Proves convexity of certain hypersurfaces with negative λ.
problem Understanding convexity of hypersurfaces with specific λ values.
method Analyzes mean convex hypersurfaces and proves convexity for λ ≤ 0.
result Closed n-dimensional mean convex λ-hypersurfaces are convex if λ≤0. New global section found for geodesic flows on convex hypersurfaces.
problem Finding global sections for geodesic flows on convex hypersurfaces.
method Constructing a global hypersurface of section with an isometric involution.
result Generalized Birkhoff annulus to higher dimensions.
The study proves the existence of k-convex hypersurfaces for specific curvature equations.
problem Proving the existence of k-convex hypersurfaces for Hessian curvature equations. method Combining a priori estimates with the continuity method, and establishing a constant rank theorem.
result Existence and uniqueness of k-convex hypersurfaces for both nonhomogeneous and homogeneous Hessian curvature equations. Local rigidity proved for convex hypersurfaces in spaces of constant curvature.
problem Proving rigidity of convex hypersurfaces in spaces of constant curvature.
method Analyzing isometric convex hypersurfaces in spaces of constant curvature of dimension n≥4. result Two convex isometric hypersurfaces are congruent locally around their corresponding under strict convexity isometries.
New compact mean convex hypersurfaces found for positive λ.
problem Finding compact embedded hypersurfaces for positive λ.
method Constructing compact mean convex hypersurfaces diffeomorphic to spheres.
result No compact convex embedded λ-hypersurfaces except a round sphere for λ > 0.
Paper solves Dirichlet problem for p-convex hypersurfaces with curvature constraints.
problem Solving the Dirichlet problem for p-convex hypersurfaces with prescribed curvature. method Proved existence of a graphic hypersurface satisfying the prescribed curvature equation with homogeneous boundary condition, obtained an interior curvature estimate.
result Existence of a graphic hypersurface satisfying the prescribed curvature equation with homogeneous boundary condition.
Lower bounds for surface area and volume of convex hypersurfaces.
problem Establishing bounds for surface area and volume of convex hypersurfaces.
method Using displacement under continuous maps to establish lower bounds.
result Proves a lower bound for the volume of a Riemannian n-sphere in all dimensions.
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.
Sharp bounds on mean curvature and geodesic lengths in convex hypersurfaces.
problem Finding sharp bounds on total mean curvature of convex hypersurfaces.
method Sharp lower bounds for mean width and Birkhoff invariant, characterizing spheres.
result Generalization of Álvarez Paiva's result to convex hypersurfaces.
Paper proves rigidity of convex hypersurfaces in various spaces.
problem Proving the uniqueness of convex hypersurfaces in multidimensional spaces.
method Generalizing Senkin's theorem to higher dimensions and constant curvature spaces.
result Rigidity of convex hypersurfaces in En+1, n≥3. Optimal inequality for free boundary hypersurfaces in convex domains.
problem Proving an optimal Heintze-Karcher inequality for free boundary hypersurfaces.
method Analyzing anisotropic free boundary hypersurfaces in convex domains.
result Optimal Heintze-Karcher-type inequality achieved for anisotropic free boundary Wulff shapes.
Paper proves a generalized Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary.
problem Proving a generalized Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary.
method Using a locally constrained nonlinear curvature flow to preserve the n-th quermassintegral and decrease the k-th quermassintegral. result Obtained the Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in Bn+1. Study of convex hypersurfaces with specific curvature properties.
problem Characterizing convex hypersurfaces with vanishing Weyl curvature and semi-parallel cubic form.
method Analyzing locally strongly convex affine hypersurfaces with vanishing Weyl curvature tensor and semi-parallel cubic form relative to the Levi-Civita connection of affine metric.
result Classification of such hypersurfaces, excluding flat affine metric cases.
The paper proves conditions for zero Gaussian curvature convex hypersurfaces to be hyperplanes.
problem Conditions for zero Gaussian curvature convex hypersurfaces to be hyperplanes.
method Proving Bernstein type theorems for entire convex graphical hypersurfaces with zero Gaussian curvature in Euclidean and Minkowski contexts.
result Zero Gaussian curvature convex hypersurfaces must be hyperplanes if the mean curvature goes to zero at infinity.
Paper estimates curvature of convex hypersurfaces with prescribed curvature.
problem Estimating curvature of p-convex hypersurfaces with prescribed curvature. method Establishes curvature estimates for p-convex hypersurfaces in Rn+1 with p≥2n. result Proves existence of star-shaped hypersurface of prescribed curvature and interior C2 estimates. 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.
The paper studies a flow for convex capillary hypersurfaces in a ball, proving smooth convergence to a spherical cap.
problem Analyzing the behavior of convex capillary hypersurfaces under mean curvature flow.
method Introduced mean curvature flow for hypersurfaces in the unit Euclidean ball with capillary boundary. Proved smooth convergence to a spherical cap for strictly convex initial hypersurfaces.
result The flow preserves strict convexity and converges smoothly to a spherical cap for all positive time.
Study shows constant curvature convex hypersurfaces on hyperboloids are parts of hyperboloids.
problem Characterizing convex hypersurfaces with constant curvature on hyperboloids.
method Analyzing hypersurfaces with constant higher order mean curvature and constant boundary angle.
result Hypersurfaces with constant curvature on hyperboloids are parts of hyperboloids.
The study proves conditions for convex hypersurfaces in Riemannian manifolds to be rational homology spheres.
problem Conditions for convex hypersurfaces to be rational homology spheres in Riemannian manifolds.
method The study proves conditions for convex hypersurfaces in Riemannian manifolds to be rational homology spheres using vanishing and estimation theorems for Betti numbers.
result Conditions for convex hypersurfaces to be rational homology spheres in Riemannian manifolds.
Study shows curvature bounds for convex hypersurfaces in specific manifolds.
problem Bounding total curvature of convex hypersurfaces in Cartan-Hadamard manifolds.
method Analyzes curvature properties and applies Borbély's theorem.
result Total curvature is bounded below by the volume of the unit sphere.
We consider compact convex hypersurfaces contracting by functions of their curvature. Under the mean curvature flow, uniformly convex smooth initial hypersurfaces evolve to remain smooth and uniformly convex, and contract to points after finite time. The same holds if the initial data is only weakly convex or non-smoot…
Convex hypersurfaces in curved spaces bound convex regions.
problem Characterizing convex hypersurfaces in curved spaces.
method Gauss-Codazzi equations, Schur comparison theorem, Alexandrov geometry.
result Closed convex hypersurfaces bound convex regions in curved spaces.
The paper proves inequalities for star-shaped and F-mean convex hypersurfaces in Rn+1.
problem Proving geometric inequalities for specific types of hypersurfaces.
method Using anisotropic p-momentum, perimeter, and volume, the paper derives inequalities for star-shaped and F-mean convex hypersurfaces. result The Wulff shape of F is the unique minimizer of the corresponding functionals among all star-shaped and F-mean convex sets. Locally convex compact immersed hypersurfaces in Finsler-Hadamard manifolds with bounded T-curvature are considered. We prove that such hypersurfaces are embedded as the boundary of convex body under certain conditions on the normal curvatures
Having in mind the well known model of Euclidean convex hypersurfaces [4], [5], and the ideas in [1] many authors defined and investigate convex hypersurfaces of a Riemannian manifold. As it was proved by the first author in [7], there follows the interdependence between convexity and Gauss curvature of the hypersurfac…
Study on convex capillary hypersurfaces with Lp curvature in half-space.
problem Prescribed Lp curvature for convex capillary hypersurfaces.
method Reduction to Hessian quotient equation with Robin boundary condition.
result Existence and uniqueness of smooth admissible solutions.
The paper proves inequalities for convex capillary hypersurfaces in a half-space.
problem Proving inequalities for convex capillary hypersurfaces in a half-space.
method Locally constrained inverse curvature flow with spherical cap convergence.
result Proves a complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces.
We introduce a class of combinatorial hypersurfaces in the complex projective space. They are submanifolds of codimension~2 in $\C P^n$ and are topologically "glued" out of algebraic hypersurfaces in $(\C^*)^n$. Our construction can be viewed as a version of the Viro gluing theorem, relating topology of algebraic hyper…
Study convex capillary hypersurfaces with prescribed curvature in a spherical cap.
problem Prescribed curvature problem for convex capillary hypersurfaces.
method Reformulated as Hessian quotient equation with Robin boundary condition.
result Existence of strictly convex capillary hypersurface with prescribed curvature.
Every convex set in a generic Riemannian manifold has peculiar properties.
problem Characterizing convex sets in Riemannian manifolds.
method Analyzing geodesics and hypersurfaces in Riemannian manifolds.
result Convex sets in generic Riemannian manifolds are strictly convex if bounded by smooth hypersurfaces.
The study examines spacelike hypersurfaces in Minkowski space with constant σn−1 curvature.
problem Characterizing spacelike hypersurfaces with constant σn−1 curvature in Minkowski space. method Analyzing hypersurfaces with bounded principal curvatures and proving properties of their convexity.
result Hypersurfaces with constant σn−1 curvature in Minkowski space are either convex or can be split into a product form. 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.
Study inverse curvature flows for capillary hypersurfaces in a unit ball.
problem Understanding the behavior of capillary hypersurfaces under inverse curvature flows.
method Investigate inverse curvature flows for strictly convex, capillary hypersurfaces in the unit Euclidean ball.
result Establish existence and convergence results for inverse curvature flows.
The paper proves the existence of specific spacelike hypersurfaces in Minkowski space.
problem Proving the existence of smooth, entire, strictly convex, spacelike hypersurfaces with constant σk curvature. method Analyzing hypersurfaces in Minkowski space, proving existence through curvature and Gauss map properties.
result Existence of smooth, entire, strictly convex, spacelike hypersurfaces with constant σk curvature. The paper extends Liebmann's Theorem to convex hypersurfaces with boundary.
problem Proving properties of convex hypersurfaces with boundary in Euclidean space.
method Analyzing locally convex, embedded, compact, connected CMC hypersurfaces bounded by a closed strictly convex submanifold.
result Spherical caps are the only such hypersurfaces with non-zero constant mean curvature bounded by a (n−1)−sphere. The paper proves geometric inequalities for pinched convex hypersurfaces in de Sitter space.
problem Geometric inequalities for convex hypersurfaces in de Sitter space.
method Locally constrained flows with initial compact spacelike hypersurfaces pinched in de Sitter space.
result Established geometric inequalities related to quermassintegrals and weighted curvature integrals.
Study anisotropic flow for capillary hypersurfaces, proving new inequalities.
problem Anisotropic capillary hypersurfaces and their properties.
method Anisotropic volume-preserving mean curvature flow, new approach for strictly convex initial hypersurfaces.
result Established new Alexandrov-Fenchel inequalities for strictly convex anisotropic capillary hypersurfaces.
Study flow on de Sitter space for convex hypersurfaces.
problem Behavior of locally constrained inverse curvature flow in de Sitter space.
method Analyze flow on de Sitter space with specific initial conditions and inequalities.
result Derive Alexandrov-Fenchel type inequalities.
Study on anisotropic curvature flow for noncompact convex hypersurfaces.
problem Anisotropic curvature flow of noncompact convex hypersurfaces.
method Flow of complete noncompact convex hypersurfaces with anisotropy determined by a Wulff shape.
result The flow exists for all positive time for initial conditions.
Survey on extending rigidity theorems to Riemannian manifolds.
problem Extending classical rigidity theorems to Riemannian manifolds.
method Review and extension of existing rigidity theorems.
result Rigidity results for convex hypersurfaces of homogeneous 3-manifolds.
We consider convex hypersurfaces for which the ratio of principal curvatures at each point is bounded by a function of the maximum principal curvature with limit 1 at infinity. We prove that the ratio of circumradius to inradius is bounded by a function of the circumradius with limit 1 at zero. We apply this result to …
Inverse curvature flows shape star-shaped hypersurfaces into spheres.
problem Evolution of star-shaped hypersurfaces inside a convex cone.
method Inverse curvature flows, convexity of the cone, gradient and Hölder estimates.
result Hypersurfaces converge to a round sphere as time goes to infinity.
In this paper, we study locally strongly convex centroaffine hypersurfaces with parallel cubic form with respect to the Levi-Civita connection of the centroaffine metric. As the main result, we obtain a complete classification of such centroaffine hypersurfaces. The result of this paper is a centroaffine version of the…
The paper studies a flow of convex hypersurfaces using anisotropic curvature functions.
problem Deforming convex hypersurfaces in Euclidean space.
method Fully nonlinear curvature flow involving k-th elementary symmetric function and support function.
result Long-time existence and convergence of the flow under certain assumptions.
The paper proves inequalities for hypersurfaces in a unit ball with specific boundary conditions.
problem Proving inequalities for hypersurfaces in a unit ball with capillary boundary conditions.
method Developed a curvature flow for θ-capillary hypersurfaces and used it to prove quermassintegral inequalities. result Proved full set of quermassintegral inequalities for θ-horocap-convex hypersurfaces. In this article, we will use the harmonic mean curvature flow to prove a new class of Alexandrov-Fenchel type inequalities for strictly convex hypersurfaces in hyperbolic space in terms of total curvature, which is the integral of Gaussian curvature on the hypersurface. We will also use the harmonic mean curvature flow…
In this paper, we establish a general inequality for locally strongly convex centroaffine hypersurfaces in Rn+1 involving the norm of the covariant derivatives of both the difference tensor K and the Tchebychev vector field T. Our result is optimal in that, applying our recent classification for local…
New constructions show stable geodesics and figure-eights in convex hypersurfaces.
problem Constructing stable geodesics and figure-eights in convex hypersurfaces.
method Explicit billiard trajectories with controlled parallel transport in convex polytopes.
result Construction of stable figure-eights and index-zero geodesics in convex hypersurfaces.