Alexandrov–Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary

Abstract

In this paper, we first introduce quermassintegrals for capillary hypersurfaces in the half-space. Then we solve the related isoperimetric type problems for the convex capillary hypersurfaces and obtain the corresponding Alexandrov–Fenchel inequalities. In order to prove these results, we construct a new locally constrained curvature flow and prove that the flow converges globally to a spherical cap.

1 Introduction

Let $\Sigma$ be a closed embedded hypersurface in $\mathbb {R}^{n+1}$ and ${\widehat{\Sigma }}$ the domain enclosed by $\Sigma$ in $\mathbb {R}^{n+1}$. The classical isoperimetric inequality states

$$\begin{aligned} \frac{|\Sigma |}{\omega _n}\ge \left( \frac{|\widehat{\Sigma }|}{{\textbf{b}}_{n+1}} \right) ^{\frac{n}{n+1}}, \end{aligned}$$

(1.1)

with equality holding if and only if $\Sigma$ is a sphere. Here ${\textbf{b}}_{n+1}=|{{\mathbb {B}}}^{n+1}|$, the volume of the unit ball ${{\mathbb {B}}}^{n+1}$, and $\omega _n=(n+1) {\textbf{b}}_{n+1}=|\mathbb {S}^n|$, the area of the unit sphere $\mathbb {S}^n$. Its natural generalization is the following classical Alexandrov–Fenchel inequality

$$\begin{aligned} \frac{ {\mathcal {V}}_{l}({\widehat{\Sigma }}) }{{\textbf{b}}_{n+1}}\ge \left( \frac{ {\mathcal {V}}_{k}({\widehat{\Sigma }}) }{{\textbf{b}}_{n+1}}\right) ^{\frac{n+1-l}{n+1-k}},\quad 0\le k<l \le n, \end{aligned}$$

(1.2)

with equality holding if and only if $\Sigma$ is a sphere, provided that $\Sigma$ is a $C^2$ convex hypersurface. Here ${\mathcal {V}}_{k+1}({\widehat{\Sigma }})$ is the quermassintegral of $\widehat{\Sigma }$ defined by

$$\begin{aligned} {\mathcal {V}}_{k+1}({\widehat{\Sigma }}){:}{=}\frac{1}{n+1} \int \limits _{\Sigma } H_{k }\, dA,\quad 0\le k\le n, \quad {\mathcal {V}}_0({\widehat{\Sigma }}) = {| {\widehat{\Sigma }}|}, \end{aligned}$$

(1.3)

where $H_k$ ($1\le k\le n$) is the k-th normalized mean curvature of $\Sigma \subset \mathbb {R}^{n+1}$ and $H_0=1$. It was proved in [29] that (1.2) holds true if $\Sigma$ is l-convex and star-shaped. Here by l-convex we mean that $H_j>0$ for all $j\le l$. The case $k=0$ was proved to be true for $(l+1)$-convex hypersurfaces in [14, 45]. The case $l=2, k=1$, in which (1.2) is called Minkowski’s inequality, was proved to be also true for outward minimizing sets in [35] (see also [24] and a very recent work [1] by using a nonlinear potential theory.) It remains still open whether (1.2) is true for all l-convex hypersurfaces, except the case $l=2, k=0$ that has been proved in [1].

In this paper we are interested in its generalization to hypersurfaces with boundary. More precisely, we consider hypersurfaces in ${\overline{\mathbb {R}}}^{n+1}_+$ with boundary supported on the hyperplane $\partial {\overline{\mathbb {R}}}^{n+1}_+$. Let $\Sigma$ be a compact manifold with boundary $\partial \Sigma$, which is properly embedded hypersurface into ${\overline{\mathbb {R}}}^{n+1}_+$. In particular, $\text {int} (\Sigma )\subset \mathbb {R}^{n+1}_+$ and $\partial \Sigma \subset \partial {\overline{\mathbb {R}}}^{n+1}_+$. Let $\widehat{\Sigma }$ be the bounded domain enclosed by $\Sigma$ and the hyperplane $\partial \overline{\mathbb {R}}^{n+1}_+$. It is clear that the following relative isoperimetric inequality follows from (1.1)

$$\begin{aligned} \frac{|\Sigma |}{|{{\mathbb {S}}}_+^{n}|}\ge \left( \frac{|{\widehat{\Sigma }}|}{|{{\mathbb {B}}}^{n+1}_+|} \right) ^{\frac{n}{n+1}}, \end{aligned}$$

(1.4)

where ${{\mathbb {B}}}^{n+1}_+$ is the upper half unit ball and ${{\mathbb {S}}}_+^{n}$ is the upper half unit sphere. For the relative isoperimetric inequality outside a convex domain, see [17, 25, 40]. As a domain in $\mathbb {R}^{n+1}$, $\widehat{\Sigma }$ has a boundary, which consists of two parts: one is $\Sigma$ and the other, which will be denoted by $\widehat{\partial \Sigma }$, lies on $\partial {\overline{\mathbb {R}}}^{n+1}_+$. Both have a common boundary, namely $\partial \Sigma$ (see Fig. 1). Instead of just considering the area of $\Sigma$, it is interesting to consider the following free energy functional

$$\begin{aligned} |\Sigma |-\cos \theta |\widehat{\partial \Sigma }| \end{aligned}$$

(1.5)

for a fixed angle constant $\theta \in (0,\pi )$. The second term $\cos \theta |\widehat{\partial \Sigma }|$ is the so-called wetting energy in the theory of capillarity (see for example [23]). If we consider to minimize this functional under the constraint that the volume $|{\widehat{\Sigma }}|$ is fixed, then we have the following optimal inequality, which is called the capillary isoperimetric inequality,

$$\begin{aligned} \frac{|\Sigma |-\cos \theta |\widehat{\partial \Sigma }|}{|{{\mathbb {S}}}_{\theta }^{n}|-\cos \theta | { \widehat{\partial \mathbb {S}^n_\theta }}|}\ge \left( \frac{|{\widehat{\Sigma }}|}{|{\mathbb B}^{n+1}_\theta |} \right) ^{\frac{n}{n+1}}, \end{aligned}$$

(1.6)

with equality holding if and only if $\Sigma$ is homothetic to $\mathbb {S}^n_\theta$, namely a spherical cap (2.3) with contact angle $\theta$. Here $\mathbb {S}_\theta ^{n}$, $\widehat{\partial \mathbb {S}_\theta ^{n}}$, ${{\mathbb {B}}}^{n+1}_\theta$ are defined by

$$\begin{aligned} \mathbb {S}^n_\theta =\{ x\in \mathbb {S}^n \,|\, \langle x, e_{n+1} \rangle>\cos \theta \}, \quad {{\mathbb {B}}}^{n+1}_\theta =\{ x\in {\mathbb B}^{n+1} \,|\, \langle x, e_{n+1} \rangle >\cos \theta \}, \\ { \widehat{\partial \mathbb {S}^n_\theta }= \{ x\in {{\mathbb {B}}}^{n+1} \,|\, \langle x, e_{n+1} \rangle =\cos \theta \},} \end{aligned}$$

where $e_{n+1}$ the $(n+1)$th standard basis in $\mathbb {R}^{n+1}_+$. For simplicity we denote

$$\begin{aligned} {{\textbf {b}}}_\theta : = {{\textbf {b}}}^\theta _{n+1}{:}{=}|{{\mathbb {B}}}^{n+1}_\theta |, \qquad \quad \omega _\theta {:}{=} \omega _{n, \theta }{:}{=} |{{\mathbb {S}}}_{\theta }^{n}|-\cos \theta | {\widehat{ \partial \mathbb {S}^n_\theta }}|. \end{aligned}$$

(1.7)

The explicit formulas for ${{\textbf {b}}}_\theta$ and $\omega _\theta$ will be given in Sect. 2.3. In particular, it is easy to check that $(n+1) {{\textbf {b}}}_\theta =\omega _\theta .$ The proof of (1.6) is not trivial, which uses the spherical symmetrization. See for example [41, Chap. 19]. For related physical problems, one can refer to the classical book of Finn [23].

The main objectives of this paper are considering the following problems

• (1) To find suitable generalizations of the quermassintergals ${\mathcal {V}}_k$ for hypersurfaces with boundary supported on $\partial \overline{\mathbb {R}}^{n+1}_+$, which are closely related to the free energy (1.5).

• (2) To establish the Alexandrov–Fenchel inequality for these new quermassintegrals.

To answer the first question, we introduce the following new geometric functionals.

$$\begin{aligned} {\mathcal {V}}_{0,\theta }({\widehat{\Sigma }}){:}{=} |{\widehat{\Sigma }}|, \qquad {\mathcal {V}}_{1,\theta }({\widehat{\Sigma }}){:}{=} \frac{1}{n+1} ( |\Sigma |-\cos \theta |\widehat{\partial \Sigma }|), \end{aligned}$$

and for $1\le k\le n$,

$$\begin{aligned} {\mathcal {V}}_{k+1,\theta }({\widehat{\Sigma }}):= & {} \frac{1}{n+1}\left( \int \limits _\Sigma H_kdA - \frac{\cos \theta \sin ^k\theta }{n}\int \limits _{\partial \Sigma } H_{k-1}^{\partial \Sigma }ds \right) , \end{aligned}$$

(1.8)

where $H_{k-1}^{\partial \Sigma }$ is the normalized $(k-1)$th mean curvature of $\partial \Sigma \subset \mathbb {R}^n$ (see Sect. 2 for details). In particular, one has

$$\begin{aligned} {\mathcal {V}}_{n+1,\theta }({\widehat{\Sigma }}) =\frac{1}{n+1} \int \limits _\Sigma H_ndA-\cos \theta \sin ^n\theta \frac{\omega _{n-1}}{n(n+1)}. \end{aligned}$$

We believe that these quantities are the suitable quermassintegrals for hypersurfaces with boundary intersecting with $\partial \overline{ \mathbb {R}}^{n+1}_+$ at angle $\theta \in (0,\pi )$, which is supported by the following result.

Theorem 1.1

Let $\Sigma _t\subset \overline{{\mathbb {R}}}_+^{n+1}$ be a family of smooth, embedded capillary hypersurfaces with a constant contact angle $\theta \in (0,\pi )$, which are given by the embedding $x(\cdot ,t):M\rightarrow \overline{\mathbb {R}}^{n+1}_+$ and satisfy

$$\begin{aligned} (\partial _t x)^\perp =f\nu , \end{aligned}$$

for some speed function f. Then for $0\le k\le n$,

$$\begin{aligned} \frac{d}{dt} {\mathcal {V}}_{k,\theta }(\widehat{\Sigma _t})=\frac{n+1-k}{n+1}\int \limits _{\Sigma _t} fH_{k}dA_t, \end{aligned}$$

(1.9)

and in particular

$$\begin{aligned} \frac{d}{dt}{\mathcal {V}}_{n+1,\theta }(\widehat{\Sigma _t})=0. \end{aligned}$$

(1.10)

A hypersurface in $\overline{{\mathbb {R}}}_+^{n+1}$ with boundary supported on $\partial \overline{{\mathbb {R}}}_+^{n+1}$ is called capillary hypersurface if it intersects with $\partial \overline{{\mathbb {R}}}_+^{n+1}$ at a constant angle. For closed hypersurfaces in $\mathbb {R}^{n+1}$, a similar variational formula as (1.9) characterizes the quermassintegrals in (1.3). This formula for the quermassintegrals is also true for closed hypersurfaces in other space forms. See for example [55].

Our second result is the generalized Alexandrov–Fenchel inequalities for convex capillary hypersurfaces.

Theorem 1.2

For $n\ge 2$, let $\Sigma \subset \overline{\mathbb {R}}^{n+1}_+$ be a convex capillary hypersurface with a constant contact angle $\theta \in (0, \frac{{\pi }}{2} ]$, then there holds

$$\begin{aligned} \frac{ {\mathcal {V}}_{n,\theta }({\widehat{\Sigma }}) }{{{\textbf {b}}}_\theta } \ge \left( \frac{ {\mathcal {V}}_{k,\theta }({\widehat{\Sigma }}) }{{{\textbf {b}}}_\theta } \right) ^{\frac{1}{n+1-k}}, \quad \forall \, 0\le k <n, \end{aligned}$$

(1.11)

with equality if and only if $\Sigma$ is a spherical cap in (2.3). Moreover,

$$\begin{aligned} {\mathcal {V}}_{n+1,\theta }({\widehat{\Sigma }})=\omega _\theta =(n+1){{\textbf {b}}}_\theta , \end{aligned}$$

(1.12)

(1.12) follows easily from (1.10), by constructing a smooth family of capillary hypersurfaces connecting to a spherical cap given in (2.3). Therefore (1.12) is true for any capillary hypersurfaces with contact angle $\theta$. Moreover, it is equivalent to

$$\begin{aligned} \int \limits _\Sigma H_ndA=(n+1)\omega _\theta +\cos \theta \sin ^n\theta \frac{\omega _{n-1}}{n} = {|\mathbb {S}^n_\theta |}, \end{aligned}$$

(1.13)

a Gauss–Bonnet type result for capillary hypersurfaces with contact angle $\theta$. When $n=2$, (1.13) implies a Willmore inequality for capillary hypersurfaces with contact angle $\theta$.

Corollary 1.3

Let $\Sigma \subset \overline{\mathbb {R}}^3_+$ be a convex capillary surface with a constant contact angle $\theta \in (0,\frac{\pi }{2}]$, then

$$\begin{aligned} \int \limits _\Sigma H^2 dA \ge 4 |\mathbb {S}^2_\theta |, \end{aligned}$$

(1.14)

with equality holding if and only if $\Sigma$ is spherical cap in (2.3).

Here $H=2H_1$ is the ordinary mean curvature for surfaces and it is obvious that $H^2 \ge 4H_2$. When $n=2$, (1.11) implies a Minkowski type inequality for convex capillary surfaces with boundary in $\overline{\mathbb {R}}^3_+$.

Corollary 1.4

Let $\Sigma \subset \overline{\mathbb {R}}^3_+$ be a convex capillary surface with a constant contact angle $\theta \in (0,\frac{\pi }{2}]$, then

$$\begin{aligned} \int \limits _\Sigma H dA \ge 2 \sqrt{\omega _{2,\theta }} \cdot (|\Sigma |-\cos \theta |\widehat{\partial \Sigma }|)^{\frac{1}{2}}+ \sin \theta \cos \theta |\partial \Sigma |, \end{aligned}$$

(1.15)

where

$$\begin{aligned} \omega _{2,\theta }= 3 {\textbf{b}}_\theta = \left( 2-3\cos \theta +\cos ^3\theta \right) \pi . \end{aligned}$$

Moreover, equality holds if and only if $\Sigma$ is a spherical cap in (2.3).

From these results, it is natural to propose

Conjecture 1.5

For $n\ge 2$, let $\Sigma \subset \overline{\mathbb {R}}^{n+1}_+$ be a convex capillary hypersurface with a contact angle $\theta \in (0, {{\pi }})$, there holds

$$\begin{aligned} \frac{ {\mathcal {V}}_{l,\theta }({\widehat{\Sigma }}) }{{{\textbf {b}}}_\theta } \ge \left( \frac{ {\mathcal {V}}_{k,\theta }({\widehat{\Sigma }}) }{{{\textbf {b}}}_\theta } \right) ^{\frac{n+1-l}{n+1-k}}, \quad \forall 0\le k <l\le n, \end{aligned}$$

(1.16)

with equality iff $\Sigma$ is a spherical cap in (2.3).

It would be also interesting to ask further if the conjecture is true for k-convex capillary hypersurfaces. In order to prove the conjecture, we introduce a suitable nonlinear curvature flow, which preserves ${\mathcal {V}}_{l,\theta }({\widehat{\Sigma }})$ and increases ${\mathcal {V}}_{k,\theta }({\widehat{\Sigma }})$, see Sect. 3. If the flow globally converges to a spherical cap, then we have the general Alexandrov–Fenchel inequaltiy (1.16). However, due to technical difficulties we are only able to prove in this paper the global convergence, for $l=n$ and $\theta \in (0, \frac{\pi }{2}]$, namely, Theorem 1.2.

To be more precise, let us first recall the related work on the proof of Alenxandrov–Fenchel inequalities by using geometric curvature flow for closed hypersurfaces in $\mathbb {R}^{n+1}$. When $\partial \Sigma =\emptyset$, and we denote ${\widehat{\Sigma }}$ be the bounded convex domain enclosed by $\Sigma$ in $\mathbb {R}^{n+1}$. In convex geometry, the Alexandrov–Fenchel inequalities (1.2) between quermassintegrals ${\mathcal {V}}_k$ and ${\mathcal {V}}_l$ play an important role. In fact there are more general inequalities. See [3, 4, 50] for instance. It is an interesting question, if one can use a curvature flow to reprove such inequalities. In [43], McCoy introduced a normalized nonlinear curvature flow to reprove the Alexandrov–Fenchel inequalities (1.2) for convex domains in Euclidean space. Later, Guan–Li [29] weakened the convexity condition and only assumed that the closed hypersurface $\Sigma$ is k-convex and star-shaped by using the inverse curvature flow, which is defined by

$$\begin{aligned} \partial _t x=\frac{H_{k-1}}{H_k}\nu . \end{aligned}$$

(1.17)

This flow was previously studied by Gerhardt [28] and Urbas [53]. One of key observations in the study of this flow is that the k-convexity and star-shaped are preserved along this flow. This flow is also equivalent to the rescaled one

$$\begin{aligned} \partial _t x=\left( \frac{H_{k-1}}{H_k}-\langle x,\nu \rangle \right) \nu , \end{aligned}$$

(1.18)

see [31, 32] for instance. The motivation to use such a flow (1.18) is its nice properties that the quermassintegrals ${\mathcal {V}}_k({\widehat{\Sigma }})$ is preserved and ${\mathcal {V}}_{k+1}({\widehat{\Sigma }})$ is non-decreasing along this flow, which follows from the well-known Minkowski formulas. Using similar geometric flows, there have been a lot of work to establish new Alexandrov–Fenchel inequalities in the hyperbolic space [6, 7, 10, 19, 26, 33, 34, 38, 49, 55] and in the sphere [15, 16, 42, 57]. For the anisotropic analogue of Alexandrov–Fenchel (Minkowski) type inequalities we refer to [8, 58, 60].

If $\partial \Sigma \ne \emptyset$, the study of geometric inequalities with free boundary or general capillary boundary has attracted much attention in the last decades. For related relative isoperimetric inequalities and the Alexandrov–Fenchel inequalities, see for instance [9, 11, 13, 22, 37, 48, 59] etc. Recently in [48] Scheuer–Wang–Xia introduced the definition of quermassintegrals for hypersurfaces with free boundary in the Euclidean unit ball $\overline{\mathbb {B}}^{n+1}$ from the viewpoint of the first variational formula, and they proved the highest order Alexandrov–Fenchel inequalities for convex hypersurfaces with free boundary in $\overline{\mathbb {B}}^{n+1}$. Very recently, the second and the third authors [59] generalized the work in [48] by introducing the corresponding quermassintegrals for general capillary hypersurfaces and established Alexandrov–Fenchel inequalities for convex capillary hypersurfaces in $\overline{\mathbb {B}}^{n+1}$. The flows introduced to establish these inequalities in [48, 59] are motivated by new Minkowski formulas proved in [56].

Now we introduce our curvature flow for capillary hypersurfaces in the half space. Let $e=-e_{n+1}$, where $e_{n+1}$ the $(n+1)$th coordinate in $\mathbb {R}^{n+1}_+$. Let $x:\Sigma \rightarrow \overline{\mathbb {R}}^{n+1}_+$ with boundary $x_{|\partial \Sigma }:\partial \Sigma \rightarrow \partial \mathbb {R}_+^{n+1}$ and $\nu$ its unit normal vector field. We introduce

$$\begin{aligned} \left( \partial _t x \right) ^\perp = \left[ \left( 1+\cos \theta \langle \nu , e \rangle \right) \frac{H_{l-1}}{H_l} -\langle x, \nu \rangle \right] \nu . \end{aligned}$$

(1.19)

Using the Minkowski formulas given in (2.9), we show that flow (1.19) preserves ${\mathcal {V}}_{l,\theta }({\widehat{\Sigma }})$, while increases ${\mathcal {V}}_{k,\theta }({\widehat{\Sigma }})$ for $k<l$. However, due to the weighted function in the flow we are only able at moment to show that flow (1.19) preserves the convexity, when $l=n$. In this case, we can further bound $\frac{H_n}{H_{n-1}}$. In order to bound all principal curvature we need to estimate the mean curvature, which satisfies a nice evolution equation (4.7). However the normal derivative of H, $\nabla _{\mu }H$, has a bad sign, if $\theta > \frac{\pi }{2}$. Hence we have to restrict ourself on the range $\theta \in (0,\frac{\pi }{2}]$. Under these conditions we then succeed to show the global convergence, and hence Alexandrov–Fenchel inequalities. It would be interesting to ask if one can also prove the global convergence for the case $\theta > \frac{\pi }{2}$. In analysis, this case is related to the worse case in the Robin boundary problem for the corresponding PDE. We remark also that a convex capillary hypersurface with contact angle $\theta \le \frac{\pi }{2}$ could have different geometry from that with $\theta > \frac{\pi }{2}$. The former was called a convex cap and was studied in [12].

Comparing with inequalities established in [48, 59], which are actually implicit inequalities and involve inverse functions of certain geometric quantities that can not be explicitly expressed by elementary functions, we have here a geometric inequality (1.11) in an explicit and clean form. An optimal inequality with an explicit form has more applications. A further good example was given very recently in an optimal insulation problem in [18], where the optimal inequalities between ${\mathcal {V}}_n$ and ${\mathcal {V}}_k$ for any $k<n$ for closed hypersurfaces have been used crucially. We expect that our results can be similarly used in an optimal insulation problem for capillary hypersurfaces.

The rest of the article is structured as follows. In Sect. 2, we introduce the quermassintegrals for capillary hypersurfaces and collect the relevant evolution equations to finish the proof of Theorem 1.1. In Sect. 3, we introduce our nonlinear inverse curvature flow and show the monotonicty of our quermassintegrals (1.8) under the flow. In Sect. 4, we obtain uniform estimates for convex capillary hypersurfaces along the flow and the global convergence. Section 5 is devoted to prove the Alexandrov–Fenchel inequalities for convex capillary hypersurfaces in the half-space, i.e. Theorem 1.2.

2 Quermassintegrals and Minkowski formulas

Since we will deform hypersurfaces by studying a geometric flow, it is convenient to use immersions. Let M denote a compact orientable smooth manifold of dimension n with boundary $\partial M$, and $x:M\rightarrow \overline{\mathbb {R}}^{n+1}_+$ be a proper smooth immersed hypersurface. In particular, $x(\text {int}(M))\subset \mathbb {R}^{n+1}_+$ and $x(\partial M)\subset \partial \mathbb {R}^{n+1}_+$. Let $\Sigma =x(M)$ and $\partial \Sigma =x(\partial M)$. If no confusion, we will do not distinguish the hypersurfaces $\Sigma$ and the immersion $x:M \rightarrow \overline{\mathbb {R}}^{n+1}_+$. Let $\widehat{\Sigma }$ be the bounded domain enclosed by $\Sigma$ and $\partial {\mathbb {R}}^{n+1}_+$. Let $\nu$ and $\overline{N}$ be the unit outward normal of $\Sigma \subset {{\widehat{\Sigma }}}$ and $\partial {\mathbb {R}}^{n+1}_+ \subset \overline{\mathbb {R}}^{n+1}_+$ respectively.

2.1 Higher order mean curvatures

For $\kappa =(\kappa _1,\kappa _2\cdots , \kappa _n) \in \mathbb {R}^n$, let $\sigma _k(\kappa ), k=1,\cdots , n,$ be the kth elementary symmetric polynomial functions and $H_k(\kappa )$ be its normalization $H_k(\kappa )=\frac{1}{\left( {\begin{array}{c}n\\ k\end{array}}\right) }\sigma _k(\kappa )$. For $i=1,2, \cdots , n$, let $\kappa |i\in \mathbb {R}^{n-1}$ (or $\kappa | \kappa _i$) denote $(n-1)$ tuple deleting the ith component from $\kappa$.

We shall use the following basic properties about $\sigma _k$.

Proposition 2.1

• (1) $\sigma _k(\kappa )=\sigma _k(\kappa |i)+\kappa _i\sigma _{k-1}(\kappa |i), \quad \forall 1\le i \le n.$

• (2) $\sum \limits _{i = 1}^n {\sigma _{k}(\kappa |i)}=(n-k)\sigma _k(\kappa ).$

• (3) $\sum \limits _{i = 1}^n {\kappa _i \sigma _{k-1}(\kappa |i)}=k\sigma _k(\kappa )$.

• (4) $\sum \limits _{i = 1}^n {\kappa _i^2 \sigma _{k-1}(\kappa |i)}=\sigma _1(\kappa )\sigma _k(\kappa )-(k+1)\sigma _{k+1}(\kappa )$.

Let $\Gamma _+{:}{=}\{\kappa \in \mathbb {R}^n:\kappa _i>0, 1\le i\le n\}$ and $\Gamma _+^k=\{\kappa \in \mathbb {R}^n \,| \, H_j(\kappa )>0, \forall 1\le j\le k\}$. It is clear that $\Gamma _+=\Gamma _+^n$.

Proposition 2.2

For $1\le k<l\le n$, we have

$$\begin{aligned} H_kH_{l-1}\ge H_{k-1}H_{l }, \quad \quad \forall \quad \kappa \in \Gamma ^l_+, \end{aligned}$$

(2.1)

with equality holding if and only if $\kappa =\lambda (1, \cdots , 1)$ for any $\lambda >0$. Moreover, $F(\kappa )=\frac{\sigma _k}{\sigma _{k-1}}(\kappa )$ is concave in $\Gamma ^k_+$.

These are well-known properties. For a proof we refer to [39, Chap. XV, Sect. 4] and [51, Lemma 2.10, Theorem 2.11] respectively.

We use D to denote the Levi–Civita connection of $\overline{\mathbb {R}}^{n+1}_+$ with respect to the Euclidean metric $\delta$, and $\nabla$ the Levi–Civita connection on $\Sigma$ with respect to the induced metric g from the immersion x. The operator $\textrm{div}, \Delta$, and $\nabla ^2$ are the divergence, Laplacian, and Hessian operator on $\Sigma$ respectively. The second fundamental form h of x is defined by

$$\begin{aligned} D_{X}Y=\nabla _{X}Y- h(X,Y)\nu . \end{aligned}$$

Let $\kappa =(\kappa _1, \kappa _2,\cdots , \kappa _n)$ be the set of principal curvatures, i.e, the set of eigenvalues of h. Then we denote $\sigma _k=\sigma _k(\kappa )$ and $H_k=H_k(\kappa )$ resp. be the kth mean curvature and the normalized kth mean curvature of $\Sigma$. We also use the convention that

$$\begin{aligned} \sigma _0=H_0=1 \quad \sigma _{n+1}=H_{n+1}=0. \end{aligned}$$

Remark 2.3

We will simplify the notation by using the following shortcuts occasionally:

• (1) When dealing with complicated evolution equations of tensors, we will use a local frame to express tensors with the help of their components, i.e. for a tensor field $T\in {\mathcal {T}}^{k,l}(\Sigma )$, the expression $T^{i_{1}\dots i_{k}}_{j_{1}\dots j_{l}}$ denotes

  $$\begin{aligned} T^{i_{1}\dots i_{k}}_{j_{1}\dots j_{l}}=T(e_{j_{1}},\dots ,e_{j_{l}},\epsilon ^{i_{1}},\dots \epsilon ^{i_{k}}), \end{aligned}$$

  where $(e_{i})$ is a local frame and $(\epsilon ^{i})$ its dual coframe.

• (2) The mth covariant derivate of a (k, l)-tensor field T, $\nabla ^m T$, is locally expressed by

  $$\begin{aligned} T^{i_1\dots i_k}_{j_1\dots j_l; j_{l+1}\dots j_{l+m}}. \end{aligned}$$

• (3) We shall use the convention of the Einstein summation. For convenience the components of the Weingarten map ${\mathcal {W}}$ are denoted by $(h^{i}_{j})=(g^{ik}h_{kj})$, and $|h|^2$ be the norm square of the second fundamental form, that is $|h|^2=g^{ik}h_{kl}h_{ij}g^{jl}$, where $(g^{ij})$ is the inverse of $(g_{ij})$. We use the metric tensor $(g_{ij})$ and its inverse $(g^{ij})$ to lower down and raise up the indices of tensor fields on $\Sigma$.

2.2 Quermassintegrals in the half-space

In order to introduce our quermassintegrals for capillary hypersurfaces in the half-space, we review first the quermassintegrals in $\mathbb {R}^{n+1}$, see e.g. [50]. Given a bounded convex domain $\widehat{\Sigma }\subset \mathbb {R}^{n+1}$ with smooth boundary $\partial \widehat{\Sigma }$, its kth quermassintegral is defined by

$$\begin{aligned} {\mathcal {V}}_0(\widehat{\Sigma }){:}{=}|\widehat{\Sigma }|, \end{aligned}$$

and for $0\le k\le n$,

$$\begin{aligned} {\mathcal {V}}_{k+1}(\widehat{\Sigma }){:}{=}\frac{1}{n+1} \int \limits _{\partial \widehat{\Sigma }}H_{k} dA, \end{aligned}$$

where $H_{k}$ is the normalized kth mean curvature of $\partial \widehat{\Sigma }\subset \mathbb {R}^{n+1}$. One can check that

$$\begin{aligned} \frac{d}{dt} {\mathcal {V}}_{k+1}(\widehat{\Sigma }_t) =\frac{n-k}{n+1} \int \limits _{\partial {\widehat{\Sigma }}_t} H_{k+1} f dA, \end{aligned}$$

(2.2)

for a family of bounded convex bodies $\{\widehat{\Sigma }_t\}$ in $\mathbb {R}^{n+1}$ whose boundary $\partial \widehat{\Sigma }_t$ evolving by a normal variation with speed function f. For a proof see e.g. [29, Lemma 5]. As mentioned above, a similar first variational formula also holds in space forms, see [46]. Therefore formula (2.2) is the characterization of the quermassintegrals for closed hypersurfaces in space forms.

Now we define the following geometric functionals for convex hypersurfaces $\Sigma$ with capillary boundary in $\overline{\mathbb {R}}^{n+1}_+$ with a constant contact angle $\theta$ along $\partial \Sigma \subset \mathbb {R}^n$. Let

$$\begin{aligned}{} & {} {\mathcal {V}}_{0,\theta }({\widehat{\Sigma }}){:}{=} |{\widehat{\Sigma }}|, \\{} & {} {\mathcal {V}}_{1,\theta }({\widehat{\Sigma }}){:}{=} \frac{1}{n+1} \left( |\Sigma |-\cos \theta |\widehat{\partial \Sigma }|\right) . \end{aligned}$$

and for $1\le k\le n$,

$$\begin{aligned} {\mathcal {V}}_{k+1,\theta }({\widehat{\Sigma }}): = \frac{1}{n+1}\left( \int \limits _\Sigma H_kdA -\frac{\cos \theta \sin ^k\theta }{n}\int \limits _{\partial \Sigma } H_{k-1}^{\partial \Sigma }ds \right) . \end{aligned}$$

Here $H_{k-1}^{\partial \Sigma }{:}{=}\frac{1}{\left( {\begin{array}{c}n-1\\ k-1\end{array}}\right) }\sigma _{k-1}({\hat{\kappa }})$ is the normalized $(k-1)$th mean curvature of $\partial \Sigma \subset \mathbb {R}^n$ and $\sigma _{k-1}({\hat{\kappa }})$ is the $(k-1)$-elementary symmetric function on $\mathbb {R}^{n-1}$ evaluating at the principal curvatures ${\hat{\kappa }}$ of $\partial \Sigma \subset \mathbb {R}^n$. In particular, we have

$$\begin{aligned} {\mathcal {V}}_{2,\theta }({\widehat{\Sigma }})=\frac{1}{n(n+1)}\left( \int \limits _\Sigma H dA-\sin \theta \cos \theta | {\partial \Sigma }|\right) . \end{aligned}$$

Here H is the (un-normalized) mean curvature, i.e. $H=n H_1$. From Gauss–Bonnet–Chern’s theorem, we know

$$\begin{aligned} {\mathcal {V}}_{n}(\widehat{\Sigma })=\frac{\omega _{n-1}}{n}, \end{aligned}$$

if $\widehat{\Sigma }\subset \mathbb {R}^{n}$ is a convex body (non-empty, compact, convex set). As a result, we see

$$\begin{aligned} {\mathcal {V}}_{n+1,\theta }({\widehat{\Sigma }}) =\frac{1}{n+1} \int \limits _\Sigma H_ndA-\cos \theta \sin ^n\theta \frac{\omega _{n-1}}{n(n+1)}. \end{aligned}$$

2.3 Spherical caps

Let $e{:}{=}-e_{n+1}=(0,\ldots ,0,-1)$. We consider a family of spherical caps lying entirely in $\overline{\mathbb {R}}^{n+1}_+$ and intersecting $\mathbb {R}^{n}$ with a constant contact angle $\theta \in (0,\pi )$ given by

$$\begin{aligned} C_{r,\theta }(e) {:}{=}\Big \{x\in \overline{\mathbb {R}}^{n+1}_+ \big | |x-r\cos \theta e|= r\Big \},\quad r\in [0,\infty ), \end{aligned}$$

(2.3)

which has radius r and centered at $r\cos \theta e$. To emphasize e and to distinguish with the center of the spherical cap, $r\cos \theta e$, we call $C_{r,\theta }(e)$ a spherical cap around e. If without confusion, we just write $C_{r,\theta }$ for $C_{r,\theta }(e)$ in the rest of this paper. One can easily check that $C_{r,\theta }$ is the static solution to flow (3.1) below, that is,

$$\begin{aligned} 1+ \cos \theta \langle \nu ,e\rangle -\frac{1}{r}\langle x,\nu \rangle =0, \end{aligned}$$

(2.4)

and it intersects with the support $\partial \overline{\mathbb {R}}^{n+1}_+$ at the constant angle $\theta$.

The volume of $\widehat{C_{r,\theta }}$

$$\begin{aligned} {\mathcal {V}}_{0,\theta } (\widehat{C_{r,\theta }})= r^{n+1} {\textbf{b}}_\theta , \end{aligned}$$

where ${\textbf{b}}_\theta$ is the volume of $C_{1,\theta }$, which is congruent to $\mathbb {S}_\theta ^{n}$, defined in the introduction. One can compute

$$\begin{aligned} {\textbf{b}}_\theta {:}{=}\frac{\omega _n}{2} I_{\sin ^2\theta }\left( \frac{n}{2},\frac{1}{2}\right) -\frac{\omega _{n-1}}{n} \cos \theta \sin ^n\theta , \end{aligned}$$

(2.5)

and $I_s(\frac{n}{2},\frac{1}{2})$ is the regularized incomplete beta function given by

$$\begin{aligned} I_s\left( \frac{n}{2},\frac{1}{2}\right) {:}{=}\frac{\int _0^s t^{\frac{n}{2}-1} (1-t)^{-\frac{1}{2} }dt}{\int _0^1 t^{\frac{n}{2}-1} (1-t)^{-\frac{1}{2} }dt}. \end{aligned}$$

(2.6)

Moreover, one can readily check that

$$\begin{aligned} {\mathcal {V}}_{1,\theta } (\widehat{C_{r,\theta }})=\frac{1}{n+1} (|C_{r,\theta }|-\cos \theta |\widehat{\partial C_{r,\theta }}|)=r^n {\textbf{b}}_\theta \end{aligned}$$

and

$$\begin{aligned} {\mathcal {V}}_{k,\theta } (\widehat{C_{r,\theta }}) = r^{n+1-k} {\textbf{b}}_\theta . \end{aligned}$$

Therefore, $C_{r,\theta }$ achieves equality in the Alexandrov–Fenchel inequalities (1.11).

2.4 Minkowski formulas

As above, $\Sigma \subset \overline{\mathbb {R}}^{n+1}_+$ is a smooth, properly embedded capillary hypersurface, given by the embedding $x:M\rightarrow \overline{\mathbb {R}}^{n+1}_+$, where M is a compact, orientable smooth manifold of dimension n with non-empty boundary. Let $\mu$ be the unit outward co-normal of $\partial \Sigma$ in $\Sigma$ and $\overline{\nu }$ be the unit normal to $\partial \Sigma$ in $\partial \mathbb {R}^{n+1}_+$ such that $\{\nu ,\mu \}$ and $\{\overline{\nu },\overline{N}\}$ have the same orientation in normal bundle of $\partial \Sigma \subset \overline{\mathbb {R}}^{n+1}_+$. We define the contact angle $\theta$ between the hypersurface $\Sigma$ and the support $\partial \overline{\mathbb {R}}^{n+1}_+$ by

$$\begin{aligned} \langle \nu , {\overline{N}} \rangle =\cos (\pi -\theta ). \end{aligned}$$

It follows

$$\begin{aligned} \begin{array} {rcl} \overline{N} &{}=&{}\sin \theta \mu -\cos \theta \nu ,\\ \overline{\nu } &{}=&{}\cos \theta \mu +\sin \theta \nu , \end{array} \end{aligned}$$

(2.7)

or equivalently

$$\begin{aligned} \begin{array} {rcl} \mu &{}=&{}\sin \theta {\overline{N}} + \cos \theta {\overline{\nu }},\\ {\nu } &{}=&{}-\cos \theta {\overline{N}} +\sin \theta {\overline{\nu }}. \end{array} \end{aligned}$$

(2.8)

$\partial \Sigma$ can be viewed as a smooth closed hypersurface in $\mathbb {R}^n$, which bounds a bounded domain $\widehat{\partial \Sigma }$ inside $\mathbb {R}^n$. By our convention, ${\bar{\nu }}$ is the unit outward normal of $\partial \Sigma$ in $\widehat{\partial \Sigma }\subset \mathbb {R}^n$. See Fig.1.

Fig. 1

A capillary hypersurface $\Sigma$ with a contact angle $\theta$

The second fundamental form of $\partial \Sigma$ in $\mathbb {R}^n$ is given by

$$\begin{aligned} {\widehat{h}}(X, Y){:}{=} -\langle \nabla ^{\mathbb {R}^n}_X Y, {\bar{\nu }} \rangle = -\langle D_X Y, {\bar{\nu }} \rangle , \quad X, Y\in T(\partial \Sigma ). \end{aligned}$$

The second equality holds since $\langle {\bar{\nu }}, {\bar{N}}\circ x \rangle =0$. The second fundamental form of $\partial \Sigma$ in $\Sigma$ is given by

$$\begin{aligned} {\widetilde{h}}(X, Y){:}{=} -\langle \nabla _X Y, \mu \rangle = -\langle D_X Y, \mu \rangle , \quad X, Y\in T(\partial \Sigma ). \end{aligned}$$

The second equality holds since $\langle \nu , \mu \rangle =0$.

Proposition 2.4

Let $\Sigma \subset \overline{\mathbb {R}}^{n+1}_+$ be a capillary hypersurface. Let $\{e_\alpha \}_{\alpha =2}^{n }$ be an orthonormal frame of $\partial \Sigma$. Then along $\partial \Sigma$,

• (1) $\mu$ is a principal direction of $\Sigma$, that is, $h_{\mu \alpha }=h(\mu , e_\alpha )=0$.

• (2) $h_{\alpha \beta }=\sin \theta {\widehat{h}}_{\alpha \beta }.$

• (3) ${\widetilde{h}}_{\alpha \beta }=\cos \theta {\widehat{h}}_{\alpha \beta } = \cot \theta h_{\alpha \beta }.$

• (4) $h_{\alpha \beta ; \mu }={\widetilde{h}}_{\beta \gamma }(h_{\mu \mu }\delta _{\alpha \gamma }-h_{\alpha \gamma })$.

Proof

The first assertion is well-known, see e.g. [47]. (2) and (3) follow from

$$\begin{aligned} h_{\alpha \beta }=-\langle D_{e_\alpha }e_\beta ,\nu \rangle =\langle {\widehat{h}}_{\alpha \beta }\overline{\nu },\nu \rangle =\sin \theta {\widehat{h}}_{\alpha \beta }, \end{aligned}$$

and

$$\begin{aligned} {\widetilde{h}}_{\alpha \beta }= -\langle D_{e_\alpha }e_\beta , \mu \rangle = \langle {\widehat{h}}_{\alpha \beta }\overline{\nu },\mu \rangle =\cos \theta {\widehat{h}}_{\alpha \beta }. \end{aligned}$$

For (4), taking derivative of $h(\mu , e_\alpha )=0$ with respect to $e_\beta$ and using the Codazzi equation and (1), we get

$$\begin{aligned} 0=e_\beta (h(\mu , e_\alpha ))= & {} h_{\alpha \mu ; \beta }+h(\nabla _{e_\beta }e_\alpha , \mu )+h(\nabla _{e_\beta }\mu , e_\alpha )\\= & {} h_{\alpha \beta ; \mu }+ \langle \nabla _{e_\beta }e_\alpha , \mu \rangle h_{\mu \mu }+\langle \nabla _{e_\beta }\mu , e_\gamma \rangle h_{\alpha \gamma }\\= & {} h_{\alpha \beta ; \mu }-{\widetilde{h}}_{\beta \gamma }(h_{\mu \mu }\delta _{\alpha \gamma }-h_{\alpha \gamma }). \end{aligned}$$

$\square$

The Proposition 2.4 has a direct conseqeunce.

Corollary 2.5

If $\Sigma$ is a convex capillary hypersurface, then $\partial \Sigma \subset \partial \overline{\mathbb {R}}_{+}^{n+1}$ is also convex, i.e., ${{\widehat{h}}}\ \ge 0$, while $\partial \Sigma \subset \Sigma$ is convex $({{\widetilde{h}}} \ge 0)$ if $\theta \in (0,\frac{\pi }{2}]$ and concave $({{\widetilde{h}}} \le 0)$ if $\theta \in [\frac{\pi }{2}, \pi )$.

The following Minkowski type formulas for capillary hypersurfaces play an important role in this paper.

Proposition 2.6

Let $x:M\rightarrow \overline{\mathbb {R}}^{n+1}_+$ be an smooth immersion of $\Sigma {:}{=}x(M)$ into the half-space, whose boundary intersects $\mathbb {R}^n$ with a constant contact angle $\theta \in (0,\pi )$ along $\partial \Sigma$. For $1\le k\le n$, it holds

$$\begin{aligned} \int \limits _\Sigma H_{k-1} ( 1+\cos \theta \langle \nu ,e \rangle )dA=\int \limits _\Sigma H_k \langle x,\nu \rangle dA, \end{aligned}$$

(2.9)

where dA is the area element of $\Sigma$ w.r.t. the induced metric g.

When $k=1$ or $\theta =\frac{\pi }{2}$, formula (2.9) is known. See e.g. [2, Proof of Theorem 5.1] and [30, Proposition 2.5]. For our purpose, we need the high order Minkowski type formulas for general $\theta$.

Proof

Denote $x^T {:}{=}x-\langle x,\nu \rangle \nu$ be the tangential projection of x on $\Sigma$, and

$$\begin{aligned} P_e{:}{=}\langle \nu ,e \rangle x-\langle x,\nu \rangle e. \end{aligned}$$

From a direct computation, we have

$$\begin{aligned} D_{e_i} \langle x,e_j\rangle = g_{ij}-\langle x,\nu \rangle h_{ij} \end{aligned}$$

(2.10)

and

$$\begin{aligned} \nabla _{i} (P_e^T)_{j} =\langle \nu ,e \rangle g_{ij}+h_{il}\langle e,e_l\rangle \langle x,e_j\rangle -h_{il}\langle x,e_l\rangle \langle e,e_j\rangle . \end{aligned}$$

(2.11)

Along $\partial \Sigma \subset \partial \mathbb {R}^{n+1}_+$, using (2.7) we see

$$\begin{aligned} \langle P_e^T,\mu \rangle= & {} \langle P_e,\mu \rangle =\langle \nu ,e\rangle \langle x,\mu \rangle -\langle x,\nu \rangle \langle e,\mu \rangle \\ {}= & {} \langle x,-\sin \theta \nu -\cos \theta \mu \rangle , \end{aligned}$$

which follows

$$\begin{aligned} \langle x^T+\cos \theta P_e^T,\mu \rangle= & {} \langle x,\mu \rangle -\cos \theta \langle x,\sin \theta \nu +\cos \theta \mu \rangle \nonumber \\= & {} \sin \theta \langle x,e\rangle =0. \end{aligned}$$

(2.12)

Denote $\sigma _{k-1}^{ij}{:}{=}\frac{\partial \sigma _k}{\partial h^i_j }$ be the kth Newton transformation. Taking contraction with (2.10), (2.11) and using Proposition 2.1 we obtain

$$\begin{aligned}{} & {} \sigma ^{ij}_{k-1}\cdot \nabla _i \left( (x^T+\cos \theta P_e^T)_j\right) \\{} & {} \quad =\sigma _{k-1}^{ij} ( g_{ij}-h_{ij}\langle x,\nu \rangle +\cos \theta \langle \nu ,e\rangle g_{ij} )\\{} & {} \quad =(n-k+1)\sigma _{k-1}( 1+\cos \theta \langle \nu ,e\rangle ) -k\sigma _k \langle x,\nu \rangle \\{} & {} \quad = \frac{n!}{(k-1)! (n-k)!} \left( H_{k-1}( 1+\cos \theta \langle \nu ,e\rangle ) -H_k \langle x,\nu \rangle \right) . \end{aligned}$$

Using integration by parts, we have

$$\begin{aligned} \int \limits _\Sigma \nabla _i\big ( \sigma ^{ij}_{k-1}(x^T+\cos \theta P_e^T)_j\big )dA = \int \limits _{\partial \Sigma } \sigma _{k-1}^{ij} (x^T+\cos \theta P_e^T)_j\cdot \mu _ids. \end{aligned}$$

From (2.12), we know that $(x^T+\cos \theta P_e^T)\perp \mu$ along $\partial \Sigma$.

Since $\mu$ is a principal direction of $\Sigma$ by Proposition 2.4, we have $\sigma _{k-1}^{ij} (x^T+\cos \theta P_e^T)_j\cdot \mu _i=0$ along $\partial \Sigma$. It is well-known that the Newton tensor is divergence-free, i.e., $\nabla _i\sigma _{k-1}^{ij}=0$. Altogether yields the conclusion. $\square$

2.5 Variational formulas

The following first variational formula motivates us to define the quermassintegrals for capillary hypersurfaces as (1.8).

Theorem 2.7

Let $\Sigma _t\subset \overline{{\mathbb {R}}}_+^{n+1}$ be a family of smooth capillary hypersurfaces supported by $\partial \mathbb {R}^{n+1}_+$ with a constant contact angle $\theta \in (0,\pi )$ along $\partial \Sigma _t$, given by the embedding $x(\cdot ,t):M\rightarrow \overline{\mathbb {R}}^{n+1}_+$, and satisfying

$$\begin{aligned} (\partial _t x)^\perp =f\nu , \end{aligned}$$

(2.13)

for a smooth function f. Then for $-1\le k\le n-1$,

$$\begin{aligned} \frac{d}{dt} {\mathcal {V}}_{k+1,\theta }(\widehat{\Sigma _t})=\frac{n-k}{n+1}\int \limits _{\Sigma _t} fH_{k+1}dA_t, \end{aligned}$$

(2.14)

and

$$\begin{aligned} \frac{d}{dt}{\mathcal {V}}_{n+1,\theta }(\widehat{\Sigma _t})=0. \end{aligned}$$

Before proving Theorem 2.7, we remark that if $\Sigma _t\subset \overline{\mathbb {R}}^{n+1}_+$ is a family of smooth capillary hypersurfaces evolving by (2.13), then the tangential component $(\partial _tx)^T$ of $\partial _t x$, which we denote by $T\in T\Sigma _t$, must satisfy

$$\begin{aligned} T |_{\partial \Sigma _t}=f\cot \theta \mu + {\widetilde{T}}, \end{aligned}$$

(2.15)

where ${\widetilde{T}}\in T(\partial \Sigma _t)$. In fact, the restriction of $x(\cdot , t)$ on $\partial M$ is contained in $\mathbb {R}^n$ and hence,

$$\begin{aligned} f\nu +T|_{\partial \Sigma _t}= \partial _t x|_{\partial M}\in T\mathbb {R}^n. \end{aligned}$$

From (2.7), we know

$$\begin{aligned} \nu =\frac{1}{\sin \theta } {\bar{\nu }} -\cot \theta \mu . \end{aligned}$$

Since ${\bar{\nu }}\in T\mathbb {R}^n$, it follows $T-f\cot \theta \mu \in T\mathbb {R}^n\cap T\Sigma _t=T(\partial \Sigma _t)$, and hence (2.15). Up to a diffeomorphism of $\partial M$, we can assume ${\widetilde{T}}=0$. For simplicity, in the following, we always assume that

$$\begin{aligned} T |_{\partial \Sigma _t}=f\cot \theta \mu . \end{aligned}$$

(2.16)

Hence, from now on, let $\Sigma _t$ be a family of smooth, embedding hypersurfaces with $\theta$-capillary boundary in $\overline{ \mathbb {R}}^{n+1}_+$, given by the embeddings $x(\cdot ,t):M\rightarrow \overline{\mathbb {R}}^{n+1}_+$, which evolves by the general flow

$$\begin{aligned} \partial _t x=f\nu +T, \end{aligned}$$

(2.17)

with $T\in T\Sigma _t$ satisfying (2.16). We emphasize that the tangential part T plays a key role in the proof of Theorem 2.7 below.

Along flow (2.17), we have the following evolution equations for the induced metric $g_{ij}$, the area element $dA_t$, the unit outward normal $\nu$, the second fundamental form $h_{ij}$, the Weingarten matrix $h^i_j$, the mean curvature H, the kth mean curvature $\sigma _k$ and $F{:}{=}F(h^j_i)$ of the hypersurfaces $\Sigma _t$. These evolution equations will be used later.

Proposition 2.8

Along flow (2.17), it holds that

• (1) $\partial _t g_{ij}=2fh_{ij}+\nabla _i T_j+\nabla _jT_i$.

• (2) $\partial _tdA_t =\left( fH+\textrm{div}(T)\right) dA_t.$

• (3) $\partial _t\nu =-\nabla f+h(e_i,T)e_i$.

• (4) $\partial _t h_{ij}=-\nabla ^2_{ij}f +fh_{ik}h_{j}^k +\nabla _T h_{ij}+h_{j}^k\nabla _iT_k+h_{i}^k\nabla _j T_k.$

• (5) $\partial _t h^i_j=-\nabla ^i\nabla _{j}f -fh_{j}^kh^{i}_k+\nabla _T h^i_j.$

• (6) $\partial _t H=-\Delta f-|h|^2 f+ \langle \nabla H, T\rangle$.

• (7) $\partial _t \sigma _k=- \frac{\partial \sigma _k}{\partial h_i^j}\nabla ^i\nabla _{j}f - f (\sigma _1\sigma _k-(k+1)\sigma _{k+1})+\langle \nabla \sigma _k, T\rangle$.

• (8) $\partial _t F= -F^{j}_i\nabla ^i\nabla _{j}f -fF^{j}_ih_{j}^kh^{i}_k+\langle \nabla F,T\rangle$, where $F^i_j{:}{=}\frac{\partial F}{\partial h^j_i}$.

The proof of Proposition 2.8 for $T=0$ can be found for example in [27, Chap. 2, Sect. 2.3] or [21, Appendix B]. A proof for a general T can be found in [59, Proposition 2.11].

Now we complete the proof of Theorem 2.7.

Proof of Theorem 2.7

Choose an orthonormal frame $\{e_\alpha \}_{\alpha =2}^{n}$ of $T{\partial \Sigma }\subset T\mathbb {R}^n$ such that $\{e_1{:}{=}\mu ,(e_\alpha )_{\alpha =2}^n\}$ forms an orthonormal frames for $T\Sigma$. First, by taking time derivative to the capillary boundary condition, $\langle \nu ,\overline{N}\circ x\rangle =-\cos \theta$ along $\partial \Sigma$, we obtain

$$\begin{aligned} 0= & {} \langle \partial _t \nu , \overline{N}(x(\cdot ,t)) \rangle +\langle \nu , d\overline{N}(f\nu +T)\rangle \\= & {} \langle -\nabla f+h(e_i,T)e_i,\overline{N}\rangle \\= & {} -\sin \theta \nabla _{\mu } f+\sin \theta h(e_i,T) \langle e_i,\mu \rangle \\= & {} -\sin \theta \nabla _\mu f+\sin \theta h(\mu ,\mu ) \cot \theta f, \end{aligned}$$

where we have used (2.7), Proposition 2.8 and $T|_{\partial M}=f\cot \theta \mu$. As a result,

$$\begin{aligned} \nabla _\mu f= \cot \theta h(\mu ,\mu ) f \text { on }\partial \Sigma _t. \end{aligned}$$

(2.18)

Next, using integration by parts and Proposition 2.8 we have

$$\begin{aligned}{} & {} \frac{d}{dt} \left( \int \limits _{\Sigma _t} \sigma _k dA_t\right) \nonumber \\{} & {} \quad =\int \limits _{\Sigma _t} \big [(\partial _t \sigma _k ) dA_t+\sigma _k \partial _t (dA_t)\big ] \nonumber \\{} & {} \quad =\int \limits _{\Sigma _t} \frac{\partial \sigma _k}{\partial h_i^j}\big ( -f^i_{;j}-fh_{ik}h^{kj}+\langle \nabla h^i_j, T\rangle \big ) dA_t+\int \limits _\Sigma \sigma _k\big (f\sigma _1+\textrm{div}(T)\big )dA_t\nonumber \\{} & {} \quad =-\int \limits _{\partial \Sigma _t} \frac{\partial \sigma _{k}}{\partial h^j_i} f^i\mu _j+\int \limits _{\partial \Sigma _t} \sigma _k \langle T,\mu \rangle +\int \limits _{\Sigma _t} f\left( \sigma _1\sigma _k-\frac{\partial \sigma _k}{\partial h^j_i} h_{ik}h^{kj}\right) dA_t\nonumber \\{} & {} \quad =\int \limits _{\partial \Sigma _t} \big (f\sigma _k \cot \theta -\frac{\partial \sigma _{k}}{\partial h^j_i} f^i\mu _j\big ) +(k+1)\int \limits _{\Sigma _t} f\sigma _{k+1}dA_t\nonumber \\{} & {} \quad =\int \limits _{\partial \Sigma _t} \cot \theta f \sigma _k (h|h_{11}) +(k+1)\int \limits _{\Sigma _t} f\sigma _{k+1}dA_t, \end{aligned}$$

(2.19)

where we have used $T|_{\partial M}=f\cot \theta \mu$, (2.18) and Lemma 2.1 (1), (4).

Moreover flow (3.1) induces a hypersurface flow $\partial \Sigma _t\subset \mathbb {R}^n$ with normal speed $\frac{f}{\sin \theta }$, that is,

$$\begin{aligned} \partial _t x|_{\partial M}=f\nu +f\cot \theta \mu =\frac{f}{\sin \theta }\overline{\nu }. \end{aligned}$$

By (2.2), we have

$$\begin{aligned} \frac{d}{dt}{\mathcal {V}}_k(\widehat{\partial \Sigma _t})=\frac{n-k}{n} \int \limits _{\partial \Sigma _t} \frac{f}{\sin \theta } H_k^{\partial \Sigma _t}({\widehat{h}}). \end{aligned}$$

From Proposition 2.4 (2), we know

$$\begin{aligned} h_{\alpha \beta } = \sin \theta {\widehat{h}}_{\alpha \beta }, \end{aligned}$$

and hence $\sigma _k(h|h_{11})=\sin ^k\theta \sigma _k({\widehat{h}})$. Substituting these formulas into (2.19), we obtain

$$\begin{aligned} \frac{d}{dt} \left( \int \limits _{\Sigma _t} H_k dA_t- \sin ^k\theta \cos \theta {\mathcal {V}}_k(\widehat{\partial \Sigma _t})\right) =(n-k)\int \limits _{\Sigma _t} f H_{k+1} dA_t. \end{aligned}$$

By the definition of ${\mathcal {V}}_{k+1,\theta }(\widehat{\Sigma _t})$ in (1.8), we get the desired formula (2.14) for $k\ge 0.$

It remains to consider the case $k=-1$. It is easy to check that

$$\begin{aligned} {\mathcal {V}}_{0,\theta }(\Sigma _t) =|{\widehat{\Sigma }}_t| =\frac{1}{n+1}\int \limits _{\Sigma _t} \langle x, \nu \rangle dA_t. \end{aligned}$$

A direct computation gives

$$\begin{aligned}{} & {} (n+1) \frac{d}{dt} {\mathcal {V}}_{0,\theta }(\Sigma _t)\\{} & {} \quad = \int \limits _{\Sigma _t} \left[ f - \langle x, \nabla f \rangle + \langle x,\nu \rangle fH + h(T, x^T) + \langle x, \nu \rangle \textrm{div}\, T \right] dA_t\\{} & {} \quad =\int \limits _{\Sigma _t} \left( (1 + \textrm{div}(x^T) )f + \langle x,\nu \rangle fH \right) dA_t + \int \limits _{\partial \Sigma _t} \left( -\langle x^T, \mu \rangle f+ \langle x,\nu \rangle \langle T,\mu \rangle \right) \\{} & {} \quad = (n+1) \int \limits _{\Sigma _t} f dA_t, \end{aligned}$$

since $-\langle x^T, \mu \rangle f +\langle x,\nu \rangle \langle T, \mu \rangle =0$ for $x\in \partial \Sigma$, which follows from

$$\begin{aligned}{} & {} \langle x,\nu \rangle \langle T, \mu \rangle = f\cot \theta \langle x,\nu \rangle = f\cos \theta \langle x,{\bar{\nu }} \rangle , \\{} & {} \langle x^T, \mu \rangle f = f\cos \theta \langle x,{\bar{\nu }} \rangle . \end{aligned}$$

Now we complete the proof. $\square$

3 Locally constrained curvature flow

In this section, we first introduce a new locally constrained curvature flow and show the monotonicity of the quermassintegral along the flow.

Let M be a compact orientable smooth n-dimensional manifold. Suppose $x_0:M\rightarrow \overline{\mathbb {R}}^{n+1}_+$ be a smooth initial embedding such that $x_0(M)$ is a convex hypersurface in $\overline{\mathbb {R}}^{n+1}_+$ and intersects with $\partial {\mathbb {R}}^{n+1}_+$ at a constant contact angle $\theta \in (0,{\pi })$. We consider the smooth family of embeddings $x:M\times [0,T)\rightarrow \overline{\mathbb {R}}^{n+1}_+,$ satisfying the following evolution equations

$$\begin{aligned} \begin{array}{rcll} ( \partial _t x(p,t) )^\perp &{}=&{}f(p,t)\nu \quad &{} \hbox { for } (p,t)\in M\times [0,T),\\ \langle \nu (p,t),\overline{N}\circ x(p,t)\rangle &{}=&{} \cos (\pi -\theta ) \quad &{} \hbox { for }(p,t)\in \partial M\times [0,T), \end{array} \end{aligned}$$

(3.1)

with $x(M,0)=x_0(M)$ and

$$\begin{aligned} f{:}{=}\frac{1+\cos \theta \langle \nu ,e\rangle }{F}-\langle x,\nu \rangle , \end{aligned}$$

(3.2)

where

$$\begin{aligned} F{:}= \frac{H_{l} }{H_{l-1} }. \end{aligned}$$

(3.3)

The following nice property of flow (3.1) is essential for us to prove Theorem 1.2 later.

Proposition 3.1

As long as flow (3.1) exists and $\Sigma _t$ is strictly l-convex, ${\mathcal {V}}_{l,\theta }(\widehat{\Sigma _t})$ is preserved and ${\mathcal {V}}_{k,\theta }(\widehat{\Sigma _t})$ is non-decreasing for $1\le k<l \le n$.

Proof

From Theorem 2.7, we see

$$\begin{aligned} \partial _t {\mathcal {V}}_{l,\theta }(\widehat{\Sigma _t})= & {} \frac{n+1-l}{n+1}\int \limits _{\Sigma _t} f H_l dA_t\\= & {} \frac{n+1-l}{n+1}\int \limits _{\Sigma _t}\left[ \left( 1+\cos \theta \langle \nu ,e\rangle \right) H_{l-1}-H_l \langle x,\nu \rangle \right] dA_t\\= & {} 0, \end{aligned}$$

where the last equality follows from (2.9). For $1\le k<l\le n$, from Theorem 2.7

$$\begin{aligned} \partial _t {\mathcal {V}}_{k,\theta }(\widehat{\Sigma _t})= & {} \frac{n+1-k}{n+1}\int \limits _{\Sigma _t} f H_{k } dA_t\\= & {} \frac{n+1-k}{n+1}\int \limits _{\Sigma _t} \left[ H_{k}\frac{H_{l-1}}{{H_l}}\big (1+\cos \theta \langle \nu ,e\rangle \big )-H_k\langle x,\nu \rangle \right] dA_t\\\ge & {} \frac{n+1-k}{n+1}\int \limits _{\Sigma _t} \left[ H_{k-1}\big (1+\cos \theta \langle \nu ,e\rangle \big )-H_k\langle x,\nu \rangle \right] dA_t\\= & {} 0, \end{aligned}$$

where we have used the Newton–MacLaurin inequality (2.1) and the Minkowski formula (2.9) in the last two steps respectively. $\square$

4 A priori estimates and convergence

The main result of this section is the following long-time existence and the convergence result of flow (3.1) with $l=n$, i.e.,

$$\begin{aligned} F=\frac{H_n}{H_{n-1}}\end{aligned}$$

(4.1)

under an angle constraint

$$\begin{aligned} \theta \in \left( 0,\frac{\pi }{2}\right] . \end{aligned}$$

Theorem 4.1

Assume $x_0:M\rightarrow \overline{\mathbb {R}}^{n+1}_+$ is an embedding of a strictly convex capillary hypersurface in the half-space with the contact angle $\theta \in ( 0,\frac{\pi }{2}]$. Then there exists $x:M\times [0,+\infty )\rightarrow \overline{\mathbb {R}}^{n+1}_+$ satisfying flow (3.1) with F given by (4.1) and the initial condition $x(M,0)=x_0(M)$. Moreover, $x(\cdot ,t)\rightarrow x_\infty (\cdot )$ in $C^\infty$ topology as $t\rightarrow +\infty$, and the limit $x_\infty :M\rightarrow \overline{\mathbb {R}}^{n+1}_+$ is a spherical cap.

In order to prove this theorem, we need to obtain a priori estimates, which will be given as follows.

4.1 The short time existence

For the short time existence, one can follow the strategy presented in the paper of Huisken–Polden [36] to give a proof for a general initial capillary hypersurface. Since our initial hypersurface is convex, one can prove the short time existence in the class of star-shaped hypersurfaces. In this class, one can in fact reduce flow (3.1) to a scalar flow. Then the short time existence follows clearly from the standard theory for parabolic equations. Therefore we first consider the reduction.

Assume that a capillary hypersurfaces $\Sigma$ is strictly star-shaped with respect to the origin. One can reparametrize it as a graph over $\overline{\mathbb {S}}^n_+$. Namely, there exists a positive function r defined on $\overline{\mathbb {S}}^n_+$ such that

$$\begin{aligned} \Sigma =\left\{ r(X )X | X\in \overline{\mathbb {S}}^n_+\right\} , \end{aligned}$$

where $X{:}{=}(X_1,\ldots , X_n)$ is a local coordinate of $\overline{\mathbb {S}}^n_+$.

We denote $\nabla ^0$ be the Levi-Civita connection on $\mathbb {S}^n_+$ with respect to the standard round metric $\sigma {:}{=}g_{_{\mathbb {S}^n_+}}$, $\partial _i{:}{=}\partial _{X_i}$, $\sigma _{ij}{:}{=}\sigma (\partial _{i},\partial _{j})$, $r_i{:}{=}\nabla ^0_{i} r$, and $r_{ij}{:}{=}\nabla ^0_{i}\nabla ^0_j r$. The induced metric g on $\Sigma$ is given by

$$\begin{aligned} g_{ij}=r^2\sigma _{ij}+ r_ir_j=e^{2\varphi }\left( \sigma _{ij}+\varphi _i\varphi _j\right) , \end{aligned}$$

where $\varphi (X){:}{=}\log r(X)$. Its inverse $g^{-1}$ is given by

$$\begin{aligned} g^{ij}=\frac{1}{r^2} \left( \sigma ^{ij}-\frac{r^ir^j}{r^2+|\nabla ^0 r|^2}\right) =e^{-2\varphi }\left( \sigma ^{ij}-\frac{\varphi ^i\varphi ^j}{v^2}\right) , \end{aligned}$$

where $r^i{:}{=}\sigma ^{ij}r_j$, $\varphi ^i{:}{=}\sigma ^{ij}\varphi _j$ and $v{:}{=}\sqrt{1+|\nabla ^0 \varphi |^2}$. The unit outward normal vector field on $\Sigma$ is given by

$$\begin{aligned} \nu =\frac{1}{v}\left( \partial _r-r^{-2}\nabla ^0 r\right) =\frac{1}{v}\left( \partial _r-r^{-1}\nabla ^0 \varphi \right) . \end{aligned}$$

The second fundamental form h on $\Sigma$ is

$$\begin{aligned} h_{ij}=\frac{e^\varphi }{v }\left( \sigma _{ij}+\varphi _i\varphi _j-\varphi _{ij}\right) , \end{aligned}$$

and its Weingarten matrix $h^i_j$ is

$$\begin{aligned} h_j^{i}=g^{ik}h_{kj}=\frac{1}{e^\varphi v }\left[ \delta ^i_j-\left( \sigma ^{ik}-\frac{\varphi ^i\varphi ^k}{v^2}\right) \varphi _{kj}\right] . \end{aligned}$$

The higher order mean curvature $H_k$ can also be expressed by $\varphi$. Moreover,

$$\begin{aligned} \langle x,\nu \rangle =\langle r\partial _r,\nu \rangle =\frac{e^\varphi }{v }. \end{aligned}$$

In order to express the capillary boundary condition in terms of the radial function $\varphi$, we use the polar coordinate in the half-space. For $x{:}{=}(x',x_{n+1})\in \mathbb {R}^n\times [0,+\infty )$ and $X{:}{=}(\beta ,\xi )\in [0,\frac{\pi }{2}]\times \mathbb {S}^{n-1}$, we have that

$$\begin{aligned} x_{n+1}=r\cos \beta ,\quad |x'|=r\sin \beta . \end{aligned}$$

Then

$$\begin{aligned} e_{n+1}= \partial _{x_{n+1}}=\cos \beta \partial _r-\frac{\sin \beta }{r}\partial _\beta . \end{aligned}$$

In these coordinates the standard Euclidean metric is given by

$$\begin{aligned} |dx|^2=dr^2+r^2\left( d\beta ^2+\sin ^2\beta g_{_{\mathbb {S}^{n-1}}}\right) . \end{aligned}$$

It follows that

$$\begin{aligned} \langle \nu ,e_{n+1}\rangle =\frac{1}{v}\left( \cos \beta +\sin \beta \nabla _{\partial _\beta }^0 \varphi \right) . \end{aligned}$$

Along $\partial \mathbb {S}^n_+$ it holds

$$\begin{aligned} \overline{N}\circ x=-e_{n+1} =\frac{1}{r}\partial _\beta , \end{aligned}$$

which yields

$$\begin{aligned} -\cos \theta = \langle \nu ,\overline{N}\circ x\rangle =\left\langle \frac{1}{v}\left( \partial _r-r^{-1}\nabla ^0 \varphi \right) , \frac{1}{r}\partial _\beta \right\rangle =-\frac{\nabla ^0_{\partial _\beta }\varphi }{v}, \end{aligned}$$

that is,

$$\begin{aligned} \nabla ^0_{\partial _\beta } \varphi =\cos \theta \sqrt{1+|\nabla ^0\varphi |^2}. \end{aligned}$$

(4.2)

Therefore, in the class of star-shaped hypersurfaces flow (3.1) is reduced to the following scalar parabolic equation with an oblique boundary condition

$$\begin{aligned} \begin{array}{rcll} \partial _t \varphi &{}=&{} \frac{v}{e^\varphi }f,\quad &{}\text { in } \mathbb {S}^n_+\times [0,T^*), \\ \nabla _{\partial _\beta }^0 \varphi &{}=&{}\cos \theta \sqrt{1+|\nabla ^0\varphi |^2},\quad &{}\text { on } \partial \mathbb {S}^n_+\times [0,T^*),\\ \varphi (\cdot ,0)&{}=&{}\varphi _0(\cdot ),\quad &{}\text { on } \mathbb {S}^n_+, \end{array} \end{aligned}$$

(4.3)

where $\varphi _0$ is the parameterization radial function of $x_0(M)$ over $\overline{\mathbb {S}}^n_+$, and

$$\begin{aligned} f=\frac{H_{n-1}}{H_n}\left[ 1- \frac{\cos \theta }{v}\left( \cos \beta +\sin \beta \nabla ^0_{\partial _\beta }\varphi \right) \right] -\frac{e^\varphi }{v}. \end{aligned}$$

Since $|\cos \theta |<1$, the oblique boundary condition (4.2) satisfies the non-degeneracy condition in [44], see also [20]. Hence the short time existence follows.

4.2 Barriers

Let $T^*$ be the maximal time of smooth existence of a solution to (3.1), more precisely in the class of star-shaped hypersurfaces. It is obvious that F can not be zero and hence F is positive in $M\times [0, T^*)$. The positivity of F implies that $\Sigma _t$ is strictly convex up to $T^*$.

The convexity of $\Sigma _0$ implies that there exists some $0<r_1<r_2<\infty$, such that

$$\begin{aligned} \Sigma _0\subset \widehat{C_{r_2,\theta }}\setminus \widehat{C_{r_1,\theta }}. \end{aligned}$$

The family of $C_{ r,\theta }$ forms natural barriers of (3.1). Therefore, we can show that the solution to (4.3) is uniformly bounded from above and below.

Proposition 4.2

For any $t\in [0, T^*)$, $\Sigma _t$ satisfies

$$\begin{aligned} \Sigma _t\subset \widehat{C_{r_2,\theta } }\setminus \widehat{C_{r_1,\theta } }. \end{aligned}$$

Proof

Recall that $C_{r,\theta }$ satisfies (2.4). Thus for each $r>0$, it is a static solution to flow (3.1). The assertion follows from the avoidance principle for strictly parabolic equation with a capillary boundary condition (see [5, Sect. 2.6] or [54, Proposition 4.2]). $\square$

4.3 Evolution equations of F and H

We first introduce a parabolic operator for (3.1)

$$\begin{aligned} {\mathcal {L}} {:}{=}\partial _t-\frac{1+\cos \theta \langle \nu ,e\rangle }{F^2}F^{ij}\nabla _{ij}^2-\left\langle T+x-\frac{\cos \theta }{F}e,\nabla \right\rangle . \end{aligned}$$

Set ${\mathcal {F}}{:}{=}\sum _{i=1}^n F^{i}_i$. Using Proposition 2.1 we have

$$\begin{aligned} {\mathcal {F}}-\frac{F^{ij}h_{ij}}{F}= & {} {\mathcal {F}}-1\ge 0,\end{aligned}$$

(4.4)

$$\begin{aligned} \frac{F^{ij}h_i^k h_{kj}}{F^2}= & {} 1. \end{aligned}$$

(4.5)

Proposition 4.3

Along flow (3.1), we have

$$\begin{aligned} {\mathcal {L}} F= & {} 2\cos \theta F^{-2}F^{ij}F_{;j} h_{ik}\langle e_k,e\rangle -2(1+\cos \theta \langle \nu ,e\rangle )F^{-3}F^{ij}F_{;i}F_{;j}\\{} & {} +F\left( 1-\frac{F^{ij}(h^2)_{ij}}{F^2}\right) , \end{aligned}$$

and

$$\begin{aligned} \nabla _\mu F=0,\quad \text { on } \partial \Sigma _t. \end{aligned}$$

(4.6)

Proof

Using the Codazzi formula, we have

$$\begin{aligned} F^{ij}\langle x,\nu \rangle _{;ij}= & {} F^{ij}\left( h_{ij}+h_{ij;k}\langle x,e_k\rangle -(h^2)_{ij}\langle x,\nu \rangle \right) \\= & {} F+F_{;k}\langle x,e_k\rangle -F^{ij}(h^2)_{ij}\langle x,\nu \rangle , \end{aligned}$$

and

$$\begin{aligned} F^{ij}\langle \nu ,e\rangle _{;ij}= & {} F^{ij}\big (h_{ik;j}\langle e_k,e\rangle -(h^2)_{ij}\langle \nu ,e\rangle \big )\\= & {} F_{;k}\langle e_k,e\rangle -F^{ij}(h^2)_{ij}\langle \nu ,e\rangle . \end{aligned}$$

Combining with Proposition 2.8, we obtain

$$\begin{aligned} \partial _t F= & {} -F^{ij}f_{;ij}-fF^{ij}(h^2)_{ij}+\langle \nabla F,T\rangle \\= & {} -F^{ij} \left( \frac{1+\cos \theta \langle \nu ,e\rangle }{F}-\langle x,\nu \rangle \right) _{;ij}-fF^{ij}(h^2)_{ij}+\langle \nabla F,T\rangle \\= & {} -\cos \theta F^{ij}F^{-1} \langle \nu ,e\rangle _{;ij}+2\cos \theta F^{-2}F^{ij}F_{;j} \langle \nu ,e\rangle _{;i}\\ {}{} & {} -2(1+\cos \theta \langle \nu ,e\rangle )F^{-3}F^{ij}F_{;i}F_{;j}\\ {}{} & {} +F^{-2}F^{ij}F_{;ij}(1+\cos \theta \langle \nu ,e\rangle )+\big (F+F_{;k}\langle x,e_k\rangle -F^{ij}(h^2)_{ij}\langle x,\nu \rangle \big )\\{} & {} -(1+\cos \theta \langle \nu ,e\rangle ) F^{-1}F^{ij}(h^2)_{ij}+\langle x,\nu \rangle F^{ij}(h^2)_{ij}+\langle \nabla F,T\rangle . \end{aligned}$$

Hence it follows

$$\begin{aligned} {\mathcal {L}}F= & {} \partial _tF -(1+\cos \theta \langle \nu ,e\rangle )F^{-2}F^{ij}F_{;ij}-\langle T+x-\cos \theta F^{-1}e,\nabla F\rangle \\= & {} \cos \theta F^{ij}(h^2)_{ij}F^{-1} \langle \nu ,e\rangle +2\cos \theta F^{-2}F^{ij}F_{;j} h_{ik}\langle e_k,e\rangle \\{} & {} -2(1+\cos \theta \langle \nu ,e\rangle )F^{-3}F^{ij}F_{;i}F_{;j}\\{} & {} +\big (F-F^{ij}(h^2)_{ij}\langle x,\nu \rangle \big ) -(1+\cos \theta \langle \nu ,e\rangle ) F^{-1}F^{ij}(h^2)_{ij}+\langle x,\nu \rangle F^{ij}(h^2)_{ij}\\= & {} 2\cos \theta F^{-2}F^{ij}F_{;j} h_{ik}\langle e_k,e\rangle -2(1+\cos \theta \langle \nu ,e\rangle )F^{-3}F^{ij}F_{;i}F_{;j}\\{} & {} + F- F^{-1}F^{ij}(h^2)_{ij}. \end{aligned}$$

Along $\partial \Sigma _t,$ from (2.18) we know

$$\begin{aligned} \nabla _\mu f=\cot \theta h(\mu ,\mu ) f, \end{aligned}$$

By (2.7) or (2.8) and Proposition 2.4 (1), we have on $\partial \Sigma _t$

$$\begin{aligned} \nabla _\mu \langle x,\nu \rangle =\langle x, h(\mu ,\mu )\mu \rangle =\cos \theta h(\mu ,\mu )\langle x, \overline{\nu }\rangle = \cot \theta h(\mu ,\mu )\langle x, \nu \rangle , \end{aligned}$$

and hence

$$\begin{aligned} \nabla _\mu (f+\langle x,\nu \rangle )=\cot \theta h(\mu ,\mu ) (f+\langle x,\nu \rangle ). \end{aligned}$$

Using (2.7) and Proposition 2.4 (1) again, we have ($e={\overline{N}}$)

$$\begin{aligned} \nabla _\mu \langle \nu ,e\rangle =h(\mu ,\mu )\langle \mu ,e\rangle =-\tan \theta h(\mu ,\mu )\langle \nu ,e\rangle \end{aligned}$$

and

$$\begin{aligned} \nabla _\mu \big (1+\cos \theta \langle \nu ,e \rangle \big )= -\sin \theta h(\mu ,\mu )\langle \nu ,e\rangle , \end{aligned}$$

where we used $\langle e,\overline{\nu }\rangle =0$ and $\langle e,x\rangle =0$ on $\partial \Sigma _t$. One can easily check that the left hand side of the previous formula equals to $\cot \theta h(\mu ,\mu ) (1+\cos \theta \langle \nu , e \rangle )$, on $\partial \Sigma$. Hence it follows that

$$\begin{aligned} \nabla _\mu F= & {} \nabla _\mu \left( \frac{1+\cos \theta \langle \nu ,e\rangle }{f+\langle x,\nu \rangle }\right) =0. \end{aligned}$$

$\square$

We remark that (4.6) plays an important role in applying the maximum principle later. This property holds for curvature flow of free boundary hypersurfaces and capillary hypersurfaces, see also [48, 59].

Proposition 4.4

Along flow (3.1), we have

$$\begin{aligned} {\mathcal {L}} H= & {} (1+\cos \theta \langle \nu ,e\rangle )F^{-2} F^{kl,st}h_{kl;i}h_{st;i} +(2+\cos \theta \langle \nu ,e\rangle ) H \nonumber \\{} & {} +\Big [ 2\cos \theta F^{-2}F_{;i} \langle \nu ,e\rangle _{;i} -2\left( 1+\cos \theta \langle \nu ,e\rangle \right) F^{-3}|\nabla F|^2\nonumber \\{} & {} -(2+\cos \theta \langle \nu ,e\rangle ) F^{-1}|h|^2 \Big ], \end{aligned}$$

(4.7)

and, while $\Sigma$ is convex,

$$\begin{aligned} \nabla _\mu H\le 0,\quad \text { on } \partial \Sigma _t. \end{aligned}$$

(4.8)

Proof

First, note that

$$\begin{aligned} \Delta \langle \nu ,e\rangle= & {} H_{;k}\langle e_k,e\rangle -|h|^2\langle \nu ,e\rangle , \\ \Delta \langle x,\nu \rangle= & {} H+H_{;k}\langle x,e_k\rangle -|h|^2\langle x,\nu \rangle . \end{aligned}$$

Applying Proposition 2.8, we obtain

$$\begin{aligned} \partial _t H= & {} -\Delta f-|h|^2 f+\langle \nabla H,T\rangle \\= & {} (1+\cos \theta \langle \nu ,e\rangle )F^{-2}\Delta F-2(1+\cos \theta \langle \nu ,e\rangle ) F^{-3}|\nabla F|^2 \\ {}{} & {} +2\cos \theta F^{-2}F_{;i} \langle \nu ,e\rangle _{;i}-\cos \theta F^{-1}\Delta \langle \nu ,e\rangle +\Delta \langle x,\nu \rangle \\{} & {} -(1+\cos \theta \langle \nu ,e\rangle )F^{-1}|h|^2+\langle x,\nu \rangle |h|^2+\langle \nabla H,T\rangle \\= & {} (1+\cos \theta \langle \nu ,e\rangle )F^{-2}\Delta F-2(1+\cos \theta \langle \nu ,e\rangle ) F^{-3}|\nabla F|^2 \\ {}{} & {} +2\cos \theta F^{-2}F_{;i} \langle \nu ,e\rangle _{;i}- F^{-1}|h|^2+H\\{} & {} + \langle x,\nabla H\rangle +\langle \nabla H,T\rangle -\cos \theta F^{-1}\langle e,\nabla H\rangle . \end{aligned}$$

The Ricci equation and the Codazzi equation yield

$$\begin{aligned} h_{kl;ii}= & {} h_{ki;li}=h_{ki;il}+R^p_{ili}h_{pk}+R^p_{kli}h_{pi}\\= & {} h_{ii;kl}+(h_{pl}H-h_{pi}h_{li})h_{pk}+(h_{pl}h_{ki}-h_{pi}h_{kl})h_{pi}\\= & {} H_{;kl}+h_{pk}h_{pl}H-|h|^2 h_{kl}, \end{aligned}$$

which implies

$$\begin{aligned} \Delta F= & {} \frac{\partial ^2 F}{\partial h_{kl}\partial h_{st}} h_{kl;i}h_{st;i}+F^{kl}h_{kl;ii}\\= & {} F^{kl,st}h_{kl;i}h_{st;i}+F^{kl} H_{;kl}+F^{kl}(h^2)_{kl}H-F|h|^2. \end{aligned}$$

Hence we have

$$\begin{aligned} {\mathcal {L}}H= & {} \partial _t H-F^{-2}(1+\cos \theta \langle \nu ,e\rangle )F^{ij}H_{;ij}-\langle T+x-\cos \theta F^{-1}e,\nabla H\rangle \\= & {} (1+\cos \theta \langle \nu ,e\rangle )F^{-2}\big [ F^{kl,st}h_{kl;i}h_{st;i} +F^{kl}(h^2)_{kl}H-F|h|^2\big ]\\ {}{} & {} -2(1+\cos \theta \langle \nu ,e\rangle ) F^{-3}|\nabla F|^2 +2\cos \theta F^{-2}F_{;i} \langle \nu ,e\rangle _{;i} - F^{-1}|h|^2+H\\= & {} (1+\cos \theta \langle \nu ,e\rangle )F^{-2} F^{kl,st}h_{kl;i}h_{st;i} +(2+\cos \theta \langle \nu ,e\rangle ) H\\{} & {} +\Big [ 2\cos \theta F^{-2}F_{;i} \langle \nu ,e\rangle _{;i} -2(1+\cos \theta \langle \nu ,a\rangle ) F^{-3}|\nabla F|^2\\{} & {} -(2+\cos \theta \langle \nu ,e\rangle ) F^{-1}|h|^2 \Big ]. \end{aligned}$$

Along $\partial \Sigma _t$, choosing an orthonormal frame $\{e_\alpha \}_{\alpha =2}^{n}$ of $T{\partial \Sigma _t}$ such that $\{e_1{:}{=}\mu ,(e_\alpha )_{\alpha =2}^n\}$ forms an orthonormal frames for $T\Sigma _t$. From Proposition 2.4, we have

$$\begin{aligned} h_{\alpha \beta ;\mu }= \cos \theta {\widehat{h}}_{\beta \gamma } (h_{11}\delta _{\alpha \gamma }-h_{\alpha \gamma }), \end{aligned}$$

for all $2\le \alpha \le n$. Equation (4.6) implies

$$\begin{aligned} 0=\nabla _\mu F=F^{11}h_{11;1}+\sum \limits _{\alpha =2}^nF^{\alpha \alpha }h_{\alpha \alpha ;1}, \end{aligned}$$

which in turn implies

$$\begin{aligned} \nabla _\mu H= & {} h_{11;1}+\sum _{\alpha =2}^n h_{\alpha \alpha ;1}\\= & {} -\sum _{\alpha =2}^n \frac{F^{\alpha \alpha }}{F^{11}}h_{\alpha \alpha ;1}+\sum _{\alpha =2}^n h_{\alpha \alpha ;1} =\sum _{\alpha =2}^n \frac{1}{F^{11}}\big ( F^{11}-F^{\alpha \alpha }\big )h_{\alpha \alpha ;1}\\= & {} \sum _{\alpha =2}^n \frac{1}{F^{11}}\big ( F^{11}-F^{\alpha \alpha }\big )(h_{11}-h_{\alpha \alpha }) {\widetilde{h}}_{\alpha \alpha }\\\le & {} 0, \end{aligned}$$

where the last inequality follows from the concavity of F, and the convexity of $\partial \Sigma \subset \Sigma$, see Corollary 2.5. Hence (4.8) is proved. $\square$

Remark 4.5

(4.8) is the only place where we have used $\theta \in (0,\frac{\pi }{2}]$.

4.4 Curvature estimates

First, we have the uniform bound of F, which follows directly from Proposition 4.3 and the maximum principle.

Proposition 4.6

Along flow (3.1), it holds

$$\begin{aligned} \min _{M} F(\cdot , 0)\le F(p,t)\le \max _{M} F(\cdot , 0), \quad \forall (p,t)\in M\times [0,T^*). \end{aligned}$$

In particular, from the uniform lower bound of $F{:}{=}\frac{H_n}{H_{n-1}}$, we get a uniform curvature positive lower bound.

Corollary 4.7

$\Sigma _t, t\in [0,T^*)$ is uniformly convex, that is, there exists $c>0$ depending only on $\Sigma _0$, such that the principal curvatures of $\Sigma _t$,

$$\begin{aligned} \min _i \kappa _i(p, t) \ge c, \end{aligned}$$

for all $(p,t)\in M\times [0,T^*)$.

Next we obtain the uniform bound of the mean curvature.

Proposition 4.8

There exists $C>0$ depending only on $\Sigma _0$, such that

$$\begin{aligned} H(p,t)\le C,\quad \forall (p,t)\in M\times [0,T^*). \end{aligned}$$

Proof

From (4.8), we know that $\nabla _\mu H\le 0$ on $\partial \Sigma _t$. Thus H attains its maximum value at some interior point, say $p_0\in \text {int}(M)$. We now compute at $p_0$.

From the concavity of $F=\frac{n\sigma _n}{\sigma _{n-1}}$ in Proposition  2.2, we know

$$\begin{aligned} \left( 1+\cos \theta \langle \nu ,e\rangle \right) F^{-2} F^{kl,st}h_{kl;i}h_{st;i} \le 0. \end{aligned}$$

Using Proposition 4.4, we have

$$\begin{aligned} {\mathcal {L}}H\le & {} (2+\cos \theta \langle \nu ,e\rangle ) H \\{} & {} +\Big [ 2\cos \theta F^{-2}F_{;i} \langle \nu ,e\rangle _{;i} -2\left( 1+\cos \theta \langle \nu ,e\rangle \right) F^{-3}|\nabla F|^2-(2+\cos \theta \langle \nu ,e\rangle ) F^{-1}|h|^2 \Big ] \\:= & {} \text {K}_1+\text {K}_2, \end{aligned}$$

The term $|\text {K}_1|$ is bounded by 3H. For the term $\text {K}_2$, we note that

$$\begin{aligned} F\text {K}_2:= & {} 2\cos \theta F^{-1}F_{;i} \langle \nu ,e\rangle _{;i} -2(1+\cos \theta \langle \nu ,e\rangle ) F^{-2}|\nabla F|^2\\ {}{} & {} -(2+\cos \theta \langle \nu ,e\rangle ) |h|^2 \\= & {} 2\cos \theta F^{-1}F_{;i}h_{ii}\langle e,e_i\rangle -2(1+\cos \theta \langle \nu ,e\rangle )F^{-2}|\nabla F|^2\\ {}{} & {} -(2+\cos \theta \langle \nu ,e\rangle ) |h|^2 \\:= & {} -\sum _{i=1}^n \big ( \text {S}_1 F_{;i}^2+ \text {S}_{2,i}F_{;i} h_{ii}+ \text {S}_3h_{ii}^2\big ) =- \text {S}_1\sum _{i=1}^n{\left( F_{;i}-\frac{ \text {S}_{2,i}}{2 \text {S}_1}h_{ii}\right) ^2}\\ {}{} & {} +\sum _{i=1}^n \left( \frac{ \text {S}_{2,i}^2}{4 \text {S}_1}- \text {S}_3\right) h_{ii}^2, \end{aligned}$$

where we have used the notations

$$\begin{aligned} \text {S}_1{:}{=}2(1+\cos \theta \langle \nu ,e\rangle ) F^{-2}, \quad {\text {S}_{2,i}{:}{=}-2\cos \theta F^{-1}\langle e,e_i \rangle }, \quad \text {S}_3{:}{=}2+\cos \theta \langle \nu ,e\rangle . \end{aligned}$$

One can check

$$\begin{aligned} \text {S}_{2,i}^2-4 \text {S}_1 \text {S}_3:= & {} 4\cos ^2\theta F^{-2} \langle e,e_i\rangle ^2-8(1+\cos \theta \langle \nu ,e\rangle ) F^{-2} (2+\cos \theta \langle \nu ,e\rangle ) \\\le & {} 4F^{-2}\Big [ \cos ^2\theta |e^T|^2-2(1+\cos \theta \langle \nu ,e\rangle )(2+\cos \theta \langle \nu ,e\rangle )\Big ] \\= & {} 4F^{-2}\Big [-3(1+\cos \theta \langle \nu ,e\rangle )^2-1+\cos ^2\theta \Big ] \\\le & {} -c_0 \end{aligned}$$

for some positive constant $c_0$. Combining with Proposition 4.6, it implies

$$\begin{aligned} \text {K}_2\le -\frac{c_0}{4F \text {S}_1}|h|^2=-\frac{c_0F}{8(1+\cos \theta \langle \nu ,e \rangle )} |h|^2\le -C |h|^2, \end{aligned}$$

for some positive constant $C>0$. Therefore,

$$\begin{aligned}{} & {} 0 \le {\mathcal {L}}H(p_0) \le \text {K}_1+\text {K}_2 \le 3H-C|h|^2, \end{aligned}$$

which yields that H is uniformly bounded from above. $\square$

Proposition 4.8 and Corollary 4.7 imply directly that

Corollary 4.9

$\Sigma _t, t\in [0,T^*)$, has a uniform curvature bound, namely, there exists $C>0$ depending only on $\Sigma _0$, such that the principal curvatures of $\Sigma _t$,

$$\begin{aligned} \max _i \kappa _i(p, t) \le C, \end{aligned}$$

for all $(p,t)\in M\times [0,T^*)$.

4.5 Convergence of the flow

First we show that the convexity implies that the star-shaped is preserved in the following sense.

Proposition 4.10

There exists $c_0>0$ depending only on $\Sigma _0$, such that

$$\begin{aligned} \langle x,\nu \rangle (p,t) \ge c_0. \end{aligned}$$

(4.9)

for all $(p,t)\in M\times [0,T^*)$.

Proof

For any $T'<T^*$, assume $\min \limits _{M\times [0,T']} \langle x,\nu \rangle (p,t)=\langle x,\nu \rangle (p_0,t_0)$. Then, either $p_0\in \partial M$ or $p_0\in M\setminus \partial M$.

If $p_0\in M\setminus \partial M$, let $\{e_i\}_{i=1}^n$ be the orthonormal frame of $\Sigma _t$, then at $p_0$,

$$\begin{aligned} 0= D_{e_i}\langle x,\nu \rangle =h_{ij}\langle x,e_j\rangle . \end{aligned}$$

Due to the strict convexity $(h_{ij})>0$, we have $\langle x,e_i\rangle =0$. It follows

$$\begin{aligned} \langle x,\nu \rangle (p_0)=|x|(p_0)\ge c_0, \end{aligned}$$

for some $c_0>0$, which depends only on the initial datum.

If $p_0\in \partial M$, by (2.7) we have

$$\begin{aligned} \langle x,\nu \rangle =\langle x,\sin \theta \overline{\nu }-\cos \theta e\rangle =\sin \theta \langle x,\overline{\nu }\rangle . \end{aligned}$$

Hence $\langle x,\overline{\nu }\rangle \big |_{\partial M}$ attains its minimum value at $p_0$. As above, choosing $\{e_\alpha \}_{\alpha =2}^n$ be the orthonormal frame of $\partial \Sigma _t$ in $\mathbb {R}^n$ such that $e_1=\overline{\nu }$, we have

$$\begin{aligned} 0= \nabla ^{\mathbb {R}^n}_{e_\alpha }\langle x,\overline{\nu }\rangle ={\widehat{h}}_{\alpha \beta }\langle x,e_\beta \rangle , \end{aligned}$$

By Proposition 2.4 (2) and Corollary 4.7, we know $({\widehat{h}}_{\alpha \beta })>0$, and hence we have $x\parallel \overline{\nu }$ at $p_0$ and

$$\begin{aligned} \langle x,\overline{\nu }\rangle (p_0) =|x|(p_0)\ge c_0, \end{aligned}$$

for some $c_0>0$, which depends only on the initial datum. Therefore, we finish the proof of (4.9). $\square$

Proposition 4.11

Flow (3.1) exists for all time with uniform $C^{\infty }$-estimates.

Proof

From Proposition 4.2, Proposition 4.10, Proposition 4.7 and Corollary 4.9, we see that $\varphi$ is uniformly bounded in $C^2(\mathbb {S}^n_+\times [0,T^*))$ and the scalar equation in (4.3) is uniformly parabolic. Since $|\cos \theta |<1$, the boundary value condition in (4.3) satisfies the uniformly oblique property. From the standard parabolic theory (see e.g. [20, Theorem 6.1, Theorem 6.4 and Theorem 6.5], also [52, Theorem 5] and [39, Theorem 14.23]), we conclude the uniform $C^\infty$-estimates and the long-time existence of solution to (4.3). $\square$

Proposition 4.12

$x(\cdot , t)$ smoothly converges to a uniquely determined spherical cap around e with capillary boundary, as $t\rightarrow \infty$.

Proof

By Proposition 3.1, we know ${\mathcal {V}}_{1,\theta }(\widehat{\Sigma _t})$ is non-decreasing, due to

$$\begin{aligned} \partial _t {\mathcal {V}}_{1,\theta }(\widehat{\Sigma _t}) =\frac{n}{n+1}\int \limits _{\Sigma _t} \left( \frac{H_1H_{n-1}}{H_n} -1\right) \left( 1+\cos \theta \langle \nu ,e\rangle \right) dA_t \ge 0. \end{aligned}$$

It follows from the long time existence and uniform $C^\infty$-estimates that

$$\begin{aligned} \int \limits _0^{\infty } \partial _t {\mathcal {V}}_{1,\theta }(\widehat{\Sigma _t}) dt \le {\mathcal {V}}_{1,\theta }(\widehat{\Sigma _\infty })<+\infty . \end{aligned}$$

Then we obtain

$$\begin{aligned} \int \limits _{\Sigma _{t_i} } \left( \frac{H_1H_{n-1}}{H_n} -1\right) \left( 1+\cos \theta \langle \nu ,e\rangle \right) dA \rightarrow 0, \hbox { as }t_i\rightarrow +\infty . \end{aligned}$$

Moreover one can show that for any sequence $t_i\rightarrow \infty$, there exists a convergent subsequence, whose limit satisfying

$$\begin{aligned} \left( \frac{H_1H_{n-1}}{H_n} -1\right) \left( 1+\cos \theta \langle \nu ,e\rangle \right) =0. \end{aligned}$$

It is easy to see that the limit is a spherical cap. Next we show that any limit of a convergent subsequence is uniquely determined, which implies the flow smoothly converges to a unique spherical cap. We shall use the argument in [48].

Note that we have proved that $x(\cdot ,t)$ subconverges smoothly to a capillary boundary spherical cap $C_{\rho _\infty ,\theta }(e_\infty )$. Since ${\mathcal {V}}_{n,\theta }$ is preserved along flow (3.1), the radius $\rho _\infty$ is independent of the choice of the subsequence of t. We now show in the following that $e_\infty =e$. Denote $\rho (\cdot ,t)$ be the radius of the unique spherical cap $C_{\rho (\cdot , t), \theta }(e)$ around e with contact angle $\theta$ passing through the point $x(\cdot ,t)$. Due to the spherical barrier estimate, i.e. Proposition 4.2, we know

$$\begin{aligned} \rho _{\max } (t){:}{=}\max \rho (\cdot ,t)=\rho (\xi _t,t), \end{aligned}$$

is non-increasing with respect to t, for some point $\xi _t\in M$. Hence the limit $\lim \limits _{t\rightarrow +\infty } \rho _{\text {max}}(t)$ exists. Next we claim that

$$\begin{aligned} \lim _{t\rightarrow +\infty } \rho _{\max }(t)=\rho _\infty . \end{aligned}$$

(4.10)

We prove this claim by contradiction. Suppose (4.10) is not true, then there exists $\varepsilon >0$ such that

$$\begin{aligned} \rho _{\max }(t)> \rho _\infty +{\varepsilon }, \hbox { for } t \hbox { large enough.} \end{aligned}$$

(4.11)

By definition, $\rho (\cdot ,t)$ satisfies

$$\begin{aligned} \rho ^2\sin ^2\theta =|x|^2-2\rho \cos \theta \langle x,e\rangle . \end{aligned}$$

(4.12)

Hence

$$\begin{aligned} \left( \rho \sin ^2\theta +\cos \theta \langle x,e\rangle \right) \partial _t \rho = \langle \partial _t x, x-\rho \cos \theta e \rangle . \end{aligned}$$

We evaluate at $(\xi _t,t)$. Since $\Sigma _t$ is tangential to $C_{\rho ,\theta }(e)$ at $x(\xi _t, t)$, we have

$$\begin{aligned} \nu _{\Sigma _t}(\xi _t, t)=\nu _{\partial C_{ r,\theta }(e)}(\xi _t, t)= {\frac{x-\rho \cos \theta e}{\rho }}. \end{aligned}$$

Thus we deduce

$$\begin{aligned} {\left( \rho _{\max }\sin ^2\theta +\cos \theta \langle x,e\rangle \right) \partial _t\rho |_{(\xi _t,t)}} = \rho _{\max }\left( \frac{1+\cos \theta \langle \nu , e\rangle }{F}-\langle x,\nu \rangle \right) .\qquad \end{aligned}$$

(4.13)

We note that there exists some $\delta >0$ such that

$$\begin{aligned} {\rho _{\max }\sin ^2\theta +\cos \theta \langle x,e\rangle }\ge \delta >0. \end{aligned}$$

(4.14)

In fact, this follows directly from (4.12), due to

$$\begin{aligned} {\rho \sin ^2\theta +\cos \theta \langle x,e\rangle }=\frac{1}{2\rho }(|x|^2+\rho ^2\sin ^2\theta )\ge \frac{\rho }{2}\sin ^2\theta >0. \end{aligned}$$

(4.15)

Since the spherical caps $C_{\rho _{\max },\theta }(e)$ are the static solutions to (3.1) and $x(\cdot ,t)$ is tangential to $C_{\rho _{\max },\theta }(e)$ at $x(\xi _t,t)$, we see from (2.4)

$$\begin{aligned} \frac{ 1+\cos \theta \langle \nu ,e\rangle }{\langle x,\nu \rangle }\Big |_{x(\xi _t,t)}= \frac{ 1+\cos \theta \langle \nu ,e\rangle }{\langle x,\nu \rangle }\Big |_{C_{\rho _{\max },\theta }(e)}=\frac{1}{\rho _{\max }(t)}. \end{aligned}$$

(4.16)

Since $x(\cdot ,t)$ subconverges to $C_{\rho _\infty ,\theta }(e_\infty )$ and $\rho _\infty$ is uniquely determined, we have

$$\begin{aligned} F=\frac{n\sigma _n}{\sigma _{n-1}}\rightarrow \frac{1}{\rho _\infty } \quad \text {uniformly}, \end{aligned}$$

as $t\rightarrow +\infty$. Thus there exists $T_0>0$ such that

$$\begin{aligned} \frac{1}{F} - {\rho _\infty } <\frac{\epsilon }{2}, \end{aligned}$$

and hence

$$\begin{aligned} \frac{1}{F} - {\rho _{\max }(t)} <-\frac{\epsilon }{2}, \end{aligned}$$

for all $t>T_0$. Taking into account of (4.16), we see

$$\begin{aligned} \left( \frac{1}{F} - {\frac{\langle x,\nu \rangle }{ 1+\cos \theta \langle \nu ,e\rangle }}\right) \Big |_{x(\xi _t,t)} <-\frac{\epsilon }{2}, \end{aligned}$$

(4.17)

for all $t>T_0$. By adopting Hamilton’s trick, we conclude from (4.13), (4.14) and (4.17) that there exists some $C>0$ such that for almost every t,

$$\begin{aligned} \frac{d}{dt}\rho _{\max }\le -C\epsilon . \end{aligned}$$

This is a contradiction to the fact that $\lim \limits _{t\rightarrow +\infty }\frac{d}{dt} \rho _{\max }=0$, and hence claim (4.10) is true. Similarly, we can obtain that

$$\begin{aligned} \lim _{t\rightarrow +\infty } \rho _{\min }(t)=\rho _\infty . \end{aligned}$$

(4.18)

Hence $\lim \limits _{t\rightarrow \infty } \rho (\cdot ,t)=\rho _\infty$. This implies that any limit of a convergent subsequence is the spherical cap around e with radius $\rho _{\infty }$. We complete the proof of Proposition 4.12. $\square$

In view of Proposition 4.11 and Proposition 4.12, Theorem 4.1 are proved.

5 Alexandrov–Fenchel inequalities

In this section, we apply the convergence result of flow (3.1) to prove Theorem 1.2.

Proof of Theorem 1.2

Remember

$$\begin{aligned} {\mathcal {V}}_{k,\theta }(\widehat{C_{ r,\theta } })= r^{n+1-k} {\textbf{b}}_\theta , \end{aligned}$$

(5.1)

where ${\textbf{b}}_\theta$ was defined by (2.5).

Assume that $\Sigma$ is strictly convex. We have proved in Sect. 4 that flow (3.1) converges a spherical cap, which we denote by $C_{r_\infty ,\theta }(e)$. By the monotonicity of ${\mathcal {V}}_{n,\theta }$ and ${\mathcal {V}}_{k,\theta }$, Proposition 3.1 we have

$$\begin{aligned} {\mathcal {V}}_{n,\theta } ({\widehat{\Sigma }}) = {\mathcal {V}}_{n,\theta } (\widehat{C_{r_\infty ,\theta }(e)} ), \qquad {\mathcal {V}}_{k,\theta } ({\widehat{\Sigma }}) \le {\mathcal {V}}_{k,\theta } (\widehat{C_{r_\infty ,\theta }(e)} ), \end{aligned}$$

moreover, equality holds iff $\Sigma$ is a spherical cap. It is clear that (5.1) is the same as (1.11).

When $\Sigma$ is convex but not strictly convex, the inequality follows by approximation. The equality characterization can be proved similar to [48, Sect. 4], by using an argument of [29]. We omit the details here. $\square$

For Corollary 1.4, one just notes that when $n=2$,

$$\begin{aligned} {\textbf{b}}_\theta =\frac{1}{3} \left( 2-3\cos \theta +\cos ^3\theta \right) \pi . \end{aligned}$$