Symmetry of hypersurfaces and the Hopf Lemma

Dedicated to Neil Trudinger on his 80th birthday with friendship and admiration.

1.   Introduction

H. Hopf established in ^[3] that an immersion of a topological $2$-sphere in $\mathbb R^3$ with constant mean curvature must be a standard sphere. He also made the conjecture that the conclusion holds for all immersed connected closed hypersurfaces in $\mathbb R^{n+1}$ with constant mean curvature. A. D. Alexandrov proved in ^[1] that if $M$ is an embedded connected closed hypersurface with constant mean curvature, then $M$ must be a standard sphere. If $M$ is immersed instead of embedded, the conclusion does not hold in general, as shown by W.-Y. Hsiang in ^[4] for $n\ge 3$ and by Wente in ^[15] for $n = 2$. A. Ros in ^[13] gave a different proof for the theorem of Alexandrov making use of the variational properties of the mean curvature.

In this note, we give exposition to some results in ^[5,6,7,8,9]. It is suggested that the reader read the introductions of ^[6,7,9].

Throughout the paper $M$ is a smooth compact connected embedded hypersurface in $\mathbb R^{n+1}$, $k(X) = (k_1(X), \cdots, k_n(X))$ denotes the principal curvatures of $M$ at $X$ with respect to the inner normal, and the mean curvature of $M$ is

We use $G$ to denote the open bounded set bounded by $M$.

Li proved in^[5] the following result saying that if the mean curvature $H: M\to \mathbb R$ has a Lipschitz extension $K: \mathbb R^{n+1} \to \mathbb R$ which is monotone in the $X_{n+1}$ direction, then $M$ is symmetric about a hyperplane $X_{n+1} = c$.

Theorem 1.1. (^[5]) Let M be a smooth compact connected embeded hypersurface without boundary embedded in $\mathbb R^{n+1}$, and let $K$ be a Lipschitz function in $\mathbb R^{n+1}$ satisfying

Suppose that at each point $X$ of $M$ the mean curvature $H(X)$ equals $K(X)$. Then $M$ is symmetric about a hyperplane $X_{n+1} = c.$

In ^[5], $K$ was assumed to be $C^1$ for the above result, but the proof there only needs $K$ being Lipschitz.

Li and Nirenberg then considered in ^[6] and ^[7] the more general question in which the condition $H(X) = K(X)$ with $K$ satisfying (1.1) is replaced by the weaker, more natural, condition:

Main Assumption. For any two points $(X', A), (X', B) \in M$ satisfying $A\le B$ and that $\{(X', \theta A+(1 - \theta)B):0 \le \theta \le 1\}$ lies in $\overline G$, we have

They showed in ^[6] that this assumption alone is not enough to guarentee the symmetry of $M$ about some hyperplane $X_{n+1} = c$. The mean curvature $H: M\to \mathbb R$ of the counterexample constructed in [6, Figure 4] has a monotone extension $K: \mathbb R^{n+1} \to \mathbb R$ which is $C^\alpha$ for every $0 < \alpha < 1$, but fails to be Lipschitz. The counterexample actually satisfies (1.2) with an equality. They also constructed a counterexample [6, Section 6] showing that the inequality (1.2) does not imply a pairwise equality.

A conjecture was made in ^[7] after the introduction of

Condition S. $M$ stays on one side of any hyperplane parallel to the $X_{n+1}$ axis that is tangent to $M$.

Conjecture 1. (^[7]) Any smooth compact connected embedded hypersurface $M$ in $\mathbb R^{n+1}$ satisfying the Main Assumption and Condition S must be symmetric about a hyperplace $X_{n+1} = c$.

The conjecture for $n = 1$ was proved in ^[6]. For $n\ge 2$, they introduced the following condition:

Condition T. Every line parallel to the $X_{n+1}$-axis that is tangent to $M$ has contact of finite order.

Note that if $M$ is real analytic then Condition T is automatically satisfied.

They proved in [7, Theorem 1] that $M$ is symmetric about a hyperplane $X_{n+1} = c$ under the Main Assumption, Condition S and T, and a local convexity condition near points where the tangent planes are parallel to the $X_{n+1}$-axis. For convex $M$, their result is

Theorem 1.2. (^[7]) Let $M$ be a smooth compact convex hypersurface in $\mathbb R^{n+1}$ satisfying the Main Assumption and Condition T. Then $M$ must be symmetric about a hyperplane $X_{n+1} = c$.

The theorem of Alexandrov is more general in that one can replace the mean curvature by a wide class of symmetric functions of the principal curvatures. Similarly, Theorems 1.1 and 1.2 (as well as the more general [7, Theorem 1]) still hold when the mean curvature function is replaced by more general curvature functions.

Consider a triple $(M, \Gamma, g)$: Let $M$ be a compact connected $C^2$ hypersurface without boundary embedded in $\mathbb R^{n+1}$, and let $g(k_1, \cdots, k_n)$ be a $C^3$ function, symmetric in $(k_1, \cdots, k_n)$, defined in an open convex neighborhood $\Gamma$ of $\{ (k_1(X), \cdots, k_n(X))\ |\ X\in M\}$, and satisfy

For convex $M$, their result ([7, Theorem 2]) is as follows.

Theorem 1.3. (^[7]) Let the triple $(M, \Gamma, g)$ satisfy (1.3). In addition, we assume that $M$ is convex and satisfies Condition T and the Main Assumption with inequality (1.2) replaced by

Then $M$ must be symmetric about a hyperplane $X_{n+1} = c$.

For $1\le m\le n$, let

be the $m$-th elementary symmetric function, and let

It is known that $g = g_m$ satisfies the above properties in

It is known that $\Gamma_1 = \{ k\in \mathbb R^n\ |\ k_1+\cdots + k_n > 0\}$, $\Gamma_n = \{ k\in \mathbb R^n\ |\ k_1, \cdots, k_n > 0\}$, $\Gamma_{m+1}\subset \Gamma_m$, and $\Gamma_m$ is the connected component of $\{ k\in \mathbb R^n\ |\ \sigma_m(k) > 0\}$ containing $\Gamma_n$.

The method of proof of Theorems 1.2 and 1.3 (as well as the more general [7, Theorems 1 and 2]) begins as in that of the theorem of Alexandrov, using the method of moving planes. Then, as indicated in the introduction of ^[6], one is led to the need for variations of the classical Hopf Lemma. The Hopf Lemma is a local result. The needed variant of the Hopf Lemma to prove Theorem 1.2 (and Conjecture 1) was raised as an open problem ([7, Open Problem 2]) which remains open. The proof of Theorems 1.2 and 1.3 (as well as the more general [7, Theorems 1 and 2]) was based on the maximum principle, but also used a global argument.

In a recent paper ^[9], Li, Yan and Yao proved Conjecture 1 using a method different from that of ^[6] and ^[7], exploiting the variational properties of the mean curvature. In fact, they proved the symmetry result under a slightly weaker assumption than Condition S:

Condition S'. There exists some constant $r > 0$, such that for every $\overline X = (\overline X', \overline X_{n+1})\in M$ with a horizontal unit outer normal (denote it by $\bar \nu = (\bar \nu', 0))$, the vertical cylinder $|X'-(\overline X'+r\bar \nu')| = r$ has an empty intersection with $G$. ($G$ is the bounded open set in $\mathbb R^{n+1}$ bounded by the hypersurface $M$.)

Theorem 1.4. (^[9]) Let $M$ be a compact connected $C^2$ hypersurface without boundary embedded in $\mathbb R^{n+1}$, which satisfies both the Main Assumption and Condition S'. Then $M$ must be symmetric about a hyperplane $X_{n+1} = c$.

Here are two conjectures, in increasing strength.

Conjecture 2. For $n\ge 2$ and $2\le m\le n$, let $M$ be a compact connected $C^2$ hypersurface without boundary embedded in $\mathbb R^{n+1}$ satisfying Condition S (or the slightly weaker Condition S') and $\{ (k_1(X), \cdots, k_n(X))\ |\ X\in M\}\subset \Gamma_m$. We assume that $M$ satisfies the Main Assumption with inequality (1.2) replaced by

Then $M$ must be symmetric about a hyperplane $X_{n+1} = c$.

The next one is for more general curvature functions.

Conjecture 3. For $n\ge 2$, let the triple $(M, \Gamma, g)$ satisfy (1.3). In addition, we assume that $M$ satisfies Condition S (or the slightly weaker Condition S') and the Main Assumption with inequality (1.2) replaced by (1.4). Then $M$ must be symmetric about a hyperplane $X_{n+1} = c$.

The above two conjectures are open even for convex $M$.

Conjecture 2 can be approached by two ways. One is by the method of moving planes, and this leads to the study of variations of the Hopf Lemma. Such variations of the Hopf Lemma are of its own interest. A number of open problems and conjectures on such variations of the Hopf Lemma has been discussed in ^[6,7,8]. For related works, see ^[11] and ^[14]. We will give some discussion on this in Section 1.

Conjecture 2 can also be approached by using the variational properties of the higher order mean curvature (i.e., the $\sigma_m$-curvature). If the answer to Conjecture 2 is affirmative, then the inequality in (1.5) must be an equality. This curvature equality was proved in the following lemma, using the variational properties of the $\sigma_m$-curvature:

Lemma 1. (Y. Y. Li, X. Yan and Y. Yao) For $n\ge 2$ and $2\le m\le n$, let $M$ be a compact connected $C^2$ hypersurface without boundary embedded in $\mathbb R^{n+1}$ satisfying Condition S'. We assume that $M$ satisfies the Main Assumption, with inequality (1.2) replaced by (1.5). Then (1.5) must be an equality for every pair of points.

The proof of Theorem 1.4 and Lemma 1 will be sketched in Section 2.

We have discussed in the above symmetry properties of hypersurfaces in the Euclidean space. It is also interesting to study symmetry properties of hypersurfaces under ordered curvature assumptions in the hyperbolic space, including the study of the counter part of Theorem 1.1, Theorem 1.4, and Conjecture 2 in the hyperbolic space. Extensions of the Alexandrov-Bernstein theorems in the hyperbolic space were given by Do Carmo and Lawson in ^[2]; see also Nelli ^[10] for a survey on Alexandrov-Bernstein-Hopf theorems.

2.   Discussion on Conjecture 2 and the proof of Theorem 1.3

Let

We use $k^u(t, y) = (k_1^u(t, y), \cdots, k_n(t, y))$ to denote the principal curvatures of the graph of $u$ at $(t, y)$. Similarly, $k^v = (k_1^v, \cdots, k_n^v)$ denotes the principal curvatures of the graph of $v$.

Here are two plausible variations of the Hopf Lemma.

Open Problem 1. For $n\ge 2$ and $1\le m\le n$, let $u$ and $v$ satisfy the above. Assume

Is it true that either

or

A weaker version is

Open Problem 2. In addition to the assumption in Open Problem 1, we further assume that

Is it true that either (2.2) or (2.3) holds?

Open Problems 1 and 2 for $m = 1$ are exactly the same as [7, Open Problems 1 and 2], where it was pointed out that an affirmative answer to Open Problem 2 for $m = 1$ would yield a proof of Conjecture 1 by modification of the arguments in ^[6,7]. This applies to $2\le m\le n$ as well: An affirmative answer to Open Problem 2 for some $2\le m\le n$ would yield a proof of Conjecture 2 (with Condition S) for the $m$.

As mentioned earlier, the answer to Open Problem 1 for n = 1 is yes, and was proved in ^[6]. For $n\ge 2$, a number of conjectures and open problems on plausible variations to the Hopf Lemma were given in ^[6,7,8]. The study of such variations of the Hopf Lemma can first be made for the Laplace operator instead of the curvature operators. The following was studied in ^[8].

Let $u\ge v$ be in $C^\infty(\overline \Omega)$ where $\Omega$ is given by (2.1). Assume that

and

We impose a main condition for the Laplace operator:

Under some conditions we wish to conclude that

The following two conjectures, in decreasing strength, were given in ^[8].

Conjecture 4. Assume, in addition to the above, that

Then (2.4) holds:

Conjecture 5. In addition to (2.5) assume that

Then

Partial results were given in ^[8] concerning these conjectures. On the other hand, the conjectures remain largely open.

3.   Discussion on Conjecture 2 and the proof of Theorem 1.4

Theorem 1.4 was proved in ^[9] by making use of the variational properties of the mean curvature operator. We sketch the proof of Theorem 1.4 below, see ^[9] for details.

For any smooth, closed hypersurface $M$ embeded in $\mathbb R^{n+1}$, let $V: \mathbb R^{n+1}\to \mathbb R^{n+1}$ be a smooth vector field. Consider, for $|t| < 1$,

and

It is well known that

where $H(x)$ is the mean curvature of $M$ at $x$ with respect to the inner unit normal $\nu$.

Define the projection map $\pi: (x', x_{n+1}) \to x'$, and set $R: = \pi (M)$.

Condition S' assures that $\nu(\bar x)$, $\bar x\in M$, is horizontal iff $\bar x'\in \partial R$; $\partial R$ is $C^{1, 1}$ (with $C^{1, 1}$ normal under control); and

where $M_1, M_2$ are respectively graphs of functions $f_1, f_2: R^\circ\to R,$ $f_1, f_2\in C^2(R^\circ), f_1 > f_2$ in $R^\circ$, and $\widehat M: = \{ (x', x_{n+1})\in M\ |\ x'\in \partial R\}\equiv M\cap \pi^{-1}(\partial R).$ Note that $f_1, f_2$ are not in $C^0(R)$ in general.

Lemma 2.

Proof. Take $V(x) = e_{n+1} = (0, ..., 0, 1)$, and let $M(t)$ and $S(t)$ be defined as above with this choice of $V(x)$. Clearly, $S(t)$ is independent of $t$. So we have, using (3.2) and the order assumption on the mean curvature, that

Using again the order assumption on the mean curvature we obtain The curvature equality (3.3).

For any $v\in C^\infty(R^{n})$, let $V(x): = v(x')e_{n+1}$, and let $M(t)$ and $S(t)$ be defined as above with this choice of $V(x)$. We have, using (3.2) and (3.3), that

Theorem 1.4 is proved by contradition as follows: If $M$ is not symmetric about a hyperplane, then $\nabla (f_1+f_2)$ is not identically zero. We will find a particular $V(x) = v(x') e_{n+1}$, $v\in C^2_{loc}(R^\circ)$, to make

which contradicts to (3.5).

Write

where $\widehat S$, the area of the vertical part of $M$, is independent of $t$ (since $v$ is zero near $\partial R$, so the vertical part of $M$ is not moved).

A calculation gives

where

We know that

So $[\nabla A(q_1)-\nabla A(q_2)] \cdot (q_1-q_2) > 0$ for any $q_1\ne q_2$.

If $\nabla (f_1+f_2)\in C^2_{loc}(R^\circ)u\setminus \{0\}$, we would take $v = \nabla (f_1+f_2)$ and obtain

In general, $\nabla (f_1+f_2)$ would not be in $C^2_{loc}(R^\circ)$. It turns out that Condition S' allows us to do a smooth cutoff near $\partial R$, and conclude the proof. We skip the crucial details, which can be found in the last few pages of ^[9].

Now we give the

Proof of Lemma 1. The proof is similar to that of Lemma 2, see also the proof of [9, Proposition 3] for more details. We still take $V(X) = e_{n+1}$ and let $M(t)$ be as in (3.1). Consider

Clearly, $S_{m-1}(t)$ is independent of $t$.

The variational properties of higher order curvature [12, Theorem B] gives

thus the same argument as (3.4) yields

We deduce from the above, using the curvature inequality (1.5), that the equality in (1.5) must hold for every pair of points. Lemma 1 is proved.

Acknowledgments

Partially supported by NSF grants DMS-1501004, DMS-2000261, and Simons Fellows Award 677077.

Conflict of interest

The author declares no conflict of interest.