Multiplicity one for min-max theory in compact manifolds with boundary and its applications

Denote by $(M^{n+1},\partial M,g)$ a compact Riemannian manifold with smooth boundary of dimension $n+1$. In [Alm62], Almgren proved that the space of mod 2 relative cycles $\mathcal{Z}_{n}(M,\partial M;\mathbb{Z}_{2})$ is weakly homotopic to $\mathbb{RP}^{\infty}$; see also [LMN16]*§2.5. By performing a min-max procedure, Gromov [Gro88] defined the volume spectrum of $M$, which is a sequence of non-decreasing positive numbers

depending only on $M$ and $g$. Moreover, Gromov [Gro88] also showed that for each $g$, $\omega_{k}$ grows like $k^{\frac{1}{n+1}}$; see also Guth [Guth09]. Recently, Liokumovich-Marques-Neves [LMN16] proved that the volume spectrum obey a uniform Weyl Law.

When $M$ is closed with no boundary, by adapting the regularity theory given by Pitts [Pi] and Schoen-Simon [SS], and the index bounds given by Marques-Neves [MN16], it is known that, when $3\leq(n+1)\leq 7$,  each $k$-width $\omega_{k}$ is realized by a disjoint collection of closed, embedded, minimal hypersurfaces with integer multiplicities, i.e. there exist integers $\{m_{j}^{k}\}$ and minimal hypersurfaces $\{\Sigma_{j}^{k}\}$ so that

see [IMN17]*Proposition 2.2. In [MN16, MN18], Marques-Neves raised the Multiplicity One Conjecture, which asserts that for bumpy metrics¹¹1Here a metric is said to be bumpy if each closed immersed minimal hypersurfaces has no Jacobi field, and the set of bumpy metrics in a closed manifold is generic in the Baire sense by White \citelist[Whi91][Whi17]., $m_{j}^{k}=1$ and $\Sigma_{j}^{k}$ is two-sided for all $k\in\mathbb{N}$ and $1\leq j\leq l_{k}$. This conjecture was later confirmed by the last-named author [Zhou19]; (see also Chodosh-Mantoulidis [CM20]). This, together with [MN18], implies the existence of a closed, embedded minimal hypersurface of Morse index $k$ for each $k\in\mathbb{N}$ under a bumpy metric. This body of works established the Morse theory for the area functional. We also refer to [MN17, IMN17, MNS17, Song18, YY.Li19, Marques-Montezuma-Neves20, Song-Zhou20] for recent developments on the existence theory of closed minimal hypersurfaces, particularly on the resolution of Yau’s conjecture.

In this paper, we focus on compact manifolds with non-empty boundary. In [LZ16], Li-Zhou proved the general existence of free boundary minimal hypersurfaces by adapting the Almgren-Pitts theory. Inspired by \citelist[MN16][IMN17], the last two authors together with Q. Guang and M. Li generalized (1.1) to compact manifolds in [GLWZ19]. Precisely, for each positive integer $k$, there exist integers $\{m_{j}^{k}\}$ and disjoint, embedded, free boundary minimal hypersurfaces $\{\Sigma_{j}^{k}\}$ so that

Inspired by [Zhou19], it is natural to conjecture that $m_{j}^{k}=1$ for generic metrics on compact manifolds with boundary. In this article we confirm this conjecture.

To solve the above challenge (a), we prove the following strong bumpy metric theorem for compact manifolds with boundary.

Here a hypersurface $\Sigma$ is proper if it has empty touching set, i.e. $\mathrm{Int}(\Sigma)\cap\partial M=\emptyset$.

As a direct corollary of Theorem 1.1, we construct free boundary minimal hypersurfaces with arbitrarily large area in the unit ball $\mathbb{B}^{n+1}$, and more generally in any strictly convex domain of $\mathbb{R}^{n+1}$. Recall that a compact Riemannian manifold with non-negative Ricci curvature and strictly convex boundary does not contain any stable free boundary minimal hypersurfaces. Moreover, by Fraser-Li [FrLi]*Lemma 2.4, any two free boundary minimal hypersurfaces in this type of manifolds intersect with each other. Making use of the compactness \citelist[ACS17][GWZ18], we have the following result.

In particular, the unit ball $\mathbb{B}^{n+1}$ with $3\leq(n+1)\leq 7$ contains a sequence of free boundary minimal hypersurfaces with areas going to $+\infty$. These are different from the hypersurfaces constructed by Fraser-Schoen [FrSch16], Kapouleas-Li [KL17] and Carlotto-Franz-Schulz [CFS20].

Besides the independent interest, the multiplicity one result for free boundary minimal hypersurfaces may also have many other applications. Here we present an application to construct new minimal hypersurfaces in certain non-compact and singular manifolds.

An embedded hypersurface in $\mathbb{R}^{n+1}$ is called a self-shrinker if

where $H$ is the mean curvature with respect to the normal vector field $\nu$. Equivalently, a self-shrinker is a minimal hypersurface in $\mathbb{R}^{n+1}$ under the Gaussian metric $\mathcal{G}=(4\pi)^{-1}e^{-\frac{|x|^{2}}{2n}}\delta_{ij}$; see [CM12_1] for more details. The area of such a hypersurface under Gaussian metric is called the entropy, defined by Colding-Minicozzi in [CM12_1].

Self-shrinkers are models of the singularities of mean curvature flows, and the index of a self-shrinker (as a minimal hypersurface in $(\mathbb{R}^{n+1},\mathcal{G})$) characterizes the instability of the mean curvature flow at the singularity; see the discussions in [CM12_1]. Understanding self-shrinkers is very important to the study of singular behaviors of mean curvature flows.

The main challenge of using min-max method to construct self-shrinkers is that the Gaussian metric space $(\mathbb{R}^{n+1},\mathcal{G})$ is non-compact, and the space becomes singular at infinity – the curvature blows up at infinity, meanwhile the spheres with increasing radii have exponentially decreasing areas. Therefore, the classical min-max method can not be used directly. We note that Ketover-Zhou [Ketover-Zhou18] have established a version of the min-max theory in $(\mathbb{R}^{3},\mathcal{G})$ using a special flow near the singularity.

Our approximation method provides the following new constructions of embedded self-shrinkers. We believe that our method can also be applied to other non-compact singular spaces.

A direct corollary is the existence of self-shrinkers with arbitrary large entropy.

There are several pioneer works which provided constructions of embedded self-shrinkers in $\mathbb{R}^{3}$ by other techniques, see \citelist[An92][Ng14][KKM18] [Mo11]. However, the entropy of all the self-shrinkers constructed in these works is bounded from above by a universal constant. In contrast, our min-max method constructs self-shrinkers with arbitrarily large entropy.

The entropy of a mean curvature flow is monotone non-increasing. As a consequence, any tangent flow of a mean curvature flow has entropy bounded from above by the entropy of the initial hypersurface. Based on this fact, there has been much research on low entropy mean curvature flow; cf. \citelist[Bernstein-Wang] [mramor2018low] [bernstein2020closed]. In contrast, our theorem implies that large entropy mean curvature flow may have very complicated tangent flows, which addresses the complication of mean curvature flow singularities.

Inspired by the proof of the Multiplicity One Theorem in closed manifolds [Zhou19], we prove Theorem 1.1 by establishing the min-max theory for free boundary $h$-hypersurfaces. Let $(M,\partial M,g)$ be a compact Riemannian manifold with boundary of dimension $3\leq(n+1)\leq 7$. Moreover, we assume that every embedded free boundary minimal hypersurface in $(M,\partial M,g)$ is proper and does not have non-trivial Jacobi fields. Then the $k$-the volume spectrum $\omega_{k}(M;g)$ is realized by the min-max value of a homotopy class of $k$-sweepouts. Note that the sweepouts are maps into $\mathcal{Z}_{n}(M,\partial M;\mathbb{Z}_{2})$. We can lift the $k$-sweepouts to the space of Cappccioppoli set $\mathcal{C}(M)$, and consider the associated relative homotopy class, denoted by $\widetilde{\Pi}$. We will show that the relative min-max width is equal to $\omega_{k}(M;g)$.

Take $h\in C^{\infty}(M)$ to be defined later. Let $\epsilon_{j}\rightarrow 0$ and denote by $\Sigma_{j}=\partial\Omega_{j}$ the free boundary $\epsilon_{j}h$-hypersurface, produced by relative min-max theory (associated with $\widetilde{\Pi}$). Letting $j\rightarrow\infty$, and using compactness results, we get a free boundary minimal hypersurface $\Sigma$ with multiplicity $m$. The aim is to prove $m=1$ by taking suitable $h$. Indeed, if $m>1$, we can construct a non-trivial positive Jacobi field $u$ so that $Lu=\pm 2h$ on $\Sigma$ and $\frac{\partial u}{\partial\bm{\eta}}=h^{\partial M}(\nu,\nu)u$, where $L$ is the Jacobi operator, $\nu$ and $\bm{\eta}$ are respectively the unit normal and co-normal vector of $\Sigma$, and $h^{\partial M}(\cdot,\cdot)$ is the second fundamental form of $\partial M$. By choosing a suitable $h$, we prove that this kind of $u$ does not exist.

However, there are several new challenges in this process. First, we need to show the genericity of the metrics under which every embedded free boundary minimal hypersurface is proper and has no non-trivial Jacobi field. Such a metric is called strongly bumpy. The denseness of such metrics can be obtained by perturbing any given metric on a closed manifold $\widetilde{M}$ (which contains $M$ as a domain) so that each minimal hypersurface in $\widetilde{M}$ with free boundary on $\partial M$ is non-degenerate and transverse to $\partial M$. To obtain openness, we use the compactness to show that given a strongly bumpy metric $g$ and a constant $C$, there is an open neighborhood of $g$ so that for each metric $g^{\prime}$ in that neighborhood, any free boundary minimal hypersurface in $(M,\partial M,g^{\prime})$ with weak Morse index and area no more than $C$ is non-degenerate and proper. By taking intersections of such open and dense sets of metrics for a sequence $C_{n}\rightarrow\infty$, we obtain the desired genericity.

Second, to establish the min-max theory for free boundary $h$-hypersurfaces, we need to generalize a number of regularity results. In Theorem 2.1, we prove the regularity for minimizing free boundary $h$-hypersurfaces by using the reflecting method in [Gr87] and the regularity results in [Mor03]. The major step for this min-max theory is to establish the regularity for $h$-almost minimizing varifolds. To finish the key gluing step therein, we apply the interior gluing procedure by by Zhou-Zhu [ZZ18], and then take the advantage of unique continuation to extend the gluing all the way to boundary. Comparing with the proof of the corresponding regularity results for free boundary minimal hypersurfaces in [LZ16], our arguments for free boundary $h$-hypersurfaces are much simpler since we only need to consider the case where the touching sets are contained in some $(n-1)$-dimensional subsets.

To obtain the aforementioned control on the touching sets, we need to restrict to “good” prescribing functions $h$, where each free boundary $h$-hypersurface has at most $(n-1)$-dimensional touching set (including self-touching and touching with $\partial M$). To show the genericity of such good functions, we extend a given smooth function $h$ on $M$ to $\widetilde{h}$ on $\widetilde{M}$ and perturb it to $\widetilde{h}_{1}$ by [ZZ18]*Proposition 3.8 so that each $\widetilde{h}_{1}$-hypersurface has small self-touching set. By genericity of Morse functions we can find another perturbed function $\widetilde{h}_{2}$ so that each $\widetilde{h}_{2}$-hypersurface touches $\partial M$ in a small subset as desired. This implies $\widetilde{h}_{2}|_{M}$ is a good choice. Moreover, each function in a neighborhood of $\widetilde{h}_{2}|_{M}$ is also a good choice. This gives the genericity of good prescribing functions.

Finally, we have to prove the index upper bounds in the min-max procedure, and a crucial ingredient is the countability of free boundary $h$-hypersurfaces for good pairs (see Subsection 2.4). Our proof of the countability follows from the following observation of the compactness of free boundary $h$-hypersurfaces. Let $\Sigma_{j}$ be a sequence of free boundary $h$-hypersurfaces with bounded index and area. Then it converges as varifolds to some limit hypersurface $\Sigma$; (see Theorem 2.9). If the convergence is smooth, then $\Sigma$ has a non-trivial Jacobi field related to $\mathcal{A}^{h}$; on the contray, if the convergence is not smooth, $\Sigma$ has strictly less index than that of $\Sigma_{j}$ for sufficiently large $j$. We refer to Lemma 2.12 for more details.

To construct self-shrinkers of large entropy, we use approximation methods. More precisely, we focus on balls with larger and larger radius in $\mathbb{R}^{n+1}$, perturb the Gaussian metric on the balls slightly to be generic, and use the theory developed in this article to construct free boundary minimal hypersurfaces of large area. By passing to a limit, these free boundary minimal hypersurfaces will converge to minimal hypersurface in the Gaussian space, or equivalently self-shrinkers. When passing to limits, there may be mass drop, and the limit shrinkers would not have large entropy. To overcome this issue, we prove a new monotonicity formula of minimal surfaces in almost Gaussian spaces, which is more close related to the monotonicity formula of minimal hypersurfaces in the Euclidean spaces compared with that for shrinkers. We also study the spectrum of the Jacobi operator for shrinkers of the form $\Sigma\times\mathbb{R}$ whenever $\Sigma$ is a lower dimensional shrinker, and we prove in Theorem 5.3 that if $\Sigma$ has finite index, so does $\Sigma\times\mathbb{R}$. This allows us to construct smooth self-shrinkers with finite index and arbitrarily large entropy in high dimensional spaces.

This paper is organized as follows. In Section 2, we first collect some notations and then provide some technical results including the regularity for $\mathcal{A}^{h}$-minimizing hypersurfaces and the compactness of free boundary $h$-hypersurfaces. We mention that Theorem 1.3 is proven in Subsection 2.5. Then in Section 3, we prove the relative min-max theory for free boundary $h$-hypersurfaces. The regularity of $h$-almost minimizing varifold with free boundary is proved in Theorem 3.10. Section 4 is devoted to the proof of Theorem 1.1. Finally, by applying our main result, we prove the existence of self-shrinkers with arbitrarily large entropy in Section 5. We also provide some well-known results and technical computations in Appendix A–D.

We would like to thank Professor Bill Minicozzi for his interest. Part of this work was carried out when Z.W. was a postdoc at Max-Planck Institute for Mathematics in Bonn and he thanks the institute for its hospitality and financial support. X. Z. is partially supported by NSF grant DMS-1811293, DMS-1945178 and an Alfred P. Sloan Research Fellowship.

We collect some notions here.

Let $(M^{n+1},\partial M^{n+1},g)$ be a compact, oriented, smooth Riemannian manifold with smooth boundary, of dimension $3\leq n+1\leq 7$. In this paper, we always isometrically embed $(M,\partial M,g)$ into a closed manifold $(\widetilde{M},\widetilde{g})$ with dimension $n+1$. Assume that $(\widetilde{M},\widetilde{g})$ is embedded in some $\mathbb{R}^{L},L\in\mathbb{N}$. {basedescript}\desclabelstyle\multilinelabel\desclabelwidth10em

the geodesic ball of $(\widetilde{M},g)$;

the space of smooth vector fields on $\widetilde{M}$;

the collection of $X\in\mathfrak{X}(\widetilde{M})$ with $X(p)\in T_{p}(\partial M)$ for $p$ in a neighborhood of $\partial\Sigma$ in $\partial M$;

$k$-dimensional Hausdorff measure;

the space of smooth Riemannian metrics on $M$;

the space of integer rectifiable $k$-currents with support in $M$;

the space of mod 2 rectifiable $k$-currents with support in $M$;

the group $\mathbb{Z}$ or $\mathbb{Z}_{2}$;

the space of $T\in\mathcal{R}_{k}(M;G)$ and $\mathrm{spt}(\partial T)\subset\partial M$;

the space of $T\in\mathcal{R}_{k}(M;G)$ and $\partial T\in\mathcal{R}_{k-1}(M;G)$;

equivalent class of $Z_{k}(M,\partial M;G)$, i.e. $T_{1}\in\tau$ if and only if $T-T_{1}\in\mathcal{R}_{k}(\partial M;G)$ for any $T\in\tau$;

the space of equivalent classes in $Z_{k}(M,\partial M;G)$;

the integer rectifiable varifold associated with $T\in\mathcal{R}_{k}(M;\mathbb{Z})$;

the Radon measure associated with $T\in\mathcal{R}_{k}(M;\mathbb{Z})$;

the mass norm on $\mathcal{R}_{k}(M;G)$;

$\mathbf{M}(\tau)=\inf\{\mathbf{M}(T):T\in\tau\}$.

the flat metric on $\mathcal{R}_{k}(M;G)$;

$\mathcal{F}(\tau_{1},\tau_{2})=\inf\{\mathcal{F}(T_{1},T_{2}):T_{1}\in\tau_{1},T_{2}\in\tau_{2}\}$ for $\tau_{1},\tau_{2}\in\mathcal{Z}_{k}(M,\partial M;G)$;

the space of sets $\Omega\subset M$ with finite perimeter (Cappccioppoli set);

the space of smooth vector field in $M$ so that $X(p)\in T_{p}(\partial M)$ for $p\in\partial M$;

the (reduced) boundary of $\llbracket\Omega\rrbracket$ restricted in the interior of $M$, (thus, as an integer rectifiable current, $\partial\Omega\llcorner\partial M=0$);

the space of $k$-dimensional integral varifolds;

the tangent space of $V$ at $p$;

denotes the $k$-dimensional Grassmannian bundle over $M$;

$\mathbf{F}(V,W)=\sup\{V(f)-W(f):f\in C_{c}(G_{k}(\mathbb{R}^{L})),|f|\leq 1,\mathrm{Lip}(f)\leq 1\},$ for $V,W\in\mathcal{V}_{k}(M)$;

$\mathbf{F}(\Omega_{1},\Omega_{2})=\mathcal{F}(\Omega_{1}-\Omega_{2})+\mathbf{F}(|\partial\Omega_{1}|,|\partial\Omega_{2}|)$ for $\Omega_{1},\Omega_{2}\in\mathcal{C}(M)$;

the integral current associated with $M$;

the integral varifold associated with $M$;

the collection of $\Omega^{\prime}\in\mathcal{C}(M)$ so that $\mathbf{F}(\Omega,\Omega^{\prime})\leq\epsilon$, where $\Omega\in\mathcal{C}(M)$.

For any $\tau\in\mathcal{Z}_{k}(M,\partial M;G)$, there is a canonical representative $T\in\tau$ where $T\in Z_{k}(M,\partial M;G)$ and $T\llcorner\partial M=0$.

Given $c>0$, a varifold $V\in\mathcal{V}_{k}(M)$ is said to have $c$-bounded first variation in an open subset $U\subset M$, if

where $\mathfrak{X}(U,U\cap\partial M)$ is defined to be the subset of $\mathfrak{X}(M,\partial M)$ compactly supported in $U$.

Let $\Sigma$ be an $n$-dimensional manifold with smooth boundary. Recall that an immersion is a map $\phi:(\Sigma,\partial\Sigma)\hookrightarrow(M,\partial M)$. In this paper, we often use $\Sigma$ to denote both the immersion and the hypersurface when there is no ambiguity.

We are interested in the following weighted area functional defined on $\mathcal{C}(M)$. Given $h\in C^{\infty}(M)$, define the $\mathcal{A}^{h}$-functional on $\mathcal{C}(M)$ by

The first variation formula for $\mathcal{A}^{h}$ along $X\in\mathfrak{X}(M,\partial M)$ is

where $\nu$ is the outward unit normal vector field of $\partial\Omega$.

When the boundary $\partial\Omega=\Sigma$ has support on a smooth immersed hypersurface, by virtue of the divergence theorem, we have

where $\bm{\eta}$ is the unit outer co-normal vector field of $\partial\Sigma$. If $\Omega$ is a critical point of $\mathcal{A}^{h}$, then (2.2) directly implies that $\Sigma$ must have mean curvature $H=h|_{\Sigma}$ and $\nu\perp\nu_{\partial M}$ along $\partial\Sigma$, where $\nu_{\partial M}$ is the outward unit normal vector field of $\partial M$.

Recall that an immersed hypersurface is called an $h$-hypersurface if its mean curvature $H(x)=h(x)$ everywhere.

In [Mor03], F. Morgan gives a general way to prove the regularity of isoperimetric hypersurfaces in Riemannian manifolds. His methods can be applied to $\mathcal{A}^{h}$-functional to prove the regularity of $h$-hypersurfaces bounding a domain which minimizes the $\mathcal{A}^{h}$-functional; (see  [ZZ18]*Theorem 2.2). In this part, we provide the regularity for $\mathcal{A}^{h}$-minimizers with free boundaries.

Let $B$ be an $(n+1)$-ball in $\mathbb{R}^{n+1}$, and $S\subset B$ a compact embedded $n$-ball such that $\partial S\cap B=\emptyset$. Denote by $B^{+}$ and $B^{-}$ the two components of $B\setminus S$. Note that the $\mathcal{A}^{h}$-functional in (2.1) is well-defined for $\Omega\in\mathcal{C}(B^{+})$.

We recall a few notations. Without loss of generality, we assume that $S\cap\operatorname{spt}\|\partial\Omega\|\neq\emptyset$. For $x\in B$, let $\xi(x)$ be the closest point to $x$ among $S$. Clearly, $\xi$ is well-defined in a small neighborhood of $S$. By possible rescaling, we assume that $B$ is contained in such a neighborhood. Then we define the reflection map $\sigma$ across $S$ by

We have $\sigma^{2}=\mathrm{id}$ and define

Thus $\widetilde{\Omega}\in\mathcal{C}(V)$, where $V=B^{+}\cup(S\cap B)\cap\sigma(B^{+})$. By possible rescaling, we may assume that $B_{3R}\subset B$. Set

We also denote by $\widetilde{B}_{r}(p)=\sigma(B_{r}(p))$. By shrinking $r$ if it is necesssary, we can assume that there exists $\kappa>1$ so that

Note that such process works for any Riemannian manifold.  Let $(M^{n+1},\partial M,g)$ be compact Riemannian manifold with boundary which is isometrically embedded into a closed manifold $(\widetilde{M},g)$. Recall that $B_{r}(p)$ is denoted by the geodesic ball of $\widetilde{M}$ with radius $r$ and center at $p$.  We then have the following regularity theorem:

An immersed hypersurface is almost embedded if it locally decomposes into smooth embedded sheets that touch other sheets or $\partial M$ but do not cross.  A hypersurface is properly embedded if it is almost embedded and no sheets touch other sheets or $\partial M$.

Let $\mathcal{S}=\mathcal{S}(g)$ be the collection of all Morse functions $h$ such that the zero set $\Sigma_{0}=\{h=0\}$ is a compact, smoothly embedded hypersurface so that

• •

  $\Sigma_{0}$ is transverse to $\partial M$ and the mean curvature of $\Sigma_{0}$ vanishes to at most finite order;

• •

  $\{x\in\partial M:H_{\partial M}(x)=h(x)\text{ or }H_{\partial M}(x)=-h(x)\}$ is contained in an $(n-1)$-dimensional submanifold of $\partial M$.

In the next, we prove that for each $h\in\mathcal{S}(g)$, the touching set of an almost embedded free boundary $h$-hypersurface has dimension less than or equal to $(n-1)$.

Recall that an almost embedded hypersurface $\Sigma$ is called a free boundary $h$-hypersurface if $H_{\Sigma}=h|_{\Sigma}$ and $\Sigma$ meets $\partial M$ orthogonally along $\partial\Sigma$. Given $h\in\mathcal{S}(g)$, denote by $\mathcal{P}^{h}$ the collection of free boundary $h$-hypersurfaces such that $\llbracket\Sigma\rrbracket=\partial\Omega$ for some open set $\Omega\subset M$.

Note that when $h\in\mathcal{S}(g)$, the min-max free boundary $h$-hypersurfaces produced in Theorem 3.10 satisfy the above requirements. Indeed, such $\llbracket\Sigma\rrbracket=\partial\Omega$ is a critical point of the weighted $\mathcal{A}^{h}$ functional:

The second variation formula for $\mathcal{A}^{h}$ along normal vector field $X=\varphi\nu$ is given by

In the above formula, $\nabla\varphi$ is the gradient of $\varphi$ on $\Sigma$; $\mathrm{Ric}$ is the Ricci curvature of $M$; $A$ and $h^{\partial M}$ are the second fundamental forms of $\Sigma$ and $\partial M$ with normal vector fields $\nu$ and $\nu_{\partial M}$, respectively.

We remark that in (2.10), $\Pi_{\Sigma}(\cdot,\cdot)$ can also be defined for any immersed free boundary $h$-hypersurfaces and it is a quadratic form on the space of $C^{\infty}$-functions on $\Sigma$ (not $\phi(\Sigma)$).

The Jacobi field of $\Sigma$ is defined to be a smooth function $\varphi$ on $\Sigma$ (not $\phi(\Sigma)$) so that

The classical Morse index for $\Sigma$ is defined to be the number of negative eigenvalues of the above quadratic form. However, since $\Sigma$ may touch itself and the boundary of $M$, a weaker version of the index is needed. Such a concept was introduced by Zhou [Zhou19, Definition 2.1, 2.3] for closed $h$-hypersurfaces based on Marques-Neves [MN16]*Definition 4.1.