Hypersurfaces In Hyperbolic Space Research Articles

Volume preserving mean curvature flow of L2-almost umbilical hypersurfaces in hyperbolic space

In this paper, we study the stability of the volume preserving mean curvature flow of closed hypersurfaces in the hyperbolic space. We prove that an [Formula: see text]-almost umbilical hypersurface will be deformed to a totally umbilical hypersurface along the flow. Our result removes the assumption on the mean curvature in the theorems of Huang-Lin-Zhang [Peking J. Math. (2023)] and Leng-Xu-Zhao [Int. J. Math. (2014)].

Relative entropy of hypersurfaces in hyperbolic space

Abstract We study a notion of relative entropy for certain hypersurfaces in hyperbolic space. We relate this quantity to the renormalized area introduced by Graham–Witten. We also obtain a monotonicity formula for relative entropy applied to mean curvature flows in hyperbolic space.

Shifted inverse curvature flows in hyperbolic space

We introduce the shifted inverse curvature flow in hyperbolic space. This is a family of hypersurfaces in hyperbolic space expanding by $F^{-p}$ with positive power $p$ for a smooth, symmetric, strictly increasing and $1$-homogeneous curvature function $f$ of the shifted principal curvatures with some concavity properties. We study the maximal existence and asymptotical behavior of the flow for horo-convex hypersurfaces. In particular, for $0<p\leq 1$ we show that the limiting shape of the solution is always round as the maximal existence time is approached. This is in contrast to the asymptotical behavior of the (non-shifted) inverse curvature flow, as Hung and Wang [18] constructed a counterexample to show that the limiting shape of inverse curvature flow in hyperbolic space is not necessarily round.

On the Gauss map of equivariant immersions in hyperbolic space

Given an oriented immersed hypersurface in hyperbolic space H n + 1 $\mathbb {H}^{n+1}$ , its Gauss map is defined with values in the space of oriented geodesics of H n + 1 $\mathbb {H}^{n+1}$ , which is endowed with a natural para-Kähler structure. In this paper, we address the question of whether an immersion G $G$ of the universal cover of an n $n$ -manifold M $M$ , equivariant for some group representation of π 1 ( M ) $\pi _1(M)$ in Isom ( H n + 1 ) $\mathrm{Isom}(\mathbb {H}^{n+1})$ , is the Gauss map of an equivariant immersion in H n + 1 $\mathbb {H}^{n+1}$ . We fully answer this question for immersions with principal curvatures in ( − 1 , 1 ) $(-1,1)$ : while the only local obstructions are the conditions that G $G$ is Lagrangian and Riemannian, the global obstruction is more subtle, and we provide two characterizations, the first in terms of the Maslov class, and the second (for M $M$ compact) in terms of the action of the group of compactly supported Hamiltonian symplectomorphisms.

On n-superharmonic functions and some geometric applications

In this paper we study asymptotic behaviors of n-superharmonic functions at singularity using the Wolff potential and capacity estimates in nonlinear potential theory. Our results are inspired by and extend [6] of Arsove–Huber and [63] of Taliaferro in 2 dimensions. To study n-superharmonic functions we use a new notion of thinness in terms of n-capacity motivated by a type of Wiener criterion in [6]. To extend [63], we employ the Adams–Moser–Trudinger’s type inequality for the Wolff potential, which is inspired by the inequality used in [15] of Brezis–Merle. For geometric applications, we study the asymptotic end behaviors of complete conformally flat manifolds as well as complete properly embedded hypersurfaces in hyperbolic space. These geometric applications seem to elevate the importance of n-Laplace equations and make a closer tie to the classic analysis developed in conformal geometry in general dimensions.

Volume preserving flow and Alexandrov–Fenchel type inequalities in hyperbolic space

In this paper, we study flows of hypersurfaces in hyperbolic space, and apply them to prove geometric inequalities. In the first part of the paper, we consider volume preserving flows by a family of curvature functions including positive powers of k -th mean curvatures with k=1,\ldots,n , and positive powers of p -th power sums S_p with p &gt; 0 . We prove that if the initial hypersurface M_0 is smooth and closed and has positive sectional curvatures, then the solution Mt of the flow has positive sectional curvature for any time t &gt; 0 , exists for all time and converges to a geodesic sphere exponentially in the smooth topology. The convergence result can be used to show that certain Alexandrov–Fenchel quermassintegral inequalities, known previously for horospherically convex hypersurfaces, also hold under the weaker condition of positive sectional curvature. In the second part of this paper, we study curvature flows for strictly horospherically convex hypersurfaces in hyperbolic space with speed given by a smooth, symmetric, increasing and degree one homogeneous function f of the shifted principal curvatures \lambda_i=\kappa_i-1 , plus a global term chosen to impose a constraint on the quermassintegrals of the enclosed domain, where f is assumed to satisfy a certain condition on the second derivatives. We prove that if the initial hypersurface is smooth, closed and strictly horospherically convex, then the solution of the flow exists for all time and converges to a geodesic sphere exponentially in the smooth topology. As applications of the convergence result, we prove a new rigidity theorem on smooth closed Weingarten hypersurfaces in hyperbolic space, and a new class of Alexandrov–Fenchel type inequalities for smooth horospherically convex hypersurfaces in hyperbolic space.

Cylindrical estimates for mean curvature flow in hyperbolic spaces

<p style='text-indent:20px;'>We consider the mean curvature flow of a closed hypersurface in hyperbolic space. Under a suitable pinching assumption on the initial data, we prove a priori estimate on the principal curvatures which implies that the asymptotic profile near a singularity is either strictly convex or cylindrical. This result generalizes the estimates obtained in the previous works of Huisken, Sinestrari and Nguyen on the mean curvature flow of hypersurfaces in Euclidean spaces and in the spheres.

Weighted Alexandrov–Fenchel inequalities in hyperbolic space and a conjecture of Ge, Wang, and Wu

We consider a conjecture made by Ge, Wang, and Wu regarding weighted Alexandrov–Fenchel inequalities for horospherically convex hypersurfaces in hyperbolic space (a bound, for some physically motivated weight function, of the weighted integral of the k k th mean curvature in terms of the area of the hypersurface). We prove an inequality very similar to the conjectured one. Moreover, when k k is zero and the ambient space has dimension three, we give a counterexample to the conjectured inequality.

Locally constrained curvature flows and geometric inequalities in hyperbolic space

In this paper, we first study the locally constrained curvature flow of hypersurfaces in hyperbolic space, which was introduced by Brendle, Guan and Li [7]. This flow preserves the $m$th quermassintegral and decreases $(m+1)$th quermassintegral, so the convergence of the flow yields sharp Alexandrov-Fenchel type inequalities in hyperbolic space. Some special cases have been studied in [7]. In the first part of this paper, we show that h-convexity of the hypersurface is preserved along the flow and then the smooth convergence of the flow for h-convex hypersurfaces follows. We then apply this result to establish some new sharp geometric inequalities comparing the integral of $k$th Gauss-Bonnet curvature of a smooth h-convex hypersurface to its $m$th quermassintegral (for $0\leq m\leq 2k+1\leq n$), and comparing the weighted integral of $k$th mean curvature to its $m$th quermassintegral (for $0\leq m\leq k\leq n$). In particular, we give an affirmative answer to a conjecture proposed by Ge, Wang and Wu in 2015. In the second part of this paper, we introduce a new locally constrained curvature flow using the shifted principal curvatures. This is natural in the context of h-convexity. We prove the smooth convergence to a geodesic sphere of the flow for h-convex hypersurfaces, and provide a new proof of the geometric inequalities proved by Andrews, Chen and the third author of this paper in 2018. We also prove a family of new sharp inequalities involving the weighted integral of $k$th shifted mean curvature for h-convex hypersurfaces, which as application implies a higher order analogue of Brendle, Hung and Wang's [8] inequality.

Harmonic mean curvature flow and geometric inequalities

We employ the harmonic mean curvature flow of strictly convex closed hypersurfaces in hyperbolic space to prove Alexandrov-Fenchel type inequalities relating quermassintegrals to the total curvature, which is the integral of Gaussian curvature on the hypersurface. The resulting inequality allows us to use the inverse mean curvature flow to prove Alexandrov-Fenchel inequalities between the total curvature and the area for strictly convex hypersurfaces. Finally, we apply the harmonic mean curvature flow to prove a new class of geometric inequalities for h-convex hypersurfaces in hyperbolic space.

On the Asymptotic Plateau Problem for CMC Hypersurfaces in Hyperbolic Space

Let $$\mathbb {R}_{+}^{n+1}$$ be the half-space model of the hyperbolic space $$\mathbb {H}^{n+1}.$$ It is proved that if $$\Gamma \subset \left\{ x_{n+1} =0\right\} \subset \partial _{\infty }\mathbb {H}^{n+1}$$ is a bounded $$C^{0}$$ Euclidean graph over $$\left\{ x_{1}=0, x_{n+1}=0\right\} $$ then, given $$\left| H\right| <1,$$ there is a complete, properly embedded, CMC H hypersurface $$\Sigma $$ of $$\mathbb {H}^{n+1}$$ such that $$\partial _{\infty }S=\Gamma \cup \left\{ x_{n+1}=+\infty \right\} .$$ This result can be seen as a limit case of the existence theorem proved by Guan and Spruck in Guan and Spruck (2000) on CMC $$\left| H\right| <1$$ radial graphs with prescribed $$C^{0}$$ asymptotic boundary data. In spite of the above presentation of our result, our proof does not use coordinates but the Killing graph approach and therefore not only does not depend on the model of $$\mathbb {H}^{n},$$ but also allows the use of natural intrinsic geometric barriers of the hyperbolic space. A simple adaptation of our proof gives a new proof of Theorem 4.8 of Guan and Spruck (2000) and Theorem 4 of Dajczer et al. (2016). By the recent result on interior gradient estimates obtained by Marcos Dajzcer, Jorge Lira and the first author in Dajczer et al. (2016) we are able to apply here Perron’s method instead of the exhaustion method traditionally used in papers dealing with asymptotic Dirichlet problems, as in Guan and Spruck (2000) and Dajczer et al. (2016). The possibility of using Perron’s method is fundamental since the exhaustion technique seems strongly to not work for “horizontal” graphs.

Quermassintegral preserving curvature flow in Hyperbolic space

We consider the quermassintegral preserving flow of closed \emph{h-convex} hypersurfaces in hyperbolic space with the speed given by any positive power of a smooth symmetric, strictly increasing, and homogeneous of degree one function $f$ of the principal curvatures which is inverse concave and has dual $f_*$ approaching zero on the boundary of the positive cone. We prove that if the initial hypersurface is \emph{h-convex}, then the solution of the flow becomes strictly \emph{h-convex} for $t>0$, the flow exists for all time and converges to a geodesic sphere exponentially in the smooth topology.

Willmore inequality on hypersurfaces in hyperbolic space

In this article, we prove a geometric inequality for star-shaped and mean-convex hypersurfaces in hyperbolic space by inverse mean curvature flow. This inequality can be considered as a generalization of Willmore inequality for closed surface in hyperbolic $3$-space.

Weakly horospherically convex hypersurfaces in hyperbolic space

In [2], the authors develop a global correspondence between immersed weakly horospherically convex hypersurfaces $\phi:M^n \to \mathbb{H}^{n+1}$ and a class of conformal metrics on domains of the round sphere $\mathbb{S}^n$. Some of the key aspects of the correspondence and its consequences have dimensional restrictions $n\geq3$ due to the reliance on an analytic proposition from [5] concerning the asymptotic behavior of conformal factors of conformal metrics on domains of $\mathbb{S}^n$. In this paper, we prove a new lemma about the asymptotic behavior of a functional combining the gradient of the conformal factor and itself, which allows us to extend the global correspondence and embeddedness theorems of [2] to all dimensions $n\geq2$ in a unified way. In the case of a single point boundary $\partial_{\infty}\phi(M)=\{x\} \subset \mathbb{S}^n$, we improve these results in one direction. As an immediate consequence of this improvement and the work on elliptic problems in [2], we have a new, stronger Bernstein type theorem. Moreover, we are able to extend the Liouville and Delaunay type theorems from [2] to the case of surfaces in $\mathbb{H}^{3}$.

On the Gauss map of Weingarten hypersurfaces in hyperbolic spaces

In this paper, we deal withWeingarten hypersurfaces in hyperbolic spaces. By supposing suitable restrictions on the Gauss map of such hypersurfaces we obtain rigidity results which extent the recent results of Barros, Aquino and de Lima.

Fundamental tone of minimal hypersurfaces with finite index in hyperbolic space

Let M be a complete minimal hypersurface in hyperbolic space $\mathbb {H}^{n+1}(-1)$ with constant sectional curvature −1. We prove that if M has a finite index and finite $L^{2}$ norm of the second fundamental form, then the fundamental tone $\lambda_{1} (M)$ is bounded above by $n^{2}$ .

Inequalities of Alexandrov–Fenchel type for convex hypersurfaces in hyperbolic space and in the sphere

In this paper, firstly, inspired by Nat\'{a}rio's recent work \cite{Na}, we use the isoperimetric inequality to derive some Alexandrov-Fenchel type inequalities for closed convex hypersurfaces in the hyperbolic space $\H^{n+1}$ and in the sphere $\SS^{n+1}$. We also get the rigidity in the spherical case. Secondly, we use the inverse mean curvature flow in sphere \cite{gerh,Mak-Sch} to prove an optimal Sobolev type inequality for closed convex hypersurfaces in the sphere.

Hypersurfaces in hyperbolic space with support function

Based on [19], we develop a global correspondence between immersed hypersurfaces ϕ:Mn→Hn+1 satisfying an exterior horosphere condition, also called here horospherically concave hypersurfaces, and complete conformal metrics e2ρgSn on domains Ω in the boundary Sn at infinity of Hn+1, where ρ is the horospherical support function, ∂∞ϕ(Mn)=∂Ω, and Ω is the image of the Gauss map G:Mn→Sn. To do so we first establish results on when the Gauss map G:Mn→Sn is injective. We also discuss when an immersed horospherically concave hypersurface can be unfolded along the normal flow into an embedded one. These results allow us to establish general Alexandrov reflection principles for elliptic problems of both immersed hypersurfaces in Hn+1 and conformal metrics on domains in Sn. Consequently, we are able to obtain, for instance, a strong Bernstein theorem for a complete, immersed, horospherically concave hypersurface in Hn+1 of constant mean curvature.

A geometric inequality on hypersurface in hyperbolic space

In this paper, we use the inverse curvature flow to prove a sharp geometric inequality on star-shaped and two-convex hypersurface in hyperbolic space.

COMPLETE REAL HYPERSURFACES AND SPECIAL 𝕂-LINE BUNDLES IN 𝕂-HYPERBOLIC SPACES

Using the geometry of geodesics, we discuss the global aspects of complete real hypersurfaces in hyperbolic spaces of constant holomorphic sectional curvature [Formula: see text] over any division algebra 𝕂. Our assumption is that the shape operator and the curvature transformation with respect to the normal unit have the same eigenspaces. Note that we do not assume constancy of the principal curvatures. Under this assumption, we give a complete global classification of such hypersurfaces. Since the argument is purely geometric, we need not vary the argument for different base algebras. The foliations of [Formula: see text] with totally geodesic leaves called 𝕂-lines play an important role.