Heintze-Karcher inequality and
capillary hypersurfaces in a wedge

The study of capillary surfaces goes back to Thomas Young, who studied in 1805 the equilibrium state of liquid fluids. It was he who first introduced the notion of mean curvature and the boundary contact angle condition of capillarity, the so-called Young’s law. This problem was reintroduced and reformulated by Laplace and by Gauss later. For the history of capillary surfaces see Finn’s survey [Finn99]. A capillary hypersurface $\Sigma$ in $\mathbb{R}^{n+1}$ with boundary $\partial\Sigma$ on a support hypersurface $S\subset\mathbb{R}^{n+1}$ is a critical point of the following functional

among all compact hypersurfaces with boundary $\partial\Sigma$ on $S$ under a volume constraint. Here $\Omega$ is the bounded domain enclosed by $\Sigma$ and $S$, and $|\Sigma|$ is the $n$-dimensional area of $\Sigma$, and $\theta\in(0,\pi)$. Equivalently, a capillary hypersurface is a constant mean curvature (CMC) hypersurface with boundary which intersects the support hypersurface $S$ at a constant angle $\theta$. There has been a lot of interdisciplinary investigations on the stationary solutions and local minimizers of the above energy. For the interested reader, we refer to Finn’s book [Finn86], which is an excellent survey on capillary surfaces.

Inspired by the recent development of the min-max theory for minimal surfaces and CMC surfaces [CD03, MN16a, MN16b], there have been a lot of works on free boundary minimal surfaces and CMC surfaces, which are a special class of capillary surfaces with $\theta=\frac{\pi}{2}$, see for example [DR18, LZ21, GLWZ21, ZZ19]. Very recently the min-max theory for capillary surfaces was developed in [DMDP21, LiZZ21].

One important application of the capillary surfaces was recently obtained by Chao Li in [Li21], where he utilizes capillary surfaces in a polyhedron to study Gromov’s dihedral rigidity conjecture. His work on capillary surfaces is related to this paper, in which we will consider capillary surfaces supported on a wedge.

The main objective of this paper is to make a complete classification of embedded capillary hypersurfaces in a wedge. Such a hypersurface can be seen as a model for capillary hypersurfaces in Riemannian polyhedra. For the precise definition of a wedge, see below in the Introduction. Our starting point is the Heintze-Karcher inequality. Let us first recall Heintze-Karcher’s theorem and Heintze-Karcher’s inequality for closed hypersurfaces.

In a seminal paper [HK78], Heintze-Karcher proved a general tubular volume comparison theorem for embedded Riemannian submanifolds, which generalizes the celebrated Bishop-Gromov’s volume comparison theorem in Riemannian geometry. For an embedded closed hypersurface $\Sigma$, which encloses a bounded domain $\Omega$, in an $(n+1)$-dimensional Riemannian manifold of nonnegative Ricci curvature, Heintze-Karcher’s theorem reads as follows,

Here $H(p)$ is the mean curvature of $\Sigma$ at $p$ and $c(p)$ is the length to reach the first focal point of $\Sigma$ from $p$ by the normal exponential map. As a direct consequence of (1), one deduces that

provided that $\Sigma$ is strictly mean convex, namely, $H>0$ on $\Sigma$. Nowadays, (2) is literately referred to as Heintze-Karcher’s inequality in hypersurfaces theory. A well-known new proof via Reilly’s formula [Reilly77] has been given by Ros [Ros87]. Moreover, Ros [Ros87] utilized the Heintze-Karcher inequality (2) to reprove the celebrated Alexandrov’s soap bubble theorem, which states that any embedded closed constant mean curvature hypersurfaces in $\mathbb{R}^{n+1}$ must be a round sphere.

Since then various Heintze-Karcher-type inequalities have been established in various circumstance. For instance, Montiel-Ros [MR91] and Brendle [Br13] established Heintze-Karcher-type inequalities in space forms and in certain warped product manifolds respectively, see also [QX15, LX19]. The Heintze-Karcher inequality in $\mathbb{R}^{n+1}$ has been also established for sets of finite perimeter, see e.g. [DM19, San19]. Like the Alexandrov-Fenchel inequalities, the Heintze-Karcher inequality becomes one of fundamental geometric inequalities in differential geometry.

Inspired by the method of Ros [Ros87], and also by the work of Brendle [Br13], we have proved a Heintze-Karcher-type inequality for hypersurfaces with free boundary in a unit ball [WX19] by using a generalized Reilly’s formula proved by Qiu-Xia [QX15]. However this method leads to a slight different inequality if we consider hypersurfaces with capillary boundary in the unit ball or in the half-space in the very recent work, [JXZ22]. Precisely we have established in the previous work [JXZ22] a version of Heintze-Karcher-type inequality for hypersurfaces in the half-space with capillary boundary, by using the solution to a mixed boundary value problem in the classical Reilly’s formula. Let $\mathbb{R}^{n+1}_{+}:=\{x\in\mathbb{R}^{n+1}:\langle x,E_{n+1}\rangle>0\}$, where $E_{n+1}=(0,\cdots,0,1)$, and for an embedded, compact, strictly mean-convex hypersurface $\Sigma\subset\overline{\mathbb{R}^{n+1}_{+}}$ with capillary boundary with a constant contact angle $\theta_{0}\in(0,\frac{\pi}{2}]$, there holds

with equality if and only if $\Sigma$ is a spherical cap.

Inequality (3) is optimal, in the sense that the spherical caps achieve equality in (3). However it is not in the best form. For example, while we are able to use (3) to reprove the Alexandrov theorem for constant mean curvature (CMC) hypersurfaces in [JXZ22], it is not very helpful to handle the case of higher order mean curvatures (see (17) for the definition). In view of the following Minkowski formula

a possible best form, which was conjectured in [JXZ22], is

It is clear by using the Cauchy-Schwarz inequality that inequality (5) implies (3), provided that $\langle\nu,E_{n+1}\rangle$ is non-negative. But without the non-negativity one does not know which one is stronger.

The first part of this paper is to establish this “best” version of the Heintze-Karcher inequality in a little more general setting and for whole range $\theta_{0}\in(0,\pi)$. Let $\Sigma$ be a hypersurface in $\overline{\mathbb{R}^{n+1}_{+}}$ with (possibly non-connected) boundary $\partial\Sigma\subset\partial\mathbb{R}^{n+1}_{+}$. The hypersurface intersects with the supported hyperplane $\partial\mathbb{R}^{n+1}_{+}$ transversely. ¹¹1In the paper we abuse a little bit the terminology of capillarity. A capillary hypersurface mentioned at the beginning of the Introduction is called a capillary CMC hypersurface in the paper.

Note that we also have removed the restriction that $\theta_{0}\leq\frac{\pi}{2}$, comparing with the previous work [JXZ22]. As a direct application, we get the Alexandrov-type theorem for embedded capillary hypersurfaces in $\overline{\mathbb{R}^{n+1}_{+}}$ with constant $r$-th mean curvature, for any $r\in\{1,\cdots,n\}$.

Our proof of Theorem 1.1 is inspired by the original idea of Heintze-Karcher [HK78] (see also Montiel-Ros [MR91]) which uses parallel hypersurfaces to estimate the enclosed volume. However the ordinary parallel hypersurfaces do not work for capillary hypersurfaces. The one of key ingredients of this paper is a correct form of parallel hypersurfaces $\zeta(\Sigma,t)$ defined in (33). To prove Theorem 1.1, we need to show the surjectivity of $\zeta$ onto the enclosed domain $\Omega$, for which we discover an appropriate foliation by round spheres with simultaneously varied center and radius.

It is interesting to see that our proof of Theorem 1.1 provides a refinement of the ordinary Heintze-Karcher inequality for closed hypersurfaces, since any closed hypersurface can be viewed as a capillary hypersurface with an empty boundary. Hence we have

In the second part of this paper, we study hypersurfaces with capillary boundary in a wedge domain. Here we simply call it a wedge. An ordinary wedge, we call it a classical wedge in this paper, is the unbounded closed region determined by two intersecting hyperplanes with dihedral angle $\alpha$, which is also called an opening angle, lying in $(0,\pi)$. There have been many works on the study of the stability of CMC capillary hypersurfaces (c.f. [CK16, LX17, Souam21, XZ21]) and on embedded CMC capillary hypersurfaces in wedges (c.f. [McCuan97, Park05, Lopez14]). Comparing with the half-space case, a big difference is that the Alexandrov’s reflection method might fail in the case of wedges, though the authors in [McCuan97, Park05, Lopez14] managed to modify Alexandrov’s reflection to obtain their classification results in certain cases. It is interesting that our method to establish the Heintze-Karcher-type inequality works in the wedge case, and even works in a more general setting. See also the recent development of this method in the anisotropic setting [JWXZ23, JWXZ23b].

In fact we shall consider generalized wedges which are determined by finite many mutually intersecting hyperplanes. To be more precise, let $\mathbf{W}$ be the unbounded closed region in $\mathbb{R}^{n+1}(n\geq 2)$, which are determined by finite many mutually intersecting hyperplanes $P_{1},\ldots,P_{L}$, for some integer $1\leq L\leq n+1$, such that the dihedral angle between $P_{i}$ and $P_{j}$, $i\neq j$, lies in $(0,\pi)$. We call such $\mathbf{W}$ a generalized wedge. $P_{i}\cap P_{j},i\neq j$ is called an edge of the wedge $\mathbf{W}$. If $L=2$, we call $\mathbf{W}$ a classical wedge. Let $\bar{N}_{i}$ be the outwards pointing unit normal to $P_{i}$ in $\mathbf{W}$ for $i=1,\cdots,L$. Thus $\{\bar{N}_{1},\ldots,\bar{N}_{L}\}$ are linearly independent. Up to a translation, we may assume that the origin $O\in\bigcap\limits_{i=1}^{L}P_{i}$. Given $\vec{\theta}_{0}=(\theta^{1}_{0},\cdots,\theta^{L}_{0})\in\prod\limits_{i=1}^{L}(0,\pi)$. Now we define an important vector $\mathbf{k}_{0}$ associated with $\mathbf{W}$ and $\vec{\theta}_{0}$ by

where $c_{i}$ is such that $\langle\mathbf{k}_{0},\bar{N}_{i}\rangle=\cos\theta_{0}^{i}$. We say that $\Sigma$ is a $\vec{\theta}$-capillary hypersurface in $\mathbf{W}$ with $\vec{\theta}=(\theta^{1},\theta^{2},\cdots,\theta^{L})$ if it intersects $\partial\mathbf{W}$ at contact angle $\theta^{i}(x)$ for $x\in\partial\Sigma\cap P_{i}$. It is easy to see that $\mathbf{k}_{0}$ is the center of the $\vec{\theta}_{0}$-capillary spherical cap with radius $1$. The following key assumption (8) has a clear geometric meaning that the unit sphere centered at $\mathbf{k}_{0}$ intersects the edge of the wedge $\mathbf{W}$.

We shall prove the following Heintze-Karcher-type inequality in a wedge.

The idea of proof of Theorem 1.4 is similar to that of Theorem 1.1, by using a suitable family of parallel hypersurfaces $\zeta(\Sigma,t)$ which relates $\mathbf{k}_{0}$. To show the surjectivity of $\zeta$, assumption (9) plays a crucial role. Actually, this condition was required in previous related papers, except [Lopez14]. See Remark 1.10. In this paper we are able to remove the additional assumption (9) in a classical wedge, i.e., $L=2$. Precisely, we have the following

The proof of Theorem 1.5 relies on a delicate analysis on the edge, which is the most technical part of this paper. A special case, $L=2$ and $\vec{\theta}_{0}=(\frac{\pi}{2},\frac{\pi}{2})$, i.e., $\Sigma$ is a free boundary hypersurface, for which $\mathbf{k}_{0}=0,$ (10) was proved by Lopez in [Lopez14] via Reilly’s formula. It is a natural question to ask if Theorem 1.4 holds true for $L>2$ without (9). Theorem 1.5 leads us to believe that assumption (9) is unnecessary.

Now we make some remarks on condition (8).

As applications of Theorem 1.4 and Theorem 1.5, we prove an Alexandrov-type theorem and a non-existence result for embedded CMC capillary hypersurfaces in a wedge.

As a consequence of Theorem 1.7, Theorem 1.8 and Remark 1.6 (iii), we have the following

Several remarks and questions are in order.

We end the Introduction with a few supplement on the study of capillary hypersurfaces in a wedge domain. A nonparametric capillary surface is a graph of a function $f$ over a domain, say $\Omega$, which satisfies the constant mean curvature equation with a corresponding capillary boundary condition. This is an equilibrium free surface of a fluid in a cylindrical container. When the domain $\Omega$ has a corner, then this nonparametric surface can be viewed as a capillary surface in a wedge (or in a wedge domain). There have been a lot of research on such a problem, especially after Concus-Finn [CF69], where it was already observed that the opening angle of the wedge and both contact angles should satisfy certain conditions for the existence. See also [CF74]. Later in [CF96] Concus-Finn proved that $f$ is continuous at a given corner if (11) holds, while if (11) does not hold there is no solution in one case and in the left case, namely $\alpha>\pi-|\theta^{1}_{0}-\theta^{2}_{0}|$, they conjectured that $f$ has a jump discontinuity at the corner. See the Concus-Finn rectangle in [Lan10], Figure 2. This conjecture was solved by Lancaster in [Lan10] with the methods developed by Allard [A72] and especially by Simon [S80]. The latter was crucially used in a very recent work of Chao Li [Li21] mentioned at the beginning of the Introduction. See also his further work [EL22] with Edelen on surfaces in a polyhedral domain, which is also closely related to surfaces in a wedge domain.

The rest of the paper is organized as follows. In Section 2, we collect some basic facts about wedges and capillary hypersurfaces in wedges. In Section 3 and Section 4, we prove the main theorems on the Heintze-Karcher inequality in the half-space and a wedge. In Section 5, we prove the Alexandrov-type theorem and the non-existence result for CMC capillary hypersurfaces in a wedge.

Let $\Omega$ be the bounded domain in $\mathbf{W}$ with piecewise smooth boundary $\partial\Omega=\Sigma\cup(\bigcup_{i=1}^{L}T_{i})$, where $\Sigma=\overline{\partial\Omega\cap\mathring{\mathbf{W}}}$ is a smooth compact embedded $\vec{\theta}$-capillary hypersurface in $\mathbf{W}$ and $T_{i}=\partial\Omega\cap P_{i}$. Denote the corners by $\Gamma_{i}=\Sigma\cap T_{i}$, which are smooth, co-dimension two submanifolds in $\mathbb{R}^{n+1}$. For the sake of simplicity, we denote by $\Gamma$ the union of $\Gamma_{i}$, i.e., $\Gamma=\bigcup_{i=1}^{L}\Gamma_{i}$. We use the following notation for normal vector fields. Let $\nu$ and $\bar{N}_{i}$ be the outward unit normal to $\Sigma$ and $P_{i}$ (with respect to $\Omega$) respectively. Let $\mu_{i}$ be the outward unit co-normal to $\Gamma_{i}=\partial\Sigma\cap P_{i}\subset\Sigma$ and $\bar{\nu}_{i}$ be the outward unit co-normal to $\Gamma_{i}\subset T_{i}$. Under this convention, along each $\Gamma_{i}$ $\{\nu,\mu_{i}\}$ and $\{\bar{\nu}_{i},\bar{N}_{i}\}$ span the same 2-dimensional plane and have the same orientation in the normal bundle of $\partial\Sigma\subset\mathbb{R}^{n+1}$. Hence one can define the contact angle function along each $\Gamma_{i}$, $\theta_{i}:\Gamma_{i}\to(0,\pi)$ by

Let $\vec{\theta}(x)$ denote the $L$-tuple $(\theta^{1}(x),\ldots,\theta^{L}(x))$. We call $\Sigma$ a $\vec{\theta}$-capillary hypersurface, if we want to emphasize the contact angle function. We also use $\vec{\theta}_{0}$-capillary hypersurface to denote such a hypersurface with $\vec{\theta}\equiv\vec{\theta}_{0}$, a vector $\vec{\theta}_{0}\in\prod_{i=1}^{L}(0,\pi).$

We denote by $\bar{\nabla}$, $\bar{\Delta}$, $\bar{\nabla}^{2}$ and $\bar{\rm div}$, the gradient, the Laplacian, the Hessian and the divergence on $\mathbb{R}^{n+1}$ respectively, while by $\nabla$, $\Delta$, $\nabla^{2}$ and ${\rm div}$, the gradient, the Laplacian, the Hessian and the divergence on the smooth part of $\partial\Omega$, respectively. Let $g$, $h$ and $H$ be the first, second fundamental forms and the mean curvature of the smooth part of $\partial\Omega$ respectively. Precisely, $h(X,Y)=\langle\bar{\nabla}_{X}\nu,Y\rangle$ and $H={\rm tr}_{g}(h)$. In particular, since $P_{i}$ is planar, the second fundamental form $h_{i}\equiv 0$, correspondingly, the mean curvature $H_{i}$ vanishes.

We need the following structural lemma for compact hypersurfaces in $\mathbb{R}^{n+1}$ with boundary, which is well-known and widely used, see [AS16, JXZ22].

The following lemma is well-known when the capillary hypersurfaces are bounded by containers with totally umbilical boundaries, in particular, a wedge in $\mathbb{R}^{n+1}$, see e.g., [AS16, LX17, WX19].

The $r$-th mean curvature $H_{r}$ of $\Sigma$ is defined by the identity:

for all real number $t$. Thus $H_{1}=\frac{H}{n}$ is the mean curvature of $\Sigma$ and $H_{n}$ is the Gaussian curvature, and we adopt the convention that $H_{0}=1$.

We have the following Minkowski-type formula for $\vec{\theta}_{0}$-capillary hypersurfaces.

We have also

which is easy to prove. If one views (32) as one of (18) with $r=0$, the Heintze-Karcher inequality (6) that we want to prove could also be viewed one of them with $r=-1$. Certainly now it is an inequality, instead of an equality.

In the sequel, $\Sigma$ will be always referred to as a smooth, embedded capillary hypersurface.

Let us close this section with a remark. In the proof of $\Omega\subset\zeta(Z)$, our choice of the touching balls is enlightened by the following observation.

In this section, we prove Theorem 1.4 and Theorem 1.5 in a wedge $\mathbf{W}$, following largely from the proof of the Heintze-Karcher-type inequality presented in the previous section.