Stability of the Quermassintegral Inequalities in Hyperbolic Space

For the quermassintegral inequalities of horospherically convex hypersurfaces in the (n+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n+1)$$\end{document}-dimensional hyperbolic space, where n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 2$$\end{document}, we prove a stability estimate relating the Hausdorff distance to a geodesic sphere by the deficit in the quermassintegral inequality. The exponent of the deficit is explicitly given and does not depend on the dimension. The estimate is valid in the class of domains with upper and lower bound on the inradius and an upper bound on a curvature quotient. This is achieved by some new initial value-independent curvature estimates for locally constrained flows of inverse type.


Introduction
The isoperimetric inequality is a fundamental result in geometry that relates the volume of a region in the Euclidean, or also in some non-flat spaces, to the surface area of its boundary.In the Euclidean setting, among all bounded domains Ω ⊂ R n+1 , n ≥ 1, there holds with equality only when Ω is a geodesic ball.Here ω n+1 is the volume of the (n + 1)-dimensional unit ball and |•| stands for the Hausdorff measure of the appropriate dimension.Equality in this inequality is attained if and only if Ω is a ball.Hence it is natural to investigate the stability question, namely how close is Ω to a geodesic ball, provided the deviation in (1.1) from the equality case is small.For the isoperimetric inequality this question has been addressed to great extent, e.g.[3,4,11] and we are not attempting a more detailed overview here.
The quermassintegral inequalities are a generalisation of the isoperimetric inequality.They are a collection of geometric inequalities that interrelate the coefficients in the Steiner formula, which is the Taylor expansion of the volume of outer parallel bodies of a convex body K ⊂ R n+1 , vol(K + ρB) see [14, p. 208].
In the Euclidean space the W m can be expressed as curvature integrals and the corresponding inequalities are written as follows: where Ω ⊂ R n+1 is a convex bounded domain and E m is the (normalised) elementary degree m symmetric polynomial of principal curvatures of ∂Ω as an embedding in R n+1 .The convexity assumption was relaxed to mconvex and starshaped in [7].In the convex class, the stability for the inequalities has been thoroughly investigated for example in [6,13], while in the non-convex case the only available result seems to be that of the second author [12].The purpose of this paper is the transfer of such investigations into the (n + 1)-dimensional hyperbolic space, where the quermassintegral inequalities were proved by Wang/Xia for horospherically convex domains [15, Thm.1.1], by using a suitable curvature flow.They proved that if Ω is a bounded smooth h-convex (i.e.all principal curvatures are greater than 1) domain in H n+1 , then there holds Equality holds if and only if Ω is a geodesic ball.Here W m is the m th quermassintegral in H n+1 (see section 2 for the definition), f m (r) = W m (B r ), and f −1 l is the inverse function of f l .Hu/Li/Wei gave an alternative proof by using a different flow [10].We will review their method later, as we are going to use the same flow for our result.In this paper we study the stability of these inequalities in the hyperbolic space.In particular we prove the following result, which controls the Hausdorff distance of an h-convex hypersurface in H n+1 to a geodesic sphere by the deviation of the inequality (1.2) from the equality case: and a geodesic sphere S H such that Here ρ − (Ω) is the inradius of the domain Ω.The dependence of C on ρ − (Ω) means that we neither control C when ρ − (Ω) tends to zero, nor when it tends to infinity.1.2.Remark.(i) Note that the curvature dependence of C does allow for curvature blowup in a certain sense.Namely, the quantity E m /E m−1 may remain bounded, even if |A| 2 becomes unbounded, as can be seen from the example n − 1 = m = 2, for which unless merely κ 2 goes to infinity.(ii) Also note that we do not assume ∂Ω to be nearly spherical, as it is done for example in the recent paper [16], where the authors a priori assume W 2,∞ closeness to a sphere and obtain stability of the Fraenkel asymmetry.
In particular, from the previous theorem we get an estimate in terms of W 2 and W 1 with exponent 1/3, if we choose m = 1 and impose a bound on the mean curvature H = nE 1 .It turns out the under the same assumption we can extend this to arbitrary m with the same exponent.
The idea of the proof combines two major inputs drawn from different directions.The first one, which is also deeply involved in the actual proof of the quermassintegral inequalities (1.2), is the use of a suitable curvature flow to be defined later, which preserves W m (Ω) and decreases W m+1 (Ω).The flow exists for all times and converges to a geodesic sphere.This proves the inequality.To characterize the equality case, it is observed that W m+1 (Ω) is only strictly decreasing, when the traceless second fundamental form is not zero.For the proof of (1.2) this was sufficient, but for the proof of (1.3) we will make this quantitative and obtain an estimate on the traceless second fundamental form.The second input is an estimate relating the Hausdorff distance to a geodesic sphere with the traceless second fundamental form.Such an estimate, in the form in which we need it, is due to De-Rosa/Gioffré [2].Combination of these two ingredients will complete the proof.
After reviewing preliminaries in section 2, we prove new a priori estimates for the locally constrained flow of h-convex hypersurfaces in section 3, which are of independent interest.In section 4 we complete the proof.space H n+1 as the warped product manifold, coming from polar coordinates around a given origin o, where λ(r) = sinh(r) and g S n is the standard round metric on the ndimensional unit sphere.We will also occasionally write •, • for ḡ.In this paper, d H n+1 will always denote the geodesic distance of two points in hyperbolic space, while denotes the Hausdorff distance of two compact sets.
The vector field λ∂ r on H n+1 is a conformal Killing field, i.e.
where ∇ is the Levi-Civita connection of ḡ.
Let M be a smooth closed hypersurface in H n+1 with outward unit normal ν, then we define the support function of the hypersurface by Writing (g ij ) for the metric induced on M with inverse (g ij ) and Levi-Civita connection ∇, h ij the second fundamental form and A = (h i j ) = (g ik h kj ) the Weingarten operator, we have the following equation, which follows from the conformal Killing property and the Weingarten equation: (2.1) Using the change of variables, where As a result, the hyperbolic space can now be viewed as a conformally flat space.We will need a simple lemma about the surface area of a submanifold of H n+1 , when viewed as a Euclidean submanifold.Then the Euclidean conformal image Proof.We have with some local parametrisation X : The notion of convexity by horospheres or short h-convexity, is crucial for our result: Such h-convex domains already enjoy a quite rigid geometry, and several of their geometric quantities are already controlled by the inradius: Let ρ − (Ω) be the inradius of Ω, i.e. the largest number, such that a ball of radius equal to that number fits into Ω.Let o be the center of that ball.In [1, Thm.1] it is shown that then Furthermore one can extract an estimate on the support function.Due to (2.1), where u attains a minimum, ∇r must be zero, since A is invertible.However, min ∂Ω r = ρ − (Ω) and hence The h-convexity of a hypersurface of H n+1 translates into convexity of the conformal image: Proof.We have Hence the second fundamental form is positive definite.Now we define the hyperbolic quermassintegrals.For any smooth body Ω in the hyperbolic space H n+1 with boundary M = ∂Ω, the k th quermassintegrals W k is defined inductively as follows: where Here E k is the normalized elementary symmetric polynomial in n-variables In this paper, we use the curvature functions For us, only the properties on the positive cone Γ + ⊂ R n matter, where these functions are monotone, i.e.
∂F ∂κ i > 0 and concave.We may also understand these functions as being defined on the Weingarten operator, or on the second fundamental form and the metric, Then we write and there holds We refer to [5, Ch. 2] for a thorough treatment.

New a priori estimates for the locally constrained flow
Wang/Xia [15] proved the quermassintegral inequalities (1.2) in the hyperbolic space by using the following flow: Let M 0 = ∂Ω be a smooth, h-convex hypersurface in H n+1 with Then the flow is defined as where ν is the outward normal to the hypersurface, and c(t) is chosen such that the l th quermassintegral is preserved under this flow.
The same inequality (1.2) was proved by Hu/Li/Wei [10] where they used a different flow: with the notation from section 2. This flow preserves the m th quermassintegral W m (Ω t ) and decreases W m+1 (Ω t ) monotonically.
We will quantify the proofs from [9] and [10] and employ the flow (3.1) to extract information on the size of the traceless second fundamental form.To exploit this further, we will use the result from De Rosa/Gioffrè's paper [2].The closeness of the hypersurface to a geodesic sphere can be controlled by the L p norm of the traceless second fundamental form Å, whenever Å is small.Their result is only for the Euclidean space, however we note that up to a term coming from the conformal factor, the traceless second fundamental form is conformally invariant, and hence the umbilicity in the Euclidean and the hyperbolic space are comparable.We will point out the necessary details whenever appropriate.We will also need some refined curvature estimates, which do not depend on their initial values.Therefore we require some evolution equations and additional a priori estimates, which we develop in the sequel.
It is known that the flow (3.1) has arbitrary spheres as barriers, i.e. for all (t, ξ) ∈ [0, ∞) × S n there holds due to (2.3), Since the flow preserves the h-convexity, we also obtain a uniform C 1 -bound via Let us define the operator 3.1.Lemma.Along the flow (3.1), the induced metric g = (g ij ) and second fundamental form (h ij ) satisfy the following equations, see [10, Lemma 3.1] Proof.We use

3.3.
Corollary.Along the flow (3.1), the curvature function satisfies the estimate Proof.The lower bound follows immediately from the h-convexity and the monotonicity of F .For the upper bound, we use the estimates from [10, Cor.2.3], which give We conclude that at maximal points of F we have LF ≤ 0 and the result follows from the maximum principle.
Proof.Using the evolution of g ij , we can easily find the evolution of Hence 3.5.Corollary.Along the flow (3.1) and up to time t = 1, the curvature function satisfies the estimate Proof.We proceed similarly to the proof of Corollary 3.3.At maximal points of H we have, using |A| 2 ≥ H 2 /n and the concavity of F , where in the last step we used Corollary 3.3.We have also used Cauchy-Schwarz to absorb ∇F and first order terms in H.The result again follows from a simple ODE comparison argument.
4. Proof of Theorem 1.1 and Theorem 1.3 In this section, we prove Theorem 1.1.In the following proof, we take C = C (n, ρ − (Ω), max ∂Ω F ) to be a generic constant depending on the quantities mentioned.
We have also used that Ω converges to a round ball at infinite time, Ω ∞ = B where (1.2) holds with equality, and W m is preserved under the flow, W m (B) = W m (Ω).
Along the flow we have and hence, using where we used (3.2) and [1, Thm.1].
Using the above bound, we want to estimate the Hausdorff distance between M t and M 0 = ∂Ω.Let X(ξ, 0) and X(ξ, t) be two points in M 0 and M t respectively.Let γ : [0, t] → H n+1 be a curve defined as γ(τ ) = X(ξ, τ ).
Note that |M t δ | is controlled from above and below in terms of ρ − (Ω), due to the convergence of the surface area-preserving curvature flow which converges to a geodesic sphere with radius between ρ − (Ω) and ρ − (Ω)+ log 2. Due to Lemma 2.1 we have γ = γ(n, ρ − (Ω)).[2, Thm.1.2] gives, provided that ǫ ≤ ǫ 0 (n, ρ − (Ω), max ∂Ω F ) with ǫ 0 sufficiently small, a parametrization and a point O ∈ R n+1 , such that and f satisfies the estimate This implies that Mt δ is Hausdorff-close to the Euclidean unit sphere, that Mt δ is close to a Euclidean sphere of radius γ −1 and that in turn M t δ is close to a hyperbolic sphere, with exactly the same error estimate, dist(M t δ , S H ) ≤ Cǫ Employing (4.4) finishes the proof for ǫ ≤ ǫ 0 .However, if ǫ > ǫ 0 , the estimate is trivial due to max ∂Ω r ≤ ρ − (Ω) + log 2.
To prove Theorem 1.3, we reconvene at (4.3) and do not estimate max E m using Corollary 3.5, but the constant itself is now allowed to depend on H. Hence the factor δ −m+1 is simply not present and in the subsequent computations we can pretend m would be one.The proof can then literally be completed as above.

2. 1 .
Lemma.Let (M, g) be the embedding of a compact smooth manifold M into H n+1 with max M r ≤ Λ 0 .

see [ 5 ,
Equ. (1.1.51)].There holds φ = log e φ = log 4 − log(4 − ρ 2 ) and hence 1.2], we view M t δ as a Riemannian submanifold of the Euclidean ball of radius 2, which is conformal to H n+1 as in (2.2).Due to Lemma 2.3 and furnishing the Euclidean geometric tensors by a tilde, we see that Mt δ is convex.Now we have to normalize Mt δ ,