Jurgen Berndt, Jose Carlos Diaz-Ramos
We study the geometry of homogeneous hypersurfaces and their focal sets in complex hyperbolic spaces. In particular, we provide a characterization of the focal set in terms of its second fundamental form and determine the principal curvatures of the homogeneous hypersurfaces together with their multiplicities.
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We construct examples of inhomogeneous isoparametric real hypersurfaces in complex hyperbolic spaces.
A Lie hypersurface in the complex hyperbolic space is an orbit of a cohomogeneity one action without singular orbit. In this paper, we classify Ricci soliton Lie hypersurfaces in the complex hyperbolic spaces.
We study the constant mean curvature (CMC) hypersurfaces in hyperbolic space whose asymptotic boundaries are closed codimension-1 submanifolds in sphere at infinity. We consider CMC hypersurfaces as generalizations of minimal hypersurfaces. We naturally generalize some notions of minimal hypersurfaces like being area minimizing, convex hull property, exchange roundoff trick to the CMC hypersurface context. We also give a generic uniqueness result for CMC hypersurfaces in hyperbolic space.
We classify all real hypersurfaces with constant principal curvatures in the complex hyperbolic plane.
We introduce curvature-adapted foliations of complex hyperbolic space and study some of their properties. Generalized pseudo-Einstein hypersurfaces of complex hyperbolic space are classified. Analogous results for curvature-adapted hypersurfaces of quaternionic hyperbolic space are also obtained.
We classify isoparametric hypersurfaces in complex hyperbolic spaces.
A Lie hypersurface in the complex hyperbolic space is a homogeneous real hypersurface without focal submanifolds. The set of all Lie hypersurfaces in the complex hyperbolic space is bijective to a closed interval, which gives a deformation of homogeneous hypersurfaces from the ruled minimal one to the horosphere. In this paper, we study intrinsic geometry of Lie hypersurfaces, such as Ricci curvatures, scalar curvatures, and sectional curvatures.
We investigate the existence, convergence and uniqueness of modified general curvature flow of convex hypersurfaces in hyperbolic space with a prescribed asymptotic boundary.
We classify all real hypersurfaces with three distinct constant principal curvatures in complex hyperbolic spaces of dimension greater than two.
We give a geometric characterization of certain hypersurfaces of cohomogeneity one in the complex projective and hyperbolic planes. We also obtain some partial classifications of austere hypersurfaces and of Levi-flat hypersurfaces with constant mean curvature in these spaces.
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[RW99]. We also obtain a monotonicity formula for relative entropy applied to mean curvature flows in hyperbolic space.
In this paper we prove the conjecture of Alexander and Currier that states, except for covering maps of equidistant surfaces in hyperbolic 3-space, a complete, nonnegatively curved immersed hypersurface in hyperbolic space is necessarily properly embedded.
We investigate the problem of finding complete strictly convex hypersurfaces of constant curvature in hyperbolic space with a prescribed asymptotic boundary at infinity for a general class of curvature functions.
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.
We prove the existence of a complete locally Lipschitz continuous hypersurface in weak sense with prescribed Weingarten curvature and asymptotic boundary at infinity in hyperbolic space under certain assumptions.
We construct new examples of embedded, complete minimal hypersurfaces in quaternionc hyperbolic space and also some minimal foliations. We introduce fans an construct analytic deformations of bisectors.
We find complete hypersurfaces of constant curvature in hyperbolic space with a prescribed asymptotic boundary at infinity for a general class of (elliptic) curvature functions which includes the higher order mean curvatures and their curvature quotients.
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.
We study the intrinsic geometry of area minimizing (and also of almost minimizing) hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. For any such hypersurface we define and construct a so-called S-structure which reveals some unexpected geometric and analytic properties of the hypersurface and its singularity set. In this paper, this is used to prove the existence of hyperbolic unfoldings: canonical conformal deformations of the regular part of these hypersurfaces into complete Gromov hyperbolic spaces of bounded geometry with Gromov boundary homeomorp…
In this paper we continue our study of finding the curvature flow of complete hypersurfaces in hyperbolic space with a prescribed asymptotic boundary at infinity. Our main results are proved by deriving a priori global gradient estimates and C^2 estimates.
In this paper, we study a stability problem of free boundary hypersurfaces, and also capillary ones whose boundary supported on a horosphere in hyperbolic space. We prove that umbilical hypersurfaces are only stable immersed capillary hypersurfaces whose boundary supported on a horosphere. Using the same method, we show that a totally geodesic hyperplane is only stable immersed type-II hypersurface whose boundary supported on a horosphere.
Based on properties of n-subharmonic functions we show that a complete, noncompact, properly embedded hypersurface with nonnegative Ricci curvature in hyperbolic space has an asymptotic boundary at infinity of at most two points. Moreover, the presence of two points in the asymptotic boundary is a rigidity condition that forces the hypersurface to be an equidistant hypersurface about a geodesic line in hyperbolic space. This gives an affirmative answer to the question raised by Alexander and Currier in 1990.
We introduce the notion of Reeb parallel structure Jacobi operator for real hypersurfaces in the complex hyperbolic quadric ${Q^*}^m=SO^0_{2,m}/SO_2 SO_m$, $m \geq 3$, and give a classification theory for real hypersurfaces in ${{Q^*}^m}$, $m \geq 3$, with Reeb parallel structure Jacobi operator.
It is extended a result due to B. Guan and J. Spruck on the asymptotic Plateau's problem for CMC radial graphs in hyperbolic space to horizontal CMC graphs.
We complete the classification of ruled real hypersurfaces with shape operator of constant norm in nonflat complex space forms by showing the existence of a unique inhomogeneous example in the complex hyperbolic space.
In this paper we develop a global correspondence between immersed horospherically convex hypersurfaces in hyperbolic space and complete conformal metrics on domains in the sphere. We establish results on when the hyperbolic Gauss map is injective and when an immersed horospherically convex hypersurface can be unfolded along the normal flow into an embedded one. These results allow us to establish general Alexandrov reflection principles of elliptic problems of both Weingarten hypersurfaces and complete conformal metrics and relations between them. Consequently, we are able to obtain, for insta…
We find the first examples of real hypersurfaces with two nonconstant principal curvatures in complex projective and hyperbolic planes, and we classify them. It turns out that each such hypersurface is foliated by equidistant Lagrangian flat surfaces with parallel mean curvature or, equivalently, by principal orbits of a cohomogeneity two polar action.
In this paper we give an upper bound of the first eigenvalue of the Laplace operator on a complete stable minimal hypersurface $M$ in the hyperbolic space which has finite $L^2$-norm of the second fundamental form on $M$. We provide some sufficient conditions for minimal hypersurface of the hyperbolic space to be stable. We also describe stability of catenoids and helicoids in the hyperbolic space. In particular, it is shown that there exists a family of stable higher-dimensional catenoids in the hyperbolic space.
Some rigidity results on compact hypersurfaces with capillary boundary in Hyperbolic space
本文证明了双曲空间中毛细超曲面的Heintze-Karcher不等式，仅在完全脐曲超曲面时成立，并应用该结果证明了嵌入毛细超曲面的Alexandrov型定理，还证明了在双曲空间中支撑于全测地平面的毛细超曲面的其他刚性结果。
Using spinorial techniques, we prove, for a class of pseudo-hyperbolic ambient manifolds, a Heintze-Karcher type inequality. We then use this inequality to show an Alexandrov type theorem in such spaces.
We establish versions of the Positive Mass and Penrose inequalities for a class of asymptotically hyperbolic hypersurfaces. In particular, under the usual dominant energy condition, we prove in all dimensions $n\geq 3$ an optimal Penrose inequality for certain graphs in hyperbolic space $\mathbb H^{n+1}$ whose boundary has constant mean curvature $n-1$.
This is the second in a series of papers where we estab- lish skin structural concepts and results for singular area minimizing hypersurfaces. Here we conformally unfold these spaces to complete Gromov hyperbolic spaces with bounded geometry and we recover their singular set as the Gromov boundary but also as the Martin boundary for many classical elliptic operators.
Remarks on the spectra of minimal hypersurfaces in the hyperbolic space
本文计算了双曲空间中具有给定渐近数据的奇异面积最小化超曲面的Laplacian谱，推广至高余维数，并探讨了谱底部的极值性质。
In this paper, we introduce a new commuting condition between the structure Jacobi operator and symmetric (1,1)-type tensor field $T$, that is, $R_ξϕT=TR_ξϕ$, where $T=A$ or $T=S$ for Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians. By using simultaneous diagonalzation for commuting symmetric operators, we give a complete classification of real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting condition respectively.
The existence of a smooth complete strictly locally convex hypersurface with prescribed scalar curvature and asymptotic boundary at infinity in $\mathbb{H}^{3}$ is proved under the assumption that there exists a strictly locally convex subsolution.
This paper is concerned with the completeness (with respect to the centroaffine metric) of hyperbolic centroaffine hypersurfaces which are closed in the ambient vector space. We show that completeness holds under generic regularity conditions on the boundary of the convex cone generated by the hypersurface. The main result is that completeness holds for hyperbolic components of level sets of homogeneous cubic polynomials. This implies that every such component defines a complete quaternionic Kähler manifold of negative scalar curvature.
We consider the problem $F=f(ν)$ for strictly convex, closed hypersurfaces in hyperbolic space and solve it for curvature functions $F$ the inverses of which are of class $(K^*)$.
It is very well known that Hopf real hypersurfaces in the complex projective space can be locally characterized as tubes over complex submanifolds. This also holds true for some, but not all, Hopf real hypersurfaces in the complex hyperbolic space. The main goal of this paper is to show, in a unified way, how to construct Hopf real hypersurfaces in the complex hyperbolic space from a horizontal submanifold in one of the three twistor spaces of the indefinite complex $2$-plane Grassmannian with respect to the natural para-quaternionic Kähler structure. We also identify these twistor spaces with…
We derive a relationship between the eigenvalues of the Weyl-Schouten tensor of a conformal representative of the conformal infinity of a hyperbolic Poincaré manifold and the principal curvatures on the level sets of its uniquely associated defining function with calculations based on [9] [10]. This relationship generalizes the result for hypersurfaces in ${\H}^{n+1}$ and their connection to the conformal geometry of ${\SS}^n$ as exhibited in [7] and gives a correspondence between Weingarten hypersurfaces in hyperbolic Poincaré manifolds and conformally invariant equations on the conformal inf…
Complete minimal hypersurfaces in the hyperbolic space $\mathbb{H}^4$ with vanishing Gauss-Kronecker curvature
本文研究双曲空间$\mathbb{H}^4$中Gauss-Kronecker曲率恒为零的3维完备极小超曲面，并在第二基本形式处处非零、数量曲率有下界的条件下给出分类。
Contraction of hypersurfaces with positive sectional curvature in hyperbolic space
研究双曲空间中正截面曲率紧致超曲面的收缩曲率流，证明初始超曲面的正截面曲率在流中保持，并在有限时间内收缩至圆点。
In this paper we prove a monotonicity formula for the integral of the mean curvature for complete and proper hypersurfaces of the hyperbolic space and, as consequences, we obtain a lower bound for the integral of the mean curvature and that the integral of the mean curvature is infinity.
On Hopf hypersurfaces of the complex hyperbolic quadric with constant principal curvatures
本文对复双曲二次型中具有常主曲率的Hopf超曲面进行分类，并确定了三或四个不同常主曲率时的曲率值及重数。
In this paper, firstly, inspired by Natário'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.
Complete minimal hypersurfaces in a hyperbolic space $H^{4}(-1)$
本文研究双曲空间中的完备极小超曲面，利用广义最大值原理证明$H^{4}(-1)$中具常数量曲率的3维完备极小超曲面的第二基本形式平方范数$S\leq \frac{21}{29}$。
Uniqueness of Hypersurfaces of Constant Higher Order Mean Curvature in Hyperbolic Space
本文研究了双曲空间中极球面与等距球面的唯一性，将 Bernstein 定理推广至常高阶平均曲率的嵌入超曲面，证明了浸入超曲面的 Bernstein 型结果，并确立了极球面与等距球面的刚性。
Hypersurfaces of constant sum Hessian curvature in Hyperbolic space
本文研究双曲空间中常和Hessian曲率的渐近Plateau问题，旨在寻找满足特定曲率方程和渐近边界条件的完备超曲面。
We classify totally geodesic and parallel hypersurfaces of four-dimensional non-reductive homogeneous pseudo-Riemannian manifolds.
We classify the non-degenerate homogeneous hypersurfaces in real and complex affine four-space whose symmetry group is at least four-dimensional.
We classify all of real hypersurfaces $M$ with Reeb invariant shape operator in complex hyperbolic two-plane Grassmannians $SU_{2,m}/S(U_2{\cdot}U_m)$, $m \geq 2$. Then it becomes a tube over a totally geodesic $SU_{2,m-1}/S(U_2{\cdot}U_{m-1})$ in $SU_{2,m}/S(U_2{\cdot}U_m)$ or a horosphere whose center at infinity is singular and of type $JN \in {\mathfrak J}N$ for a unit normal vector field $N$ of $M$.
On rigidity of hypersurfaces with constant shifted curvature functions in hyperbolic space
本文研究双曲空间中具有常值位移曲率函数的超曲面刚性，通过常值加权位移平均曲率及其线性组合等条件给出测地球新刻画，并在更弱条件下推广了常值 Gauss-Bonnet 曲率 $L_k$ 的刚性结果。
This paper deals with a limiting case motivated by contact geometry. The limiting case of a tensorial characterization of contact hypersurfaces in Kahler manifolds leads to Hopf hypersurfaces whose maximal complex subbundle of the tangent bundle is integrable. It is known that in non-flat complex space forms and in complex quadrics such real hypersurfaces do not exist, but the existence problem in other irreducible Kahler manifolds is open. In this paper we construct explicitly a one-parameter family of homogeneous Hopf hypersurfaces, whose maximal complex subbundle of the tangent bundle is in…
CMC hypersurface with finite index in hyperbolic space $\mathbb{H}^4$
本文利用μ-bubble方法证明了双曲空间$\u210D^4$及满足曲率条件的4维黎曼流形中，不存在特定的完备非紧常平均曲率/极小稳定超曲面，推进了几何测度论相关不存在性问题的研究。
In this paper, we study biharmonic hypersurfaces in Einstein manifolds. Then, we determine all the biharmonic hypersurfaces in irreducible symmetric spaces of compact type which are regular orbits of commutative Hermann actions of cohomogeneity one.
In this paper we give a characterization of real hypersurfaces in noncompact complex two-plane Grassmannian $SU_{2,m}/S(U_2 U_m)$, $m \geq 2$ with Reeb vector field $ξ$ belonging to the maximal quaternionic subbundle $\mathcal Q$. Then it becomes a tube over a totally real totally geodesic ${\mathbb H}H^n$, $m=2n$, in noncompact complex two-plane Grassmannian $SU_{2,m}/S(U_2 U_m)$, a horosphere whose center at the infinity is singular or another exceptional case.
Hypersurfaces of any homogeneous $\mathbb{C}P^3$
本文在任意齐性 $\mathbb{C}P^3$（含近凯勒情形）中分类了外在齐性超曲面和 Hopf 超曲面，并证明 Codazzi 型、全脐及常截面曲率超曲面均不存在。
We study the existence of starshaped compact hypersurfaces with prescribed m-th mean curvature in hyperbolic space.
We develop a min-max theory for certain complete minimal hypersurfaces in hyperbolic space. In particular, we show that given two strictly stable minimal hypersurfaces that are both asymptotic to the same ideal boundary, there is a new one trapped between the two. As an application, we show that under a low entropy condition, all the minimal hypersurfaces asymptotic to the same ideal boundary are isotopic.
Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space
本文研究双曲空间中凸超曲面的保体积高斯曲率流，证明解长期存在且指数收敛到测地球，是首个仅要求初始凸性的非局部型保体积曲率流结果。
A Heintze-Karcher type inequality in hyperbolic space
本文在双曲空间中证明了平移平均凸超曲面的Heintze-Karcher型不等式，并由此推出常平移第k阶平均曲率闭嵌入超曲面的Alexandrov型定理及h-凸超曲面的唯一性结果。
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