of metrics.

Corollary 3.5.

Assume $(\Omega,\hat{g})$ is the manifold from Lemma 3.5, for two family of metrics $\{g_{i}\}_{i=1,2}$ close to $\hat{g}$ indexed both by a small parameter $s$ , we have

(3.19)			$\displaystyle 2\left[-\int_{\Sigma}(H_{g_{2}}-H_{g_{1}})+\int_{\partial\Sigma}\tfrac{1}{\sin\hat{\gamma}}(\cos\gamma_{g_{1}}-\cos\gamma_{g_{2}})\right]$
(3.20)		$\displaystyle=$	$\displaystyle\int_{\Omega}((R_{g_{2}}-R_{g_{1}})+\langle\operatorname{Ric}_{\hat{g}},g_{2}-g_{1}\rangle_{\hat{g}})+2\int_{\partial\Omega\backslash\Sigma}(H_{g_{2}}-H_{g_{1}})+\int_{\partial\Omega}\langle g_{2}-g_{1},A_{\hat{g}}\rangle+O(s^{2}).$

Now we are ready to prove Proposition 3.3.

Proof of Proposition 3.3.

The proof is similar to [CW23]. We bring up only the main differences.

Because the right hand of both (3.3) and (3.9) converge to $\Delta u$ and $\tfrac{\partial u}{\partial\nu}$ (up to a constant) respectively, so we can first follow [CW23, Proposition 4.2] to construct a foliation $\{\Sigma_{t,t^{2}u}\}_{t\in[0,\varepsilon)}$ near $p_{0}$ with constant $H-\bar{h}$ and $\gamma_{t,t^{2}u}=\bar{\gamma}_{t,t^{2}u}$ along $\partial\Sigma_{t,t^{2}u}$ , and then [CW23, Lemma 4.3] to obtain that

(3.21)

-\lambda_{t}|\Sigma_{t}|=\int_{\Sigma_{t}}(H_{t}-\bar{h}_{t})+\int_{\partial\Sigma_{t}}\tfrac{1}{\sin\gamma_{t}}(\cos\bar{\gamma}_{t}-\cos\gamma_{t})+O(t^{3}).

Now we show that $\lim_{t\to 0}\lambda_{t}\leqslant 0$ .

We consider the rescaled set $t^{-1}\Omega_{t}$ with two rescaled metrics $t^{-2}g$ and $t^{-2}\bar{g}$ . Since $\bar{g}=\mathrm{d}t^{2}+\psi(t)^{2}g_{{S}^{2}}$ and $\psi(t)=at+o(t)$ , it is easy to see that $(t^{-1}\Omega_{t},t^{-2}\bar{g})$ converges to a truncated metric cone $\Lambda=(0,1]\times D$ with the metric $\varrho:=\mathrm{d}s^{2}+a^{2}s^{2}g_{{S}^{2}}$ where $s\in(0,1]$ and $(D,a^{2}g_{{S}^{2}})$ is some convex disk in a 2-sphere $({S}^{2},a^{2}g_{{S}^{2}})$ . We set $D_{s}=\{s\}\times D$ . Since $g$ and $\bar{g}$ has isometric tangent cone at $p_{0}$ , $(t^{-1}\Omega_{t},t^{-2}g)$ converges to $(\Lambda,\varrho)$ as well. Therefore, we can view $g_{1}=t^{-2}g$ and $g_{2}=t^{-2}\bar{g}$ (indexed by $t$ ) as two metrics on $\Lambda$ getting closer to $\varrho$ as $t\to 0$ . We rescale (3.21) by a factor of $t^{-2}$ , we obtain

-\lambda_{t}|\Sigma_{t}|t^{-2}=\int_{\Sigma_{t}}(H_{t}-\bar{h}_{t})t^{-2}+\int_{\partial\Sigma_{t}}\tfrac{1}{\sin\gamma_{t}}(\cos\bar{\gamma}_{t}-\cos\gamma_{t})t^{-2}+O(t)

which is equivalent to

-\lambda_{t}|D|_{g_{1}}=\int_{D}(H_{g_{2}}-H_{g_{1}})+\int_{\partial D}\tfrac{1}{\sin\gamma_{t}}(\cos\bar{\gamma}_{t}-\cos\gamma_{t})+O(t).

In the above the integration done is with respect to the metric $g_{1}$ and $H_{g_{i}}$ are the mean curvature of $\{1\}\times D$ in $(\Lambda,g_{i})$ computed with respect to the normal pointing inside of $\Lambda$ .

All the comparisons in item (3) of Theorem 1.2 carry over to the rescaled metrics $g_{1}$ and $g_{2}$ on $\Lambda$ , and that $(\Lambda,\varrho)$ has non-negative Ricci curvature by the assumptions of item (3) of Theorem 1.2. We use Corollary 3.5 and arrive that $\lambda_{t}\leqslant O(t)$ , that is,

\lim_{t\to 0}\lambda(t)\leqslant 0.

Since $\lambda_{t}$ satisfies the differential inequality (2.79) and considering the asymptotics $u(\cdot,t)=1+O(t)$ , $\cot\bar{\gamma}=O(t)$ and $\bar{h}=2/t+O(1)$ in (2.78), we see that $\lambda_{t}\leqslant 0$ for all $t\in(0,\varepsilon)$ . ∎

Remark 3.6.

The Ricci curvature in Corollary 3.5 blows up near $\{0\}\times D$ , however, because we are integrating with respect to the metric $\varrho$ , the volume near $\{0\}\times D$ is small. Also, the difference $g_{2}-g_{1}$ is small. So the blowing up of the Ricci curvature will not cause an issue.

3.3. Barrier construction with non-isometric tangent cones

Since $\bar{g}=\mathrm{d}t^{2}+\psi(t)^{2}g_{{S}^{2}}$ , the manifold $(M,\bar{g})$ is topologically a cone near $t=0$ and it is a point at $t=0$ . According to the assumptions of item (3) of Theorem 1.2, $(M,g)$ at $p_{0}$ also locally resembles a cone, that is,

(3.22)

g=\mathrm{d}s^{2}+s^{2}g_{0}+g_{1},

where $s$ is a parameter, $g_{0}$ is a metric on a two dimensional disk $D$ and $g_{1}$ is small compare to $\mathrm{d}s^{2}+s^{2}g_{0}$ . In other words, the tangent cone at $p_{0}$ is a cone with the metric $\mathrm{d}s^{2}+s^{2}g_{0}$ .

Now we can also identify $M$ near $p_{0}$ as $(0,\varepsilon)\times D$ and $t$ as a function on $(0,\varepsilon)\times D$ . Let $(s,x)\in(0,\varepsilon)\times D$ , we see that $\tau:=s/t$ as a function on $M$ only depends on $x\in D$ . So we view $\tau$ as a function on $D$ . Since $g\geqslant\bar{g}$ on $M$ , we have that $\tau(x)\geqslant 1$ . Now we discuss the case that $\tau(x)\equiv 1$ on $D$ .

Lemma 3.7.

If $\tau\equiv 1$ on $D$ , then $g_{0}=a^{2}g_{{S}^{2}}$ . That is, $(M,g)$ and $(M,\bar{g})$ have isometric tangent cones at $p_{0}$ .

Proof.

Since $\tau\equiv 1$ , so we can rescale $(M,\bar{g})$ and $(M,g)$ by the same scale to obtain a cone $\mathcal{C}=(0,\infty)\times D$ but with two different metrics $\chi_{1}=\mathrm{d}t^{2}+a^{2}t^{2}g_{{S}^{2}}$ and $\chi_{2}=\mathrm{d}t^{2}+t^{2}g_{0}$ . For $s>0$ , set $D_{s}=\{s\}\times D\subset\mathcal{C}$ . Since the metric comparison, the mean curvature and the scalar curvature comparison are preserved by rescaling, so $g_{0}\geqslant a^{2}g_{{S}^{2}}$ , the scalar curvature $R_{\chi_{2}}\geqslant R_{\chi_{1}}$ and the mean curvature of $\partial\mathcal{C}$ at $\partial D_{1}$ satisfies $H_{\chi_{2}}\geqslant H_{\chi_{1}}$ .

Since both $\chi_{i}$ , $i=1,2$ are warped product metrics, the comparison $R_{\chi_{2}}\geqslant R_{\chi_{1}}$ reduces to Gaussian curvature comparison $K_{2}\geqslant K_{1}=a^{-2}$ of $(D_{1},g_{0})$ and $(D_{1},a^{2}g_{{S}^{2}})$ by a direct computation of scalar curvature (or Gauss equation). Let $\kappa_{i}$ be the geodesic curvatures of $\partial D_{1}$ with respect to $\chi_{i}|_{D_{1}}$ . By direct calculation, the second fundamental form $A^{(i)}_{\partial\mathcal{C}}$ of $\partial\mathcal{C}$ in the direction $\partial_{t}$ vanishes with respect to both metrics $\chi_{i}$ and the second fundamental form $A_{D_{1}}^{(i)}$ of $D_{1}$ in $\mathcal{C}$ with respect to $\chi_{i}$ agree. It then follows from $H_{\chi_{2}}\geqslant H_{\chi_{1}}$ and (2.14) that $\kappa_{2}\geqslant\kappa_{1}$ .

To summarize, we have comparisons on $D_{1}$ that $g_{0}\geqslant a^{2}g_{{S}^{2}}$ , $K_{2}\geqslant K_{1}$ and $\kappa_{2}\geqslant\kappa_{1}$ along $\partial D_{1}$ . By Gauss-Bonnet theorem, $g_{0}\equiv a^{2}g_{{S}^{2}}$ on $D_{1}$ and it follows that $\chi_{1}\equiv\chi_{2}$ . Therefore, $(M,g)$ and $(M,\bar{g})$ have isometric tangent cones at $p_{0}$ . ∎

By the above lemma, the case $\tau\equiv 1$ is the case which implies isometric tangent cones of $(M,g)$ and $(M,\bar{g})$ at $p_{0}$ . This is the case we have already addressed in Subsection 3.2. Without loss of generality, we assume that $\tau\mathrel{\not\equiv}1$ .

We first consider the difference of $H-\bar{h}$ of the perturbation for $D_{s}$ . We now represent $\bar{h}$ at $D_{s}$ and its value at the graphical perturbations of $D_{s}$ by $\zeta$ to avoid notational confusion. By the first variation of the mean curvatures,

(3.23)			$\displaystyle(H_{s,s^{2}u}-\zeta_{s,s^{2}u})-(H_{s}-\zeta_{s})$
(3.24)		$\displaystyle=$	$\displaystyle-\Delta_{s}u-s^{2}(\operatorname{Ric}(N_{s})+\|A_{s}\|^{2}+s^{-2}(\zeta_{s,s^{2}u}-\zeta_{s}))u+O(s),$

where $\Delta_{s}$ is the Laplacian with respect to the metric $s^{-2}g|_{D_{s}}$ .

Remark 3.8.

We have $\{(s^{-1}D_{s},s^{-2}g|_{D_{s}})\}_{s>0}$ converges to $(D,g_{0})$ as $s\to 0$ by the metric (3.22) near $p_{0}$ , and to indicate that the limit carries the metric $g_{0}$ , we use $D_{0}$ instead of $D$ only.

Lemma 3.9.

We have that

s^{2}(|A_{s}|^{2}-s^{-2}(\zeta_{s,s^{2}u}-\zeta_{s}))=(2-2\tau)+O(s).

Proof.

Since $\{(s^{-1}\Lambda_{s},s^{-2}g)\}_{s>0}$ converges to a truncated radial cone and $\{(s^{-1}D_{s},s^{-2}g|_{D_{s}})\}_{s>0}$ converges to the section of the radial cone with unit distance to $p_{0}$ , so the section has second fundamental form $-2$ and by rescaling,

|A_{s}|^{2}=2s^{-2}+O(s^{-1})

as $s\to 0$ .

At a point $p=(s,x)\in D_{s}$ , the value of $t$ is given by $t=s\tau(x)$ where $x$ is the projection of $p$ to the second coordinate. Since $\tau$ as a function on $M$ only depends on $x$ , we see that the value of the function $t$ at the graphical perturbation $s+s^{2}u$ of $D_{s}$ is given by $(s+s^{2}u)\tau$ . Since $\bar{h}(t)=2t^{-1}+O(1)$ , so

\zeta_{s,s^{2}u}-\zeta_{s}=\tfrac{2}{(s+s^{2}u)\tau}-\tfrac{2}{s\tau}+O(1)=-\tfrac{2\tau}{s^{2}}(s^{2}u)+O(1).

Hence

s^{2}(|A_{s}|^{2}+s^{-2}(\zeta_{s,s^{2}u}-\zeta_{s}))=(2-2\tau)+O(s),

which proves the lemma. ∎

Let $f=\lim_{s\to 0}s^{2}(\operatorname{Ric}(N_{s})+|A_{s}|^{2}+s^{-2}(\zeta_{s,s^{2}u}-\zeta_{s}))$ which is a function on the limit $D_{0}$ , so

(3.25)

\lim_{s\to 0}[(H_{s,s^{2}u}-\zeta_{s,s^{2}u})-(H_{s}-\zeta_{s})]=-\Delta_{0}u-fu,

where $\Delta_{0}$ is the Laplacian of $D_{0}$ . Recall that $\operatorname{Ric}(N_{s})=O(s^{-1})$ , so

f=2-2\tau\text{ on }D_{0}.

Let $\alpha_{s}$ be the dihedral angles formed by $\partial M$ and $D_{s}$ , and $\alpha_{s,s^{2}u}$ be the angles formed by $\partial M$ and the graphical perturbation of $D_{s}$ .

Lemma 3.10.

The dihedral angles $\alpha_{s}$ formed by $\partial M$ and $D_{s}$ approach $\pi/2$ as $s\to 0$ .

Proof.

Since $\{(s^{-1}\Lambda_{s},s^{-2}g)\}_{s>0}$ converges to a truncated radial cone, $\{(s^{-1}D_{s},s^{-2}g|_{D_{s}})\}_{s>0}$ converges to the section of the radial cone with unit distance to $p_{0}$ , and this section is orthogonal to the radial direction in the limit, so the intersection angles of $\partial M$ and $D_{s}$ approaches $\pi/2$ as $s\to 0$ . ∎

Lemma 3.11.

We have that $A_{\partial M}(\eta,\eta)=O(1)$ .

Proof.

The lemma can be deduced from that $\eta$ is approximately the radial direction $\partial_{s}$ as $s\to 0$ , the scaling property of $A_{\partial M}$ and the following lemma. ∎

Lemma 3.12.

Let $(S,\sigma)$ be a 2-surface with boundary and $(C=[0,\infty)\times S,\mathrm{d}s^{2}+s^{2}\sigma)$ be the cone over $(S,\sigma)$ . Then the second fundamental form of $\partial C$ in $C$ in the direction $\partial_{t}$ vanishes.

Proof.

Let $Z$ be a tangent vector field over $\Sigma$ , then by direct calculation $\nabla_{\partial_{t}}Z=\nabla_{X}\partial_{t}=s^{-1}Z$ . So $\langle\nabla_{\partial_{t}}Z,\partial_{t}\rangle=0$ since on $C$ the metric is $\mathrm{d}t^{2}+t^{2}\sigma$ . Due to the same reason, the unit normal vector $Z$ of $\partial C$ in $M$ is tangent to $\Sigma$ , so the claim is proved. ∎

We are interested in the difference between $\alpha_{s,s^{2}u}$ and the value of $\bar{\gamma}$ which to avoid confusion we denote by $\beta_{s}$ ( $\beta_{s,s^{2}u}$ ) at (the graphical perturbation $s^{2}u$ of) $D_{s}$ . Using the relation of $s$ and $t$ , $\beta=\bar{\gamma}_{s/\tau,s^{2}u/\tau}$ . By the expansion of angles (see (3.5)), we see

\cos\alpha_{s,s^{2}u}-\cos\alpha_{s}=-\sin\alpha_{s}\tfrac{\partial u}{\partial\nu_{s}}+s(-\cos\alpha_{s}A(s^{-1}\nu_{s},s^{-1}\nu_{s})+A_{\partial M}(\eta_{s}{,}\eta_{s}))u+O(s^{2}).

And

s^{-1}(\cos\beta_{s,s^{2}u}-\cos\beta_{s})=su\tau^{-1}\nabla_{\eta_{s/\tau}}\cos\bar{\gamma}_{s/\tau,s^{2}u/\tau}+O(s^{2})

Since each $\Sigma_{t}$ is stable capillary minimal surface under the metric $\bar{g}$ , so we know that

\tfrac{1}{\sin\bar{\gamma}}\nabla_{\eta_{t}}\cos\bar{\gamma}=-\cos\bar{\gamma}A(\nu_{t},\nu_{t})+A_{\partial M}(\eta_{t}{,}\eta_{t}).

Based on the above asymptotic analysis and Lemmas 3.10 and 3.11, we see

(3.26)

\lim_{s\to 0}[s^{-1}(\cos\alpha_{s,s^{2}u}-\cos\alpha_{s})-s^{-1}(\cos\beta_{s,s^{2}u}-\cos\beta_{s})]=-\tfrac{\partial u}{\partial\nu_{0}}

on $\partial D_{0}$ where $\nu_{0}$ is the outward normal of $\partial D_{0}$ in $D_{0}$ . By the elliptic strong maximum principle, the operator

(-\Delta_{0}-f,-\tfrac{\partial}{\partial\nu_{0}}):C^{2,\alpha}(D_{0})\cap C^{1,\alpha}(\bar{D}_{0})\to C^{0,\alpha}(D_{0})\times C^{0,\alpha}(\partial D_{0})

is an isomorphism since $f\leqslant 0$ in $D_{0}$ due to Lemma 3.9 and $\tau\gneqq 1$ . In other words, we can specify the limits

(3.27)		$\displaystyle\lim_{s\to 0}[(H_{s,s^{2}u}-\zeta_{s,s^{2}u})-(H_{s}-\zeta_{s})]$
(3.28)		$\displaystyle\quad\text{ and }\lim_{s\to 0}[s^{-1}(\cos\alpha_{s,s^{2}u}-\cos\alpha_{s})-s^{-1}(\cos\beta_{s,s^{2}u}-\cos\beta_{s})]$

by choosing a suitable $u\in C^{2,\alpha}(D_{0})\cap C^{1,\alpha}(\bar{D}_{0})$ .

We have these facts: by Lemma 3.10, both $\alpha_{s}$ and $\beta_{s}$ tend to $\pi/2$ as $s\to 0$ , so $\lim_{s\to 0}s^{-1}(\alpha_{s}-\beta_{s})$ is a function on $\partial D_{0}$ ; $H_{s}-\zeta_{s}=(2-2\tau)s^{-1}+O(1)$ ;

(3.29)

\text{ }H_{s,s^{2}u}-\zeta_{s,s^{2}u}=(2-2\tau)s^{-1}+O(1)

for small $s>0$ with a remainder term depending on $u$ . Hence, we can specify a function $u$ to counter-effect the $O(1)$ remainder term in $H_{s}-\zeta_{s}$ and make the remainder term in (3.29) strictly negative. That is, we can specify a function $u$ such that

(3.30)		$\displaystyle\lim_{s\to 0}(H_{s,s^{2}u}-\zeta_{s,s^{2}u}-(2-2\tau)s^{-1})$	$\displaystyle=u_{0}\text{ in }D_{0},$
(3.31)		$\displaystyle\quad\lim_{s\to 0}s^{-1}(\cos\alpha_{s,s^{2}u}-\cos\beta_{s,s^{2}u})$	$\displaystyle<0\text{ along }\partial D_{0},$

for some negative function $u_{0}\in C^{0,\alpha}(\bar{D}_{0})$ . Recall the definitions of $\zeta$ , $\tau$ , $\beta$ , and by continuity, there exists a surface $\Sigma_{-}\subset M$ satisfying

H-\bar{h}<0\text{ in }\Sigma_{-}\text{ and }\alpha>\bar{\gamma}\text{ along }\partial\Sigma_{-}.

This surface $\Sigma_{-}$ is a lower barrier in the sense of Definition 1.5.

Now we can prove item (3) of Theorem 1.2.

Proof of item (3) of Theorem 1.2.

Assume that $g$ and $\bar{g}$ do not have isometric tangent cone at $p_{0}$ , then we can construct a barrier $\Sigma_{-}$ such that $H-\bar{h}<0$ in $\Sigma_{-}$ and the angle $\alpha>\bar{\gamma}$ along $\partial\Sigma_{-}$ . But due to item (4) of Theorem 1.2, this is not possible. So $g$ and $\bar{g}$ have isometric tangent cones at $p_{0}$ , then by the construction of the foliation in Theorem 3.3, again we have a barrier near $t=0$ , but the barrier condition is now not strict. We can extend the rigidity $g=\bar{g}$ in item (4) of Theorem 1.2 beyond the barrier and to all of $M$ . ∎

Remark 3.13.

By considering only the mean curvature, this provides an alternative proof of Theorem 1.1 in dimension 3. Moreover, we allow conical metrics of $(\mathbb{S}^{3},g)$ at two antipodal points.

Remark 3.14.

During the construction of barriers in the case of non-isometric cones, the Gauss-Bonnet theorem is only used in Lemma 3.7.

4. Construction of barriers (II)

In this section, we prove items (2) and (1) of Theorem 1.2. Our method is similar to the previous work [CW23].

4.1. Proof of item (2) of Theorem 1.2

For convenience, we set $t_{-}=0$ . We will construct a lower barrier near $t=0$ . As before, for any $t>0$ , we set $\Sigma_{t}$ to be the $t$ -level set of $t$ and $\Omega_{t}$ to be the $t$ -sublevel set, that is, all points of $M$ which lie below $\Sigma_{t}$ . We see from the assumption on the tangent cone at $p_{-}$ that the sequence $\{(t^{-1}M,t^{-2}\bar{g})\}_{t>0}$ converges to some right circular cone $\bar{C}$ in $\mathbb{R}^{3}$ equipped with a flat metric $g_{\mathbb{R}^{3}}$ as $t\to 0$ . Then $\{(t^{-1}M,t^{-2}g)\}_{t>0}$ converges to the same cone $\bar{C}$ but with a different constant metric $g_{0}$ . The cone $(\bar{C},g_{0})$ can be represented as a cone in the Euclidean space $(\mathbb{R}^{3},g_{\mathbb{R}^{3}})$ .

We have the existence of a barrier if $(M,g)$ and $(M,\bar{g})$ have non-isometric tangent cones at $p_{0}$ .

Lemma 4.1.

Let $M$ be given as in item (2) of Theorem 1.2. If the tangent cones of $(M,g)$ and $(M,\bar{g})$ at $p_{-}$ are not isometric, assume that the mean curvature comparison and the metric comparison hold near $p_{-}$ , then there exists a surface $\Sigma_{-}$ satisfying

H-\bar{h}<0\text{ in }\Sigma_{-}\text{ and }\gamma_{\Sigma_{-}}>\bar{\gamma}\text{ along }\partial\Sigma_{-}

as the above. This surface $\Sigma_{-}$ is a barrier in the sense of Definition 1.5.

Proof.

First, we note that the mean curvature comparison and the metric comparison (we only need boundary metric comparison) are preserved in the limits. Because that the tangent cone of $(M,\bar{g})$ at $p_{-}$ is a round (solid) circular cone and that $(t^{-1}\Sigma_{t},t^{-2}\bar{g})$ converges to its axial section as $t\to 0$ , it follows from the angle comparison of [CW23, Proposition 4.9] that there exists a plane $P$ in $\bar{C}$ such that the the dihedral angles formed by $\partial\bar{C}$ and $P$ in the metric $g_{0}$ are everywhere larger than $\bar{\gamma}(t_{-})$ .

We gain a lot of freedom to construct the barrier from the strict comparison of angles. The rest of the argument is analogous to [CW23, Proposition 4.10]. ∎

Remark 4.2.

Note that the scalar curvature comparison is not needed here.

Proof of item (2) of Theorem 1.2.

First, the tangent cones of $(M,g)$ and $(M,\bar{g})$ at $p_{-}$ must be isometric. Indeed, by Lemma 4.1 and item (4) of Theorem 1.2, the barrier constructed in Lemma 4.1 cannot have $H-\bar{h}<0$ in $\Sigma_{-}$ and $\gamma_{\Sigma_{-}}<\bar{\gamma}$ hold strictly along $\partial\Sigma_{-}$ .

By following Subsection 3.2, we can construct graphical perturbations $\Sigma_{t,t^{2}u}$ of $\Sigma_{t}$ which satisfy Proposition 3.3. For every sufficiently small $t>0$ , $\Sigma_{t,t^{2}u}$ is a barrier in the sense of Definition 1.5, we conclude that $g=\bar{g}$ for the region bounded by $\Sigma_{t,t^{2}u}$ and $P_{+}\cap\partial M$ for every $t>0$ from item (4) of Theorem 1.2. Hence, we finish the proof of item (2) of Theorem 1.2. ∎

4.2. Proof of item (1) of Theorem 1.2

This part is a slightly extension of the argument in Section 5 in our previous paper [CW23] and Ko-Yao’s paper [KY24]. So we only sketch the key steps here and refer to the above papers for more details.

Again, we set $s_{-}=0$ and we construct a lower barrier near $s=0$ . Suppose $M$ is given by

M=\{(s,p)\in[0,\varepsilon)\times{S}^{2}:s\geq f(p)\},

near $p_{-}=O$ , where $f$ is a smooth function such that $f(p_{-})=0$ and $\mathrm{Hess}f$ is positive definite at $p_{-}$ under metric $g_{S^{2}}$ . Note that we need to assume $\psi(0)\neq 0$ , otherwise, the manifold $M$ will have a cusp at point $O=(0,0,0)$ . For simplicity, we assume $\psi(t(0))=1$ , where $t(s)$ is determined by (1.5).

To better illustrate the situation, we can choose the coordinate $(x_{1},x_{2})$ on ${S}^{2}$ such that the expansion of metric $\bar{g}$ in (1.5) at $O$ is given by

\bar{g}=ds^{2}+dx_{1}^{2}+dx_{2}^{2}+O(s)+O(|x|^{2}),

where $|x|=\sqrt{x_{1}^{2}+x_{2}^{2}}$ .

For simplicity, we denote $\bar{g}_{0}=ds^{2}+dx_{1}^{2}+dx_{2}^{2}$ as the linearised part of $\bar{g}$ at $O$ . After a suitable rotation, we can write $g=g_{0}+sh+O(s^{2})$ for some constant metric $g_{0}$ defined as

(4.1)

g_{0}=a_{33}ds^{2}+(a_{11}dx_{1}^{2}+a_{22}dx_{2}^{2})+2a_{13}dx_{1}ds+2a_{23}dx_{2}ds,

where the matrix

\begin{bmatrix}a_{11}&0&a_{13}\\ 0&a_{22}&a_{23}\\ a_{13}&a_{23}&a_{33}\end{bmatrix}

is positive definite and satisfies $a_{11},a_{22},a_{33}\geq 1$ .

We assume the manifold $M$ can be written as

M=\{(s,x_{1},x_{2}):s\in[0,\varepsilon),s\leq\zeta(x_{1},x_{2})\},

where $\zeta(x_{1},x_{2})=c_{11}x_{1}^{2}+2c_{12}x_{1}x_{2}+c_{22}x_{2}^{2}+O(|x|^{3})$ is a smooth function with $c_{11},c_{22},c_{11}c_{22}-c_{12}^{2}>0$ . Here, we have used the fact that $\mathrm{Hess}f$ is positive definite at $p_{-}$ under metric $g_{S^{2}}$ .

We write $a^{ij}$ as the inverse matrix of $a_{ij}$ , and define several constants as

	$\displaystyle B={}$	$\displaystyle\sqrt{a^{33}((\sqrt{a_{11}}c_{22}+\sqrt{a_{22}}c_{11})^{2}+(\sqrt{a_{11}}-\sqrt{a_{22}})^{2}c_{12}^{2})}$
	$\displaystyle b_{11}={}$	$\displaystyle a^{33}B^{-1}c_{11}^{-1}(a_{11}(c_{11}c_{22}-c_{12}^{2})+\sqrt{a_{11}a_{22}}(c_{11}^{2}+c_{12}^{2}))$
	$\displaystyle b_{12}={}$	$\displaystyle b_{21}=a^{33}B^{-1}\sqrt{a_{11}a_{22}}(c_{11}+c_{22})$
	$\displaystyle b_{22}={}$	$\displaystyle a^{33}B^{-1}c_{22}^{-1}(a_{22}(c_{11}c_{22}-c_{12}^{2})+\sqrt{a_{11}a_{22}}(c_{12}^{2}+c_{22}^{2})).$

and consider the function $G_{\lambda,s}$ defined by

	$\displaystyle G_{\lambda,s}(x_{1},x_{2})={}$	$\displaystyle c_{11}(b_{11}(1+\lambda)-1)x_{1}^{2}+c_{22}(b_{22}(1+\lambda)-1)x_{2}^{2}$
		$\displaystyle+2c_{12}(b_{12}(1+\lambda)-1)x_{1}x_{2}-s^{2}.$

and the surface $\Sigma_{\lambda,s}$ is defined by

\Sigma_{\lambda,s}=\left\{(G_{\lambda,s}(x),x):x\in\mathbb{R}^{2}\text{ and }G_{\lambda,s}(x)\geq\zeta(|x|^{2})\right\}.

We use an ellipse $E_{\lambda}$ to parameterize $\Sigma_{\lambda,s}$ where $E_{\lambda}\subset\mathbb{R}^{2}$ is given by

E_{\lambda}:=\{\hat{x}\in\mathbb{R}^{2}:c_{11}b_{11}\hat{x}_{1}^{2}+c_{22}b_{22}\hat{x}_{2}^{2}+2c_{12}b_{12}\hat{x}_{1}\hat{x}_{2}<\frac{1}{1+\lambda}\}.

Then, the surface $\Sigma_{\lambda,s}$ can be written as a map $E_{\lambda}\rightarrow\Sigma_{\lambda,s}$ such that

\Sigma_{\lambda,s}(\hat{x}):=(G_{\lambda,s}(\Phi_{\lambda,s}(\hat{x})),\Phi_{\lambda,s}(\hat{x}))

where $\Phi_{\lambda,s}:E_{\lambda}\rightarrow\mathbb{R}^{2}$ satisfies

\Phi_{\lambda,s}(\hat{x})=s\hat{x}+O(s^{3}).

We also use $\Sigma_{s}=\Sigma_{0,s}$ for short. We have the following result by the argument in [CW23].

Proposition 4.3.

Suppose the metric $g$ can be written as $g=g_{0}+sh+O(s^{2})$ where $g_{0}$ is the constant metric defined in (4.1) and $h$ is a bounded symmetric two-tensor. Then, we have

	$\displaystyle\cos\gamma_{\lambda,s}(\hat{x})={}$	$\displaystyle\cos\bar{\gamma}_{\lambda,s}(\hat{x})-4\lambda s^{2}\left[\frac{(\hat{x}_{1}c_{11}b_{11}+\hat{x}_{2}c_{12}b_{12})^{2}}{a_{11}a^{33}}+\frac{(\hat{x}_{1}c_{12}b_{12}+\hat{x}_{2}c_{22}b_{22})^{2}}{a_{22}a^{33}}\right]$
		$\displaystyle+s^{2}O(\lambda^{2})+A(\hat{x})s^{3}+L(h)s^{3}+O(s^{4}).$

for any $\hat{x}\in E_{\lambda}$ . Here, $A(\hat{x})$ is a bounded term (not related to $s$ and $h$ ) which is also odd symmetric with respect to $\hat{x}$ , $L(h)$ is a bounded term (not related to $s$ ) relying on $h$ linearly.

Sketch of the proof.

We use the same argument for Proposition 5.1 in [KY24] and also track the term $s^{3}$ to get the following expansions

	$\displaystyle\cos\measuredangle_{g_{0}}(\Sigma_{\lambda,s},\partial M)={}$	$\displaystyle 1-\frac{2(\hat{x}_{\beta}c_{\alpha\beta}b_{\alpha\beta}(1+\lambda))^{2}}{a_{\alpha\alpha}a^{33}}s^{2}+A(\hat{x})s^{3}+O(s^{4})$
	$\displaystyle\cos\bar{\gamma}_{\lambda,s}(\hat{x})={}$	$\displaystyle 1-2s^{2}\left[(\hat{x}_{1}c_{11}+\hat{x}_{2}c_{12})^{2}+(\hat{x}_{1}c_{12}+\hat{x}_{2}c_{22})^{2}\right]+A(\hat{x})s^{3}+O(s^{4}),$

where we assume $c_{21}=c_{12}$ . Together with the remaining computation for Proposition 5.1 in [KY24] and the Corollary 5.5 in [CW23] (see the proof for Corollary 5.17 in [CW23]), we can establish the result. ∎

As a corollary of Proposition 4.3, we can easily establish the following results for $\sin\gamma_{s}$ ,

(4.2)		$\displaystyle\sin\gamma_{s}(\hat{x})={}$	$\displaystyle\sin\bar{\gamma}_{s}(\hat{x})+O(s^{2})$
(4.3)		$\displaystyle={}$	$\displaystyle 2s\sqrt{(\hat{x}_{1}c_{11}+\hat{x}_{2}c_{12})^{2}+(\hat{x}_{1}c_{12}+\hat{x}_{2}c_{22})^{2}}+O(s^{2}),$

and the following proposition.

Proposition 4.4.

Suppose the conditions in Proposition 4.3 hold. Then, for any $\lambda>0$ , we can find $s_{0}>0$ (might rely on $\lambda$ ) such that for any $s<s_{0}$ , we have

(4.4)

\gamma_{\lambda,s}(\hat{x})>\bar{\gamma}_{\lambda,s}(\hat{x})

for any $\hat{x}\in\partial E_{\lambda}$ .

We need to analyze the asymptotic behavior of mean curvature. We define the following mean curvatures:

	$\displaystyle H_{\lambda,s}^{g}(\hat{x}):={}$	$\displaystyle\text{Mean curvature of $\Sigma_{\lambda,s}$ at $\Sigma_{\lambda,s}(\hat{x})$ under metric $g$},$
	$\displaystyle H_{\lambda,s,\partial M}^{g}(\hat{x}):={}$	$\displaystyle\text{Mean curvature of $\partial M$ at $(\varphi(\|\Phi_{\lambda,s}(\hat{x})\|^{2}),\Phi_{\lambda,s}(\hat{x}))$ under metric $g$}.$

Using the same computation for Corollary 5.2 in [KY24], we have

Proposition 4.5.

Suppose the metric $g$ can be written as $g=g_{0}+sh+O(s^{2})$ where $g_{0}$ is a constant metric defined in (4.1), and $h$ is a bounded symmetric two-tensor. Then, we have the following formula for the behavior of mean curvature

(4.5)

H^{g}_{s}(\hat{x})=H^{g}_{s,\partial M}(\hat{x})-H^{\bar{g}}_{s,\partial M}(\hat{x})-2(c_{11}+c_{22})+\frac{2B}{\sqrt{a_{11}a_{22}a^{33}}}+sL(\hat{x})+O(s^{2}),

for any $\hat{x}\in E$ . Here, we write $H^{g}_{s}=H^{g}_{0,s}$ and $H^{g}_{s,\partial M}=H^{g}_{0,s,\partial M}$ for short.

Now, we consider

H_{0}:=\lim_{s\rightarrow 0}H_{s}^{g}(\hat{x}),

which is well-defined by (4.5) (the limit does not depend on the choice of $\hat{x}$ .)

We have two subcases to consider.

If $H_{0}<\bar{h}(0)$ , then we can use the continuation of $H_{\lambda,s}^{g}$ with respect to $\lambda$ and $s$ , together with Proposition 4.4, we can show the following results (cf. Proposition 5.10 in [CW23]).

Proposition 4.6.

Suppose the metric $g$ can be written as $g=g_{0}+sh+O(s^{2})$ where $g_{0}$ is a constant metric defined in (4.1), and $h$ is a bounded symmetric two-tensor. If $H_{0}<\bar{h}(0)$ , we can choose some $\lambda>0,s>0$ small such that $H_{\lambda,s}^{g}(\hat{x})>\bar{h}(\Sigma_{\lambda,s}(\hat{x}))$ for any $\hat{x}\in\partial E_{\lambda}$ and $\gamma_{\lambda,s}(\hat{x})<\bar{\gamma}_{\lambda,s}(\hat{x})$ for each $\hat{x}\in\partial E_{\lambda}$ .

Now, we focus on the case $H_{0}=\bar{h}(0)$ . In particular, it implies $a_{11}=a_{22}=1$ and $H^{g}_{\partial M}(O)=H^{\bar{g}}_{\partial M}(O)$ .

Then, we need to construct a foliation near $O$ . We define the vector field $Y_{s}(\hat{x}):=\frac{\partial}{\partial s}\Sigma_{s}(\hat{x})$ . Given $u\in C^{1,\alpha}(\bar{E})\cap C^{2,\alpha}(E)$ where $E=E_{0}$ , we can define the perturbation surface $\Sigma_{s,u}$ by

\Sigma_{s,u}:=\left\{\Sigma_{s+\frac{u}{\left<Y_{s}(\hat{x}),N_{s}(\hat{x})\right>}}(\hat{x}):\hat{x}\in E\right\}

where $N_{s}(\hat{x})$ is the unit normal vector field of $\Sigma_{s}$ .

We write $E=E_{0}$ . Using the variational formula for mean curvatures and contact angles, we have

	$\displaystyle\frac{H_{s,s^{3}u}-\bar{h}_{s,s^{3}u}}{s}={}$	$\displaystyle-\Delta_{s}^{E}u+\frac{H_{s}-\bar{h}_{s}}{s}+O(s),$
	$\displaystyle\frac{\cos\gamma_{s,s^{3}u}-\cos\bar{\gamma}_{s,s^{3}u}}{s^{3}}={}$	$\displaystyle-2s\sqrt{(\hat{x}_{1}c_{11}+\hat{x}_{2}c_{12})^{2}+(\hat{x}_{1}c_{12}+\hat{x}_{2}c_{22})^{2}}\frac{\partial u}{\partial\nu_{s}^{E}}$
		$\displaystyle+(A_{\partial M}(\eta_{s},\eta_{s})-\cos\gamma_{s}A(\nu_{s},\nu_{s})$
		$\displaystyle-\bar{A}_{\partial M}(\bar{\eta}_{s},\bar{\eta}_{s}))u+\frac{\cos\gamma_{s}-\cos\bar{\gamma}_{s}}{s^{3}}+O(s),$

where $\Delta_{s}^{E}$ denotes the Laplacian-Beltrami operator on $E$ under the metric $\frac{1}{s^{2}}\Sigma^{*}_{s}(g)$ , and $\nu_{s}^{E}$ is the unit normal vector field of $\partial E$ under the metric $\frac{1}{s^{2}}\Sigma^{*}_{s}(g)$ . Here, we have used (4.3).

By using the same argument for Proposition 5.27 in [CW23], together with the asymptotic behavior of mean curvature, for each $s\in(0,\varepsilon)$ sufficiently small, we can find $u_{s}(\cdot)=u(\cdot,s)$ such that the mean curvature $H_{s,s^{3}u_{s}}$ is $\bar{h}_{s,s^{3}u}+s\lambda(s)$ where $\lambda(s)$ is a function only depends on $s$ , the contact angle $\gamma_{s,s^{3}u_{s}}=\bar{\gamma}_{s,s^{3}u_{s}}$ , and $u$ satisfies the following

\lim_{s\rightarrow 0}(u(\hat{x},s)+u(-\hat{x},s))=0

for any $\hat{x}\in E$ . A finer analysis of $\lambda(s)$ will give $\lambda(s)<0$ for $s$ sufficiently small (cf. Proposition 5.28 in [CW23]), and it leads to the following.

Proposition 4.7.

We can construct a surface $\Sigma_{-}$ near $O$ such that the mean curvature of $\Sigma_{-}$ is not greater than $\bar{h}$ and it has prescribed contact angle $\bar{\gamma}$ with $\partial M$ .

Proof of item (1) of Theorem 1.2.

We can use Proposition 4.6 or Proposition 4.7 depending on the value of $H_{0}$ to construct a barrier surface $\Sigma_{-}$ with mean curvature not greater than $\bar{h}$ and prescribed contact angle $\bar{\gamma}$ with $\partial M$ . Then, we can use item (4) of Theorem 1.2 to extend the rigidity to all of $M$ . ∎

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Scalar curvature rigidity of domains in a 3-dimensional warped product

Abstract.

Key words and phrases:

1991 Mathematics Subject Classification:

1. Introduction

Theorem 1.1 ([Lla98]).

Theorem 1.2.

Remark 1.3.

Theorem 1.4.

Definition 1.5.

Theorem 1.6.

2. Stable capillary μ\mu-bubble

2.1. Notations

2.2. Functional and first variation

2.3. Analysis of stability

Lemma 2.1.

Proof.

Lemma 2.2.

Proof.

Remark 2.3.

2.4. Infinitesimally rigid surface

Definition 2.4.

2.5. Capillary foliation of constant H−h¯H-\bar{h}

Theorem 2.5.

Proof.

Theorem 2.6.

Proof.

2.6. From local foliation to rigidity

Proposition 2.7.

Proof.

Proof of item (4) of Theorem 1.2.

3. Construction of barriers (I)

3.1. Setting up coordinates and notations

3.2. Capillary foliation with constant H−h¯H-\bar{h}

Remark 3.1.

Remark 3.2.

Proposition 3.3.

Lemma 3.4.

Proof.

Corollary 3.5.

Proof of Proposition 3.3.

Remark 3.6.

3.3. Barrier construction with non-isometric tangent cones

Lemma 3.7.

Proof.

Remark 3.8.

Lemma 3.9.

Proof.

Lemma 3.10.

Proof.

Lemma 3.11.

Proof.

Lemma 3.12.

Proof.

Proof of item (3) of Theorem 1.2.

Remark 3.13.

Remark 3.14.

4. Construction of barriers (II)

4.1. Proof of item (2) of Theorem 1.2

Lemma 4.1.

Proof.

Remark 4.2.

Proof of item (2) of Theorem 1.2.

4.2. Proof of item (1) of Theorem 1.2

Proposition 4.3.

Sketch of the proof.

Proposition 4.4.

Proposition 4.5.

Proposition 4.6.

Proposition 4.7.

Proof of item (1) of Theorem 1.2.

References

2. Stable capillary $\mu$ -bubble

2.5. Capillary foliation of constant $H-\bar{h}$

3.2. Capillary foliation with constant $H-\bar{h}$