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

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.


Introduction
Let Σ be a closed embedded hypersurface in R n+1 and Σ the domain enclosed by Σ in R n+1 .The classical isoperimetric inequality states with equality holds if and only if Σ is a sphere.Here b n+1 = |B n+1 |, the volume of the unit ball B n+1 , and ω n = (n+1)b n+1 = |S n |, the area of the unit sphere S n .Its natural generalization is the following classical Alexandrov-Fenchel inequality with equality holds if and only if Σ is a sphere, provided that Σ is a C 2 convex hypersurface.Here V k+1 ( Σ) is the quermassintegral of Σ defined by where H k (1 ≤ k ≤ n) is the k-th normalized mean curvature of Σ ⊂ R n+1 and H 0 = 1.It was proved in [29] that (1.2) holds true if Σ is l-convex and star-shaped.Here by l-convex we mean that H j > 0 for all j ≤ 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 R where B n+1 + is the upper half unit ball and S n + is the upper half unit sphere.For the relative isoperimetric inequality outside a convex domain, see [17,40,25].As a domain in R n+1 , Σ has a boundary, which consists of two parts: one is Σ and the other, which will be denoted by ∂Σ, lies on ∂R n+1 + .Both have a common boundary, namely ∂Σ.(See Figure 1 below.)Instead of just considering the area of Σ, it is interesting to consider the following free energy functional |Σ| − cos θ| ∂Σ| (1.5) for a fixed angle constant θ ∈ (0, π).The second term cos θ| ∂Σ| 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 | Σ| is fixed, then we have the following optimal inequality, which is called the capillary isoperimetric inequality, |, ω θ := ω n,θ := |S n θ | − cos θ| ∂S n θ |. (1.7) The explicit formulas for b θ and ω θ will be given in Section 2.3 below.In particular, it is easy to check that (n + 1)b θ = ω θ .The proof of (1.6) is not trivial, which uses the spherical symmetrization.See for example [41,Chapter 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 V k for hypersurfaces with boundary supported on ∂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.
and for 1 ≤ k ≤ n, where H ∂Σ k−1 is the normalized (k − 1)-th mean curvature of ∂Σ ⊂ R n (see Section 2 for details).In particular, one has .
We believe that these quantities are the suitable quermassintegrals for hypersurfaces with boundary intersecting with ∂R n+1 + at angle θ ∈ (0, π), which is supported by the following result.
be a family of smooth, embedded capillary hypersurfaces with a constant contact angle θ ∈ (0, π), which are given by the embedding x(•, t) : M → R n+1 + and satisfy and in particular with boundary supported on ∂R n+1 + is called capillary hypersurface if it intersects with ∂R n+1 + at a constant angle.For closed hypersurfaces in 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.
be a convex capillary hypersurface with a constant contact angle θ ∈ (0, π 2 ], then there holds with equality if and only if Σ is a spherical cap in (2.3).Moreover, (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 θ.Moreover, it is equivalent to a Gauss-Bonnet type result for capillary hypersurfaces with contact angle θ.
with equality holds if and only if Σ is spherical cap in (2.3).
Here H = 2H 1 is the ordinary mean curvature for surfaces and it is obvious that H 2 ≥ 4H 2 .When n = 2, (1.11) implies a Minkowski type inequality for convex capillary surfaces with boundary in R where Moreover, equality holds if and only if Σ is a spherical cap in (2.3).
From these results, it is natural to propose be a convex capillary hypersurface with a contact angle θ ∈ (0, π), there holds with equality iff Σ is a spherical cap in (2.3).
It would be also interesting to ask further if the conjecture is true for kconvex capillary hypersurfaces.In order to prove the conjecture, we introduce a suitable nonlinear curvature flow, which preserves V l,θ ( Σ) and increases V k,θ ( Σ), see Section 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 θ ∈ (0, π 2 ], namely, Theorem 1.2.To be more precise, let us first recall the related work on the proof of Alensandrov-Fenchel inequalities by using geometric curvature flow for closed hypersurfaces in R n+1 .When ∂Σ = ∅, and we denote Σ be the bounded convex domain enclosed by Σ in R n+1 .In convex geometry, the Alexandrov-Fenchel inequalities (1.2) between quermassintegrals V k and 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 Σ is k-convex and star-shaped by using the inverse curvature flow, which is defined by (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 see [31,32] for instance.The motivation to use such a flow (1.18) is its nice properties that the quermassintegrals V k ( Σ) is preserved and V k+1 ( Σ) is nondecreasing 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 [7,6,10,19,26,33,34,38,49,55] 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 ∂Σ = ∅, 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 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 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 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 R n+1 + and ν its unit normal vector field.We introduce Using the Minkowski formulas given in (2.9), we show that flow (1.19) preserves V l,θ ( Σ), while increases V k,θ ( Σ) 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 Hn 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, ∇ µ H, has a bad sign, if θ > π 2 .Hence we have to restrict ourself on the range θ ∈ (0, π 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 θ > π 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 θ ≤ π 2 could have different geometry from that with θ > π 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 V n and 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 Section 2, we introduce the quermassintegrals for capillary hypersurfaces and collect the relevant evolution equations to finish the proof of Theorem 1.1.In Section 3, we introduce our nonlinear inverse curvature flow and show the monotonicty of our quermassintegrals (1.8) under the flow.In Section 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.

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 ∂M , and x : M → R n+1 + be a proper smooth immersed hypersurface.In particular, x(int(M )) ⊂ R n+1 + and x(∂M ) ⊂ ∂R n+1 + .Let Σ = x(M ) and ∂Σ = x(∂M ).If no confusion, we will do not distinguish the hypersurfaces Σ and the immersion x : M → R n+1 + .Let Σ be the bounded domain enclosed by Σ and ∂R n+1 + .Let ν and N be the unit outward normal of Σ ⊂ Σ and ∂R n+1 2.1.Higher order mean curvatures.
We shall use the following basic properties about σ k .
Proposition 2.1. ( with equality holds if and only if These are well-known properties.For a proof we refer to [39, Chapter XV, Section 4] and [51, Lemma 2.10, Theorem 2.11] respectively.
We use D to denote the Levi-Civita connection of R n+1 + with respect to the Euclidean metric δ, and ∇ the Levi-Civita connection on Σ with respect to the induced metric g from the immersion x.The operator div, ∆, and ∇ 2 are the divergence, Laplacian, and Hessian operator on Σ respectively.The second fundamental form h of x is defined by the set of principal curvatures, i.e, the set of eigenvalues of h.Then we denote σ k = σ k (κ) and H k = H k (κ) resp.be the k-th mean curvature and the normalized k-th mean curvature of Σ.We also use the convention that 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 ∈ T k,l (Σ), the expression T i 1 ...i k j 1 ...j l denotes T i 1 ...i k j 1 ...j l = T (e j 1 , . . ., e j l , i 1 , . . .i k ), where (e i ) is a local frame and ( i ) its dual coframe.
(3) We shall use the convention of the Einstein summation.For convenience the components of the Weingarten map 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 Σ.
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 R n+1 , see e.g.[50].Given a bounded convex domain Σ ⊂ R n+1 with smooth boundary ∂ Σ, its k-th quermassintegral is defined by where for a family of bounded convex bodies { Σ t } in R n+1 whose boundary ∂ Σ 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 Σ with capillary boundary in Here In particular, we have Here H is the (un-normalized) mean curvature, i.e.H = nH 1 .From Gauss-Bonnet-Chern's theorem, we know As a result, we see .
2.3.Spherical caps.Let e := −e n+1 = (0, . . ., 0, −1).We consider a family of spherical caps lying entirely in R n+1 + and intersecting R n with a constant contact angle θ ∈ (0, π) given by which has radius r and centered at r cos θe.To emphasize e and to distinguish with the center of the spherical cap, r cos θe, we call C r,θ (e) a spherical cap around e. If without confusion, we just write C r,θ for C r,θ (e) in the rest of this paper.One can easily check that C r,θ is the static solution to flow (3.1) below, that is, and it intersects with the support where b θ is the volume of C 1,θ , which is congruent to S θ , defined in the introduction.One can compute and 2 ) is the regularized incomplete beta function given by Moreover, one can readily check that Therefore, C r,θ achieves equality in the Alexandrov-Fenchel inequalities (1.11).The second fundamental form of ∂Σ in R n is given by The second equality holds since ν, N • x = 0.The second fundamental form of ∂Σ in Σ is given by The second equality holds since ν, µ = 0.
Corollary 2.5.If Σ is a convex capillary hypersurface, then ∂Σ ⊂ ∂R n+1 is also convex, i.e., h ≥ 0, while The following Minkowski type formulas for capillary hypersurfaces play an important role in this paper.
where dA is the area element of Σ w.r.t. the induced metric g.
Using integration by parts, we have From (2.12), we know that (x T + cos θP T e ) ⊥ µ along ∂Σ.Since µ is a principal direction of Σ by Proposition 2.4, we have σ ij k−1 (x T + cos θP T e ) j • µ i = 0 along ∂Σ.It is well-known that the Newton tensor is divergence-free, i.e., ∇ i σ ij k−1 = 0. Altogether yields the conclusion.

Variational formulas. The following first variational formula motivates us to define the quermassintegrals for capillary hypersurfaces as (1.8).
Theorem 2.7.Let Σ t ⊂ R n+1 + be a family of smooth capillary hypersurfaces supported by ∂R n+1 + with a constant contact angle θ ∈ (0, π) along ∂Σ t , given by the embedding x(•, t) : M → R n+1 + , and satisfying for a smooth function f .Then for ) Before proving Theorem 2.7, we remark that if is a family of smooth capillary hypersurfaces evolving by (2.13), then the tangential component (∂ t x) T of ∂ t x, which we denote by T ∈ T Σ t , must satisfy where T ∈ T (∂Σ t ).In fact, the restriction of x(•, t) on ∂M is contained in R n and hence, , and hence (2.15).Up to a diffeomorphism of ∂M , we can assume T = 0.For simplicity, in the following, we always assume that Hence, from now on, let Σ t be a family of smooth, embedding hypersurfaces with θ-capillary boundary in R n+1 + , given by the embeddings x(•, t) : M → R n+1 + , which evolves by the general flow with T ∈ T Σ 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 ν, the second fundamental form h ij , the Weingarten matrix h i j , the mean curvature H, the k-th mean curvature σ k and F := F (h j i ) of the hypersurfaces Σ t .These evolution equations will be used later.
Proposition 2.8.Along flow (2.17), it holds that (1) ( The proof of Proposition 2.8 for T = 0 can be found for example in [27, Chapter 2, Section 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 α } n α=2 of T ∂Σ ⊂ T R n such that {e 1 := µ, (e α ) n α=2 } forms an orthonormal frames for T Σ.First, by taking time derivative to the capillary boundary condition, ν, N • x = − cos θ along ∂Σ, we obtain where we have used (2.7), Proposition 2.8 and T | ∂M = f cot θµ.As a result, (2.18) Next, using integration by parts and Proposition 2.8 we have where we have used T | ∂M = f cot θµ, (2.18) and Lemma 2.1 (1), ( 4).Moreover flow (3.1) induces a hypersurface flow ∂Σ t ⊂ R n with normal speed f sin θ , that is, By (2.2), we have From Proposition 2.4 (2), we know and hence σ k (h|h 11 ) = sin k θσ k ( h).Substituting these formulas into (2.19),we obtain By the definition of V k+1,θ ( Σ t ) in (1.8), we get the desired formula (2.14) for k ≥ 0. It remains to consider the case k = −1.It is easy to check that x, ν dA t .
A direct computation gives Now we complete the proof.

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 and intersects with ∂R n+1 + at a constant contact angle θ ∈ (0, π).We consider the smooth family of embeddings x : M × [0, T ) → R n+1 + , satisfying the following evolution equations with x(M, 0) = x 0 (M ) and where 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 Proof.From Theorem 2.7, we see where the last equality follows from (2.9).For 1 ≤ k < l ≤ n, from Theorem 2.7 where we have used the Newton-MacLaurin inequality (2.1) and the Minkowski formula (2.9) in the last two steps respectively.

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., under an angle constraint θ ∈ (0, is an embedding of a strictly convex capillary hypersurface in the half-space with the contact angle θ ∈ (0, π 2 ].Then there exists x : M × [0, +∞) → 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(•, t) → x ∞ (•) in C ∞ topology as t → +∞, and the limit 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 Σ is strictly star-shaped with respect to the origin.One can reparametrize it as a graph over S n + .Namely, there exists a positive function r defined on We denote ∇ 0 be the Levi-Civita connection on S n + with respect to the standard round metric σ := g i r, and r ij := ∇ 0 i ∇ 0 j r.The induced metric g on Σ is given by where ϕ(X) := log r(X).Its inverse g −1 is given by where r i := σ ij r j , ϕ i := σ ij ϕ j and v := 1 + |∇ 0 ϕ| 2 .The unit outward normal vector field on Σ is given by The second fundamental form h on Σ is and its Weingarten matrix h i j is The higher order mean curvature H k can also be expressed by ϕ.Moreover, In order to express the capillary boundary condition in terms of the radial function ϕ, we use the polar coordinate in the half-space.For In these coordinates the standard Euclidean metric is given by Along ∂S n + it holds that is, Therefore, in the class of star-shaped hypersurfaces flow (3.1) is reduced to the following scalar parabolic equation with an oblique boundary condition where ϕ 0 is the parameterization radial function of x 0 (M ) over S n + , and Since | cos θ| < 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 × [0, T * ).The positivity of F implies that Σ t is strictly convex up to T * .The convexity of Σ 0 implies that there exists some 0 The family of C r,θ 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 ∈ [0, T * ), Σ t satisfies Proof.Recall that C r,θ 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 [ Set F := n i=1 F i i .Using Proposition 2.1 we have and Proof.Using the Codazzi formula, we have Combining with Proposition 2.8, we obtain Hence it follows Along ∂Σ t , from (2.18) we know Using (2.7) and Proposition 2. where we used e, ν = 0 and e, x = 0 on ∂Σ t .One can easily check that the left hand side of the previous formula equals to cot θh(µ, µ)(1 + cos θ ν, e ), on ∂Σ.Hence it follows that 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].The Ricci equation and the Codazzi equation yield Hence we have where the last inequality follows from the concavity of F , and the convexity of ∂Σ ⊂ Σ, see Corollary 2.5.Hence (4.8) is proved.
In particular, from the uniform lower bound of F := Hn H n−1 , we get a uniform curvature positive lower bound.
Corollary 4.7.Σ t , t ∈ [0, T * ) is uniformly convex, that is, there exists c > 0 depending only on Σ 0 , such that the principal curvatures of Σ t , Next we obtain the uniform bound of the mean curvature.Proposition 4.8.There exists C > 0 depending only on Σ 0 , such that Proof.From (4.8), we know that ∇ µ H ≤ 0 on ∂Σ t .Thus H attains its maximum value at some interior point, say p 0 ∈ int(M ).We now compute at p 0 .

One can check
for some positive constant c 0 .Combining with Proposition 4.6, it implies for some positive constant C > 0. Therefore, which yields that H is uniformly bounded from above.
It follows from the long time existence and uniform C ∞ -estimates that

Then we obtain
Σt Moreover one can show that for any sequence t i → ∞, there exists a convergent subsequence, whose limit satisfying 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(•, t) subconverges smoothly to a capillary boundary spherical cap C ρ∞,θ (e ∞ ).Since V n,θ is preserved along flow (3.1), the radius ρ ∞ is independent of the choice of the subsequence of t.We now show in the following that e ∞ = e.Denote ρ(•, t) be the radius of the unique spherical cap C ρ(•,t),θ (e) around e with contact angle θ passing through the point x(•, t).Due to the spherical barrier estimate, i.e.Proposition 4.2, we know ρ max (t) := max ρ(•, t) = ρ(ξ t , t), is non-increasing with respect to t, for some point ξ t ∈ M .Hence the limit lim t→+∞ ρ max (t) exists.Next we claim that We prove this claim by contradiction.Suppose (4.10) is not true, then there exists ε > 0 such that ρ max (t) > ρ ∞ + ε, for t large enough.We evaluate at (ξ t , t).Since Σ t is tangential to C ρ,θ (e) at x(ξ t , t), we have ν Σt (ξ t , t) = ν ∂C r,θ (e) (ξ t , t) = x − ρ cos θe ρ .
In view of Proposition 4.11 and Proposition 4.12, Theorem 4.1 are proved.

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 where b θ was defined by (2.5).Assume that Σ is strictly convex.We have proved in Section 4 that flow (3.1) converges a spherical cap, which we denote by C r∞,θ (e).By the monotonicity of V n,θ and V k,θ , Proposition 3.1 we have V k,θ ( Σ) ≤ V k,θ ( C r∞,θ (e)), moreover, equality holds iff Σ is a spherical cap.It is clear that (5.1) is the same as (1.11).When Σ is convex but not strictly convex, the inequality follows by approximation.The equality characterization can be proved similar to [48,Section 4], by using an argument of [29].We omit the details here.
Conflict of interest.On behalf of all authors, the corresponding author states that there is no conflict of interest.
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
The authors have no relevant financial or non-financial interests to disclose.

Figure 1 .
Figure 1.A capillary hypersurface Σ with a contact angle θ

4. 4 .Proposition 4 . 6 .
Curvature estimates.First, we have the uniform bound of F , which follows directly from Proposition 4.3 and the maximum principle.Along flow (3.1), it holds min