Symmetric mean curvature flow on the n-sphere

Using the rotation invariant property, the dimension reduction argument, and an inductive argument, we are able to generalize Cheng-Wei-Zeng’s result [7] to the stationary integral varifold case. The existence of stationary $SO(n)$ invariant solutions $u_{\epsilon}^{-\infty}$ to (PAC) which have $T_{1,n-1}$ as their limit interface was shown by [3, 15]. Using the classification of low area $SO(n)$ invariant stationary integral varifold, we can prove a rigidity result of low energy $SO(n)$ invariant critical points of $E_{\epsilon}$.

Using the techniques from the previous work [6], we are able to construct $SO(n)$ invariant solutions $\{u_{\epsilon}(\cdot,t)\}$ to (PAC), connecting $u_{\epsilon}^{-\infty}$ and a ground state solution (least energy unstable solutions of the Allen-Cahn equation, which are symmetric critical points with nodal sets exactly along equatorial spheres $S^{n}$ by [4]).

The convergence of $\{u_{\epsilon}(\cdot,t)\}$ to an integral Brakke flow on $S^{n+1}$ can be derived from [18] and [26, 30]. This flow is cyclic mod 2, in the sense of White [33]. We then use the area bounds for the limit interfaces obtained from $u_{\epsilon}^{\pm\infty}$, and the $SO(n)$ invariant property, to describe the forward and backward limit of this flow using the classification of low area $SO(n)$ invariant stationary integral varifolds in $S^{n+1}$.

In Section 2, we state some results concerning the Allen-Cahn equation and give a brief description of the results from the previous paper. In Section 3, we study the symmetry and rigidity of solutions to (AC) with low energy. In Section 4, we show the existence of $SO(n)$ invariant solutions to (PAC) connecting the Allen-Cahn approximation of the Clifford hypersurface and the equatorial sphere, and the corresponding Brakke flow. In the Appendix, we give proofs of several inequalities regarding the area of the minimal Clifford hypersurfaces.

We would like to express our gratitude to André Neves for his support and numerous invaluable discussions and suggestions. Additionally, we extend our thanks to Pedro Gaspar, Yangyang Li, Daniel Stern, and Ao Sun for their insightful conversations. We are also appreciative of all the constructive comments provided by the referee.

Here $W(\cdot)$ is a double-well potential, the standard example is the function $W(t)=\frac{1}{4}(1-t^{2})^{2}$. Hereafter, we fix such a potential $W$.

One can check that $u$ is a critical point of $E_{\epsilon}$ on a closed manifold $(M^{n+1},g)$ if and only if $u$ (weakly) solves the elliptic Allen-Cahn equation:

We write $\sigma=\int_{-1}^{1}\sqrt{W(t)/2}dt$. This is the energy of the heteroclinic solution $\mathbb{H}_{\epsilon}(t)$ of (AC) on $\mathbb{R}$, namely, the unique bounded solution in $\mathbb{R}$ (modulo translation) such that $\mathbb{H}_{\epsilon}(t)\to\pm 1$ when $t\to\pm\infty$.

For the quadratic form given by the second variation of the energy $E_{\epsilon}$ at $u$, we can define the Morse index and the nullity of a solution $u$ of (AC) (as a critical point of $E_{\epsilon}$), denoted $\operatorname{ind}_{\epsilon}(u)$ and $\text{nul}_{\epsilon}(u)$, as the number of negative eigenvalues and the dimension of the kernel of the linear operator

counted with multiplicity, respectively.

We note that given $\epsilon>0$ and a sufficiently regular function $u$ on $M$ (so that almost every level set is a regular hypersurface), we can consider the associated $n$-varifolds $V_{\epsilon,u}$ defined by

for any continuous function $\phi$ defined in the Grassmannian manifold $G_{n-1}(M)$, where $V_{\epsilon,u}(\phi)$ denotes the integral of $\phi$ on $G_{n-1}(M)$ with respect to $V_{\epsilon,u}$. We write $\mu_{\epsilon,u}=\|V_{\epsilon,u}\|$ for the associated Radon measure on $M$ (the weight measure of $V_{\epsilon,u}$). In the case where $\{u_{j}\}$ are solutions to (AC) or (PAC), with $\epsilon=\epsilon_{j}\downarrow 0$, we will write ${V_{\epsilon_{j},t}=V_{\epsilon_{j},u_{j}(\cdot,t)}}$ and ${\mu_{\epsilon_{j},t}:=\mu_{\epsilon_{j},u_{j}(\cdot,t)}}$.

We have the following convergence results for solutions to (AC) and (PAC).

The minimal surface $\operatorname{supp}\|V\|$ is often called a limit interface obtained from $u_{j}$. Solutions with limit interface $\Gamma$ are called the Allen-Cahn approximation of $\Gamma$.

We state below the main convergence result we will use in the present article, which follows from [18] and the work of Tonegawa [30] (see also [26] and [29]):

We summarize the results from the previous paper [6] joint with P. Gaspar. Some results and techniques will be adapted to be used in this article.

For a solution $u_{\epsilon}^{-\infty}$ on $S^{n+1}$ to (AC) with Morse index $I$, by the result of Choi-Mantoulidis [9], where they show the existence of ancient gradient flows and uniqueness under integrability conditions, and the theory of parabolic PDEs, we show that there is a family of gradient flows of $E_{\epsilon}$,

such that $\lim\limits_{t\to-\infty}\mathscr{S}_{\epsilon}(\cdot,t)=u_{\epsilon}^{-\infty}$. The curve $t\to\mathscr{S}_{\epsilon}(a)(\cdot,t)$ is tangent at $u_{\epsilon}^{-\infty}$ to eigenfunctions $\{\varphi_{j}\}_{j=1}^{I}$ of

with coefficients $a=(a_{1},\cdots,a_{I})$.

When $I\geq 2$, for small $r<\eta$, using the fact that the first eigenfunction $\varphi_{1}>0$, the maximum principle for parabolic equations, and the Frankel-type property proved by Hiesmayr [15], we prove that $\mathscr{S}_{\epsilon}(\pm r,0,\cdots,0)$ converge to $\pm 1$ as $t\to+\infty$.

Let $\mathcal{S}=\{u\in C^{2}(M)\mid|u|\leq 1\}$. By Lemma 2.3 in [12] (and the continuous dependence of initial data, see e.g. Cazenave-Haraux [5]), there is a continuous map

such that $\Phi(u,\cdot):M\times[0,\infty)\to\mathbb{R}$ is a solution of (PAC) defined for all $t\geq 0$ with $\Phi(u,0)=u$, and such that $\Phi(u,t)\in\mathcal{S}$, for all such $t$.

By an analysis argument, we show that the sets

are open. By connectedness, we show that along any curve connecting $(\pm r,0,\cdots,0)\in B_{\eta}$, there exists a point $a$ on it such that $\mathscr{S}_{\epsilon}(a)$ does not converge to $\pm 1$ as $t\to+\infty$.

In the previous paper, we study solutions and flows on $S^{3}$. By analyzing the limit interfaces of the backward limit and the forward limit of $\mathscr{S}_{\epsilon}(a)$ (on $S^{3}$) we mentioned above, which uses a parity argument and the classification of low area stationary integral varifold in $S^{3}$ based on the resolution of the Willmore conjecture by Marques-Neves [20], we show the existence of gradient flow of $E_{\epsilon}$ connecting the Allen-Cahn approximations of a Clifford torus and an equatorial sphere.

Using the convergence results for solutions to (PAC) mentioned above, we show the limit of the gradient flows of $E_{\epsilon}$ described above is a Brakke flow $\{V_{t}\}$. Using the energy estimate and the parity argument again, we show that the backward limit and the forward limit are a Clifford torus and an equatorial sphere, respectively.

By Choi-Mantoulidis [9] rigidity of ancient gradient flows, we proved a symmetry argument for solutions to (PAC). We described a relation between the isometries on $S^{3}$ and the isometries on the negative eigenspaces of the linearized Allen-Cahn operator at $u_{\epsilon}^{-\infty}$. From this relation, we observe that if $a\in B_{\eta}$ is fixed by an isometry that preserves $u_{\epsilon}^{-\infty}$, then so are $\mathscr{S}_{\epsilon}(a)(\cdot,t)$, the limit Brakke flow $V_{t}$ and the limits $V_{\pm\infty}$ (the limit of $V_{t}$ as $t\to\pm\infty$).

Using the symmetry argument and a continuity argument, we give a precise description of the backward limit and the forward limit, and therefore obtain a $2-$parameter family of Brakke flows generated by rotations on $S^{3}$.

In the previous paper, we study solutions and flows on $S^{3}$ that are invariant under the following $3$ isometries:

We show the existence of a $2-$parameter family of weak mean curvature flows connecting a Clifford torus and equatorial spheres that are invariant under these isometries on $S^{3}$. In this article, we study solutions and flows on $S^{n+1}$ ($n\geq 2$) under a stronger symmetry assumption, that is invariant under $SO(n)$ (for instance, let $n=2$, this condition means that the flows are invariant under all rotations acting on $x_{3},x_{4}$ coordinates), and show the existence of a $1-$parameter family of weak mean curvature flows connecting $T_{1,n-1}$ and the equators $S^{n}$ in $S^{n+1}$ that are $SO(n)$ invariant.

Recall that $T_{1,n-1}$ is the Clifford hypersurface $S^{1}(\sqrt{\frac{1}{n}})\times S^{n-1}(\sqrt{\frac{n-1}{n}})$ in $S^{n+1}$.

Cheng-Wei-Zeng [7, Theorem 1.1] proved that for a $n-$dimensional compact minimal rotational hypersurface in $S^{n+1}$, if it is neither a equatorial sphere $S^{n}$ nor a Clifford hypersurface $T_{1,n-1}$, then its area $>2(1-\frac{1}{\pi})\text{Area}(T_{1,n-1})$.

We generalize their result to the stationary integral varifold case using an inductive argument and the dimension reduction argument. To do this, we establish the following monotonicity and boundedness result concerning the density ratio of $T_{1,n-1}$.

The proof is computational, for detailed proof, see Appendix A.

Caju-Gaspar [3, Theorem $1.1$] proved the existence of a solution $u_{\epsilon}$ of (AC) whose nodal set converges to a minimal, separating hypersurface $\Gamma$, for sufficiently small $\epsilon$, under the assumption that all Jacobi fields are generated by global isometries. They also proved that the Morse index and the nullity of $u_{\epsilon}$ equal the Morse index and the nullity of $\Gamma$.

We here give a brief explanation of the index and the nullity estimate. We can get a Jacobi field for the Allen-Cahn approximation of $\Gamma$ by pairing a Killing field with the gradient of this solution. These Jacobi fields for $u_{\epsilon}$ produced by the Killing fields are linearly independent for sufficiently small $\epsilon$, which implies that the nullity of $\Gamma\leq$ the nullity of the Allen-Cahn approximation of $\Gamma$. Then by the lower semicontinuity from [11] and the upper semicontinuity in the multiplicity one case from [8], we claim that the Morse index and the nullity of $\Gamma$ and the Allen-Cahn approximation of $\Gamma$ are equal.

By Hsiang-Lawson [16], for the minimal hypersurface $T_{p,q}$ in $S^{n+1}$ ($n=p+q$), all Jacobi fields of $T_{p,q}$ are Killing-Jacobi fields. The nullity estimate above implies that the Jacobi fields for the Allen-Cahn approximation of $T_{p,q}$ produced by the Killing fields make up all the nullity of this solution for all sufficiently small $\epsilon$.

Now we show the symmetry property of the solution $u_{\epsilon}$. We include the proof from Hiesmayr [15] for completeness.

By using a change of variables argument as in [15], we see that for any isometry $P$ of $S^{n+1}$ and any $C^{1}$ function $u$ defined on $S^{n+1}$, the pushforward $P_{\#}$ by the isometry $P$ satisfies

Here we mention some common notations. We write $|\Sigma|$ for the unit density varifold associated to $\Sigma$. Given a surface $\Sigma\subset S^{n+1}$, and a function $u$ on $S^{n+1}$, denote their stabilizer group by $\text{Stab}\Sigma$, $\text{Stab}u$, respectively. In other words, $\text{Stab}\Sigma=\{P\in SO(n+2)|P(\Sigma)=\Sigma\},\text{Stab}u=\{P\in SO(n+2)|u\circ P=u\}$. These stabilizer groups are closed Lie subgroups of $SO(n+2)$, and we can therefore discuss the dimensions.

Recall from [4], ground state solutions are unstable solutions of least energy. In the Allen-Cahn setting, the ground state solution on $S^{n+1}$ is unique up to rigid motions, and its nodal set is exactly along an equator $S^{n}$ for sufficiently small $\epsilon$.

One may combine [14] with [2] to obtain the following general result about the nodal sets: Let $(M,g)$ be closed, let $\epsilon>0$ and $u_{\epsilon}^{1},u_{\epsilon}^{2}$ be two solutions of (AC) on $M$. If the nodal sets of $u_{\epsilon}^{1},u_{\epsilon}^{2}$ are same, then $u_{\epsilon}^{1}=\pm u_{\epsilon}^{2}$.

From now on we focus on the Clifford hypersurface $T_{1,n-1}$, that is, up to isometry, the minimal hypersurface

We study the specific Clifford hypersurface $T_{c}$ (as defined in equation (3)) and its Allen-Cahn approximation, as all other Clifford hypersurfaces $T_{1,n-1}$ (and their Allen-Cahn approximation) in $S^{n+1}$ are just different by a rotation. Denote the Allen-Cahn approximation of $T_{c}$ as $u_{\epsilon}^{-\infty}$, and we know that $E_{\epsilon}(u_{\epsilon}^{-\infty})\to(2\sigma)\cdot\text{Area}(T_{c})$ as $\epsilon\downarrow 0$.

Let the group $G_{1}$ denote all rotations acting on the $x_{1},x_{2}$ coordinates, and the group $G_{2}$ denote all rotations acting on the $x_{3},\cdots,x_{n+2}$ coordinates. It is clear that $T_{c}$ is $G_{1}\times G_{2}$ invariant.

Next, we show that the only $SO(n)$ invariant solutions of (AC) with energy level under $u_{\epsilon}^{-\infty}$ are ground states.

For the critical point $u_{\epsilon}^{-\infty}$ which is the Allen-Cahn approximation of $T_{c}$, we use the results of [6] to construct $G_{2}$ invariant gradient flow of $E_{\epsilon}$ with $u_{\epsilon}^{-\infty}$ as its backward limit.

For sufficiently small $\epsilon$, $u_{\epsilon}^{-\infty}$ has Morse index $n+3$, we denote by $\{\varphi_{i}\}_{i=1}^{n+3}$ an $L^{2}$-orthonormal basis for the eigenspaces of the linearized Allen-Cahn operator at $u_{\epsilon}^{-\infty}$ corresponding to negative eigenvalues, where we assume $\varphi_{1}>0$.

We also recall the solution map $\mathscr{S}_{\epsilon}=\mathscr{S}\colon B_{\eta}(0)\subset\mathbb{R}^{n+3}\to C^{2,\alpha}(S^{n+1}\times\mathbb{R})$, defined for some $\eta=\eta_{\epsilon}>0$. Using the same argument as in [6] Section $5$, we can choose the eigenfunctions $\varphi_{i}$ in a way that $G_{2}$ (rotations on $S^{n+1}$ acting on $x_{3},\cdots,x_{n+2}$ coordinates) acts on $\varphi_{4},\cdots,\varphi_{n+3}$ by rotations and fixes $\varphi_{2},\varphi_{3}$ pointwise. Note that $\varphi_{1}$ is fixed since $\varphi_{1}>0$ and the rotations $G_{2}$ fix $T_{c}$.