Characterizes hypersurfaces in complex projective and hyperbolic planes.
problem Geometric characterization of hypersurfaces.
method Geometric characterization and partial classifications.
result Characterizes certain hypersurfaces of cohomogeneity one.
The article proves inequalities for capillary hypersurfaces in hyperbolic space.
problem Proving inequalities for capillary hypersurfaces in hyperbolic space.
method Constructing a new locally constrained inverse curvature flow.
result Obtained Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in hyperbolic space.
Low-degree hyperbolic hypersurfaces in complex projective space constructed.
problem Constructing hyperbolic hypersurfaces of low degree in complex projective space.
method Families of hyperbolic hypersurfaces Xd constructed with degree d. result Families of hyperbolic hypersurfaces Xd of degree d≥(2n+3)2 constructed in Pn+1(C). The paper proves rigidity results for capillary hypersurfaces in hyperbolic space.
problem Understanding the rigidity of capillary hypersurfaces in hyperbolic space.
method Proving a Heintze-Karcher type inequality and applying it to Alexandrov type theorems.
result Rigidity results for capillary hypersurfaces, including totally umbilical and totally geodesic cases.
Proves nonnegatively curved hypersurfaces in hyperbolic space are usually properly embedded.
problem Proving nonnegatively curved hypersurfaces in hyperbolic space are usually properly embedded.
method Analyzes Alexander and Currier's conjecture and proves it for most cases.
result Proves a conjecture about nonnegatively curved hypersurfaces in hyperbolic space.
Study on relative entropy for hypersurfaces in hyperbolic space.
problem Understanding relative entropy for hypersurfaces in hyperbolic space.
method Relate relative entropy to renormalized area and apply monotonicity formula to mean curvature flows.
result Obtained a monotonicity formula for relative entropy in hyperbolic space.
The paper examines stable capillary hypersurfaces in hyperbolic space.
problem Stability of capillary hypersurfaces with free boundary on a horosphere.
method Analysis of umbilical and totally geodesic hypersurfaces using stability criteria.
result Umbilical and totally geodesic hypersurfaces are the only stable capillary hypersurfaces with boundary on a horosphere.
The Willmore inequality is extended to star-shaped and mean-convex hypersurfaces in hyperbolic space.
problem Proving a geometric inequality for specific hypersurfaces in hyperbolic space.
method Inverse mean curvature flow to prove the inequality.
result The Willmore inequality is generalized to star-shaped and mean-convex hypersurfaces in hyperbolic space.
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 m…
Study shows properties of noncompact hypersurfaces in hyperbolic space.
problem Characterize noncompact hypersurfaces in hyperbolic space with nonnegative Ricci curvature.
method Utilized properties of n-subharmonic functions to analyze asymptotic boundaries.
result Hypersurfaces with nonnegative Ricci curvature in hyperbolic space have at most two points in their asymptotic boundary.
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.
Analyzes singularities of convex hypersurfaces in hyperbolic space.
problem Understanding singularities of convex hypersurfaces with constant curvature.
method Analyzes the structure of singular sets using convex curvature functions.
result Describes the structure of singular sets in hyperbolic space.
Study calculates spectra of minimal hypersurfaces in hyperbolic space.
problem Computing Laplacian spectra of minimal hypersurfaces.
method Analyzes hypersurfaces in hyperbolic space with specific asymptotic data.
result Obtains spectra and extremal properties of the bottom of the spectrum.
Proves new inequality for hyperbolic space hypersurfaces.
problem Finding inequalities for hypersurfaces in hyperbolic space.
method Proves a Heintze-Karcher type inequality for shifted mean convex hypersurfaces.
result Proves Alexandrov type theorem and uniqueness result for hypersurfaces.
Survey solves curvature problems with hyperbolic spaces.
problem Singularities in hypersurface geometry.
method Hyperbolic unfolding correspondence linking hypersurfaces to Gromov hyperbolic spaces.
result Eliminates hypersurface singularities in scalar curvature geometry.
Unique inhomogeneous ruled hypersurface found in complex hyperbolic space.
problem Classifying ruled real hypersurfaces with constant norm.
method Analyzing nonflat complex space forms, proving existence and uniqueness.
result Existence of a unique inhomogeneous example in complex hyperbolic space.
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 th…
Study classifies real hypersurfaces with special Jacobi operator in complex hyperbolic quadric.
problem Characterizing real hypersurfaces with specific properties in complex hyperbolic quadric.
method Introduced Reeb parallel structure Jacobi operator and classified real hypersurfaces.
result Classification of real hypersurfaces with Reeb parallel structure Jacobi operator.
The paper finds the maximum scalar curvature for 3D minimal hypersurfaces in hyperbolic 4-space.
problem Finding the maximum scalar curvature for 3D minimal hypersurfaces in hyperbolic 4-space.
method Using the Generalized Maximum Principle, the paper proves that a 3D complete minimal hypersurface with constant scalar curvature in H4(−1) satisfies S≤2921. result A 3D complete minimal hypersurface in H4(−1) with constant scalar curvature satisfies S≤2921. New method reveals geometric properties of minimal hypersurfaces.
problem Understanding the intrinsic geometry of minimal hypersurfaces.
method Defining S-structures and hyperbolic unfoldings.
result Existence of hyperbolic unfoldings of minimal hypersurfaces.
Study classifies hypersurfaces in complex hyperbolic spaces with specific operator conditions.
problem Classifying real hypersurfaces with commuting structure Jacobi operators.
method Used simultaneous diagonalization for commuting symmetric operators to classify hypersurfaces.
result Complete classification of real hypersurfaces with commuting condition.
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.
The paper classifies and determines properties of specific hypersurfaces in complex hyperbolic quadrics.
problem Analyzing Hopf hypersurfaces with constant principal curvatures in complex hyperbolic quadrics.
method Classification and determination of principal curvatures for hypersurfaces with different numbers of distinct curvatures.
result Classification and determination of principal curvatures for Hopf hypersurfaces with up to four distinct values.
Proves existence of curved surfaces in hyperbolic space.
problem Finding surfaces with specific curvature and boundary conditions.
method Proves existence using Weingarten curvature and asymptotic boundary conditions.
result Proves existence of locally Lipschitz continuous hypersurfaces.
The paper proves new inequalities for convex hypersurfaces in hyperbolic and spherical spaces.
problem Proving new inequalities for convex hypersurfaces in hyperbolic and spherical spaces.
method Locally constrained inverse curvature flows in hyperbolic and spherical spaces.
result Established new Alexandrov-Fenchel and Minkowski inequalities involving general convex weight functions.
New theorem finds new minimal hypersurfaces in hyperbolic space.
problem Finding new minimal hypersurfaces in hyperbolic space.
method Developed a min-max theory for complete minimal hypersurfaces.
result Shows existence of a new minimal hypersurface between two given ones.
Paper constructs Hopf real hypersurfaces in complex hyperbolic space.
problem Existence of Hopf real hypersurfaces in complex hyperbolic space.
method Twistor spaces and para-quaternionic Kähler structure.
result Unified construction of Hopf real hypersurfaces in complex hyperbolic space.
Study on uniqueness of hypersurfaces in hyperbolic space with constant mean curvature.
problem Uniqueness of hypersurfaces with constant higher order mean curvature in hyperbolic space.
method Generalization of Bernstein theorem and proof of Bernstein type results for immersed hypersurfaces.
result Rigidity of horospheres and equidistant spheres in terms of their higher order mean curvatures.
Paper proves no specific CMC hypersurfaces in hyperbolic space.
problem Existence of complete noncompact CMC hypersurfaces in hyperbolic space.
method Uses μ-bubbles developed by Gromov and Chodosh-Li-Stryker.
result No complete noncompact CMC hypersurfaces with mean curvature > 1 in hyperbolic space.
We classify isoparametric hypersurfaces in complex hyperbolic spaces.
New examples of hypersurfaces found in quaternionic hyperbolic spaces.
problem Classifying actions on symmetric spaces of rank one.
method Cohomogeneity one actions and orbit equivalence.
result Uncountably many inhomogeneous isoparametric families of hypersurfaces.
The article uses harmonic mean curvature flow to prove new geometric inequalities for convex hypersurfaces.
problem Proving new geometric inequalities for convex hypersurfaces in hyperbolic space.
method Harmonic mean curvature flow, Alexandrov-Fenchel inequalities, inverse mean curvature flow, Heintze-Karcher type inequality.
result New geometric inequalities for convex hypersurfaces in hyperbolic space.
Study transforms singular area minimizing hypersurfaces into hyperbolic spaces.
problem Understanding singular area minimizing hypersurfaces.
method Conformal unfolding to Gromov hyperbolic spaces.
result Recover singular set as Gromov and Martin boundaries.
We show that the constant mean curvature hypersurfaces in the hyperbolic n-space spanning the boundary of a star shaped C^{1,1} domain in the asymptotic sphere give a foliation of the hyperbolic n-space. We also show that if C is a closed codimension-1 C^{2,a} submanifold in the asymptotic sphere bounding a unique cons…
Curvature estimates prove existence of smooth hypersurfaces in hyperbolic space.
problem Existence of smooth complete hypersurfaces with constant curvature in hyperbolic space.
method Deriving curvature estimates to prove existence for all curvature values.
result Existence of smooth hypersurfaces for all possible curvature values.
Study on deformable hypersurfaces in hyperbolic and spherical products.
problem Characterizing deformable hypersurfaces in HkimesSn−k+1. method Analyzing the geometric properties and deformations of hypersurfaces in the given Riemannian products.
result Characterized the deformations of hypersurfaces in HkimesSn−k+1. Maximally hyperbolic solutions contain future neighborhoods of intersecting hypersurfaces.
problem Maximally globally hyperbolic solutions of higher-dimensional vacuum Einstein equations.
method Analyzing intersections of characteristic hypersurfaces.
result Contains a future neighborhood of intersecting hypersurfaces.
Sharp stability estimates for hypersurfaces in hyperbolic space.
problem Stability of hypersurfaces with almost constant mean curvature in hyperbolic space.
method Method of moving planes
result Quantitative pinching result for hypersurfaces in hyperbolic space.
Convex hypersurfaces in hyperbolic space evolve to geodesic spheres.
problem Volume preserving Gauss curvature flow of convex hypersurfaces in hyperbolic space.
method Volume preserving flow with speed given by Gauss curvature power α, using Alexandrov reflection and hyperbolic curvature measures.
result Smooth solution remains convex and converges to a geodesic sphere exponentially.
We construct examples of inhomogeneous isoparametric real hypersurfaces in complex hyperbolic spaces.
Spinorial methods prove inequalities for special hypersurfaces.
problem Proving inequalities for hypersurfaces in pseudo-hyperbolic manifolds.
method Spinorial techniques to prove inequalities.
result Proves a Heintze-Karcher type inequality.
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 …
The paper explores rigidity of hypersurfaces with constant shifted curvature functions in hyperbolic space.
problem Rigidity of hypersurfaces with constant shifted curvature functions in hyperbolic space.
method Characterizations and rigidity investigations for hypersurfaces with constant weighted shifted mean curvatures or ratios.
result Rigidity results for hypersurfaces with constant linear combinations of weighted shifted mean curvatures and radially symmetric shifted mean curvatures.
Study on flows in hyperbolic and de Sitter spaces, linking shrinking and expanding surfaces.
problem Understanding flows in hyperbolic and de Sitter spaces.
method Analyzing strictly convex hypersurfaces in hyperbolic and de Sitter spaces, relating contracting and expanding flows.
result Rescaled contracting and expanding hypersurfaces converge to specific geometric shapes.
New proof classifies contact real hypersurfaces in complex hyperbolic quadric.
problem Classifying contact real hypersurfaces with constant mean curvature.
method New proof using tubes and horospheres.
result Contact real hypersurfaces are congruent to tubes or horospheres.
The study classifies real hypersurfaces in complex hyperbolic quadrics with isometric Reeb flow.
problem Classifying real hypersurfaces with isometric Reeb flow in complex hyperbolic quadrics.
method Classification based on the properties of the hypersurfaces and their embeddings.
result The existence and properties of real hypersurfaces with isometric Reeb flow are classified, leading to the non-existence in odd-dimensional cases.
We classify all real hypersurfaces with constant principal curvatures in the complex hyperbolic plane.
Classifies hypersurfaces with positive constant mean curvature in hyperbolic space.
problem Classifying hypersurfaces with positive constant mean curvature in hyperbolic space.
method Classifies hypersurfaces with rotational symmetry and positive constant r-th mean curvature in HnimesR. result Compact connected hypersurfaces of constant r-th mean curvature embedded in Hnimes[0,∞) with boundary in the slice Hnimes{0} are topological disks under suitable assumptions.