Boundary of fiber convex domains is a cohomological sphere.
problem Understanding the boundary properties of fiber convex domains.
method Analyzing smooth fiber convex domains with smooth boundaries.
result The boundary is a cohomological sphere.
Classifies ancient convex curves in convex domains.
problem Ancient convex curve shortening flows on convex domains.
method Classification of convex ancient solutions.
result Ancient convex curves in convex domains classified.
Unique compact Fuchsian manifolds with convex boundary are determined by their boundary.
problem Identifying compact Fuchsian manifolds with convex boundaries.
method Proving uniqueness based on the induced path metric on the boundary.
result Compact Fuchsian manifolds with convex boundaries are uniquely determined by the induced path metric on the boundary.
The study proves the existence of free boundary minimal disks in convex regions.
problem Proving the existence of free boundary minimal disks in convex regions.
method Based on a multiplicity-one theorem for the free boundary Simon-Smith min-max theory.
result Existence of at least three embedded free boundary minimal disks in strictly convex domains with nonnegative Ricci curvature.
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.
Study shows non-symmetric convex sets have full boundary limits.
problem Understanding boundaries of non-symmetric convex sets.
method Proved using proximal limit set analysis.
result Proximal limit set equals full projective boundary for non-symmetric irreducible divisible convex sets.
Anti-de Sitter spacetimes with convex boundaries are uniquely determined by their boundary metrics.
problem Identifying anti-de Sitter spacetimes based on boundary properties.
method Proving rigidity for spacetimes with holonomy close to Fuchsian and convex boundaries.
result Globally hyperbolic compact anti-de Sitter spacetimes are determined by their boundary metrics.
Geometric data uniquely determines convex subsets in hyperbolic manifolds.
problem Determining convex subsets in hyperbolic manifolds based on boundary data.
method Using conformal structure, induced metric, and third fundamental form on boundary components.
result Convex subsets are uniquely determined by boundary data.
Classifies ancient flows in a disc with boundary.
problem Ancient convex flows in a disc with boundary.
method Classifies flows using curve shortening.
result Ancient convex flows in a disc are classified.
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. Unique ancient convex flow in a ball with free boundary found.
problem Classifying convex ancient free boundary mean curvature flows in the ball.
method Proof of existence and uniqueness in every dimension.
result A unique (modulo rotations and translations) convex ancient mean curvature flow found.
The paper proves the existence and properties of geodesics on convex surfaces.
problem Existence and properties of geodesics on convex surfaces with free boundaries.
method Free boundary curve shortening flow on closed surfaces with strictly convex boundary.
result Existence of two free boundary embedded geodesics and geodesics with Morse Index 1 and 2.
Compact Special Weingarten surfaces with planar convex boundaries are disks.
problem Characterizing Special Weingarten surfaces with specific boundary conditions.
method Proved a Ros-Rosenberg theorem in the context of Special Weingarten surfaces.
result Compact Special Weingarten surfaces with planar convex boundaries are topological disks.
Study ancient solutions to free boundary mean curvature flow in convex manifolds.
problem Understanding ancient solutions to free boundary mean curvature flow in convex manifolds.
method Establish rigidity results and construct foliations to describe ancient solutions.
result Ancient solutions to free boundary mean curvature flow are rigid and exhaust all possibilities under certain conditions.
In this paper we find strictly locally convex hypersurfaces in Rn+1 with prescribed curvature and boundary. The main result is that if the given data admits a strictly locally convex radial graph as a subsolution, we can find a radial graph realizing the prescribed curvature and boundary. As an applicatio…
We study topological obstructions to the existence of a Riemannian metric on manifolds with boundary such that the scalar curvature is non-negative and the boundary is mean convex. We construct many compact manifolds with boundary which admit no Riemannian metric with non-negative scalar curvature and mean convex bound…
Estimates volume of convex Alexandrov spaces with boundary.
problem Estimating volume of Alexandrov spaces with convex boundaries.
method Gradient flow of semi-concave functions.
result Volume upper bound achieved implies Boundary Conjecture.
Study contractibility of boundaries in convex sets and limit sets of subgroups.
problem Understanding contractibility of boundaries and wildness of limit sets in geometric structures.
method Use sufficient conditions for contractibility, study coarse upper curvature bounds, and analyze interpolation in geodesic metric spaces.
result Conditions for contractibility of boundaries and properties of limit sets are established.
Study coning totally geodesic boundaries of hyperbolic manifolds.
problem Understanding metrics on coned-off spaces of hyperbolic manifolds.
method Analyzing the geometric and group-theoretic properties of coned-off spaces.
result Explicit conditions for negatively curved metrics and locally convex subsets.
New theorem on 3-manifolds with curvature and convex boundary.
problem Understanding 3-manifolds with specific curvature and boundary properties.
method Analyzes properties of Riemannian 3-manifolds with nonnegative scalar curvature and mean-convex boundary.
result Shows flatness of certain 3-manifolds containing specific geometric objects.
The study proves splitting theorems for manifolds with specific curvature and boundary conditions.
problem Proving splitting theorems for manifolds with specific curvature and boundary conditions.
method Warped product splitting theorem in manifolds with Ricci curvature bounded from below, requiring parabolic and convex boundary.
result Established splitting results for various manifolds with specific curvature and boundary conditions.
Unique hyperbolic manifolds identified by boundary pleating.
problem Identifying hyperbolic manifolds from their boundary pleating.
method Used pleating measured lamination on the boundary of convex cores.
result Convex co-compact hyperbolic manifolds are uniquely determined by their pleating lamination.
The study finds at least 2 free-boundary minimal disks in convex 3-balls.
problem Finding minimal disks in convex 3-balls.
method Combining mean curvature flow, min-max theory, and degree theory.
result Existence of at least 2 free-boundary minimal disks in convex 3-balls for generic metrics.
Compact hypersurfaces minimize area in convex cones with free boundary.
problem Finding compact hypersurfaces minimizing area in convex cones with free boundary.
method Minimizing an anisotropic area functional under a volume constraint.
result Compact hypersurfaces are contained in a Wulff-shape.
A mean-convex set can be regarded as a barrier for the construction of minimal surfaces. Namely, if we are given a mean-convex set and a null-homotopic Jordan curve on its boundary, then there exists an embedded minimal disk with boundary the given curve contained in the starting mean-convex set. Does a mean-convex set…
Deep learning models generalize by extending decision boundaries outside the convex hull of training data.
problem Understanding how deep learning models generalize beyond their training data.
method Investigation of decision boundaries inside and outside the convex hull of training sets, using various neural network architectures and training regimes.
result Over-parameterization is necessary for deep learning models to extend decision boundaries outside the convex hull of their training data.
In this article a class of closed convex sets in the Euclidean n-space which are the convex hull of their profiles is described. Thus a generalization of Krein-Milman theorem\cite{Lay:1982} to a class of closed non-compact convex sets is obtained. Sufficient and necessary conditions for convexity, affinity and starsh…
We propose a new approach to the study of compact Riemannian manifolds with nonnegative Ricci curvature and strictly convex boundary or positive Ricci curvature and convex boundary. Several conjectures are formulated. Some partial results that support these conjectures are established.
3-manifolds with convex boundary are rigid in certain curvature conditions.
problem Characterizing the rigidity of nonpositively curved manifolds with convex boundaries.
method Using a comparison formula for total curvature of Riemannian hypersurfaces, the authors prove the rigidity of the manifolds.
result Compact Riemannian 3-manifolds with strictly convex simply connected boundary and sectional curvature K≤a≤0 are isometric to a convex domain in a complete simply connected space of constant curvature a.
Let $L=\DD+Z$ for a C1 vector field Z on a complete Riemannian manifold possibly with a boundary. By using the uniform distance, a number of transportation-cost inequalities on the path space for the (reflecting) L-diffusion process are proved to be equivalent to the curvature condition $\Ric-\nn Z\ge - K$ and t…
We prove the convexity estimates of Huisken-Sinestrari for finite-time singularities of mean-convex, mean curvature flow with free boundary in a barrier S. Here S can be any properly embedded, oriented surface in Rn+1 of bounded geometry. We also give an alternative proof that convex mean curvature flows with …
A characterization of the proximal normal cone is obtained and a separation theorem for convex subsets of Riemannian manifolds is established. Moreover, the convexity of the distance function dS for a convex subset S in the cases where the boundary of S contains a geodesic segment, the boundary of S is C2 o…
This paper extends boundary embedding results to coarsely convex spaces.
problem Generalizing boundary embedding results to coarsely convex spaces.
method Generalizing Dydak and Virk's work on Gromov hyperbolic spaces to coarsely convex spaces.
result Maps between coarsely convex spaces induce continuous maps between their boundaries.
Generalizes rigidity of scalar curvature for convex domains.
problem Rigidity of scalar curvature for convex domains.
method Harmonic spinors on convex domains with boundary conditions constructed by Brendle.
result Rigidity results on comparison of scalar curvature and scaled mean curvature on the boundary for any convex domain.
Extends Alexandrov's result to unbounded convex domains in hyperbolic 3-space.
problem Determining unbounded convex domains in hyperbolic 3-space from boundary data.
method Using conformal structure and induced metric on the boundary.
result A wide range of boundary data can be realized on unbounded convex domains in hyperbolic 3-space.
Manifolds uniquely identified by boundary distance differences.
problem Identifying Riemannian manifolds by their boundary distances.
method Distance difference representation on non-convex boundaries without restrictions.
result Complete Riemannian manifolds uniquely determined by their boundary distances.
Constructs obstructions and deformation principles for positive scalar curvature metrics with mean convex boundaries.
problem Obstructing the existence of positive scalar curvature metrics with mean convex boundaries.
method Atiyah-Patodi-Singer index formula, deformation principle, homotopy equivalences, higher homotopy groups.
result Construction of compact manifolds with nontrivial higher homotopy groups for positive scalar curvature metrics with mean convex boundaries.
Study proves rigidity of minimal hypersurfaces in specific manifolds.
problem Proving rigidity of complete free boundary minimal hypersurfaces.
method Warped θ-bubble method, generalizing capillary surfaces. result No complete two-sided stable free boundary immersions in unit ball of R4. Study convex hyperbolic cone-metrics on 3-manifold boundaries, proving unique bent realizations.
problem Convex hyperbolic cone-metrics on 3-manifold boundaries and their bent realizations.
method Alexandrov-Weyl-type problem, bent metrics, controllably polyhedral, Lipschitz topology.
result Unique bent realizations for convex hyperbolic cone-metrics on 3-manifold boundaries.
Study finds geodesic networks for surfaces with convex boundary.
problem Finding geodesic networks for surfaces with convex boundary.
method Investigates free boundary geodesic networks in surfaces with non-negative sectional curvature and convex boundary.
result Existence of a geodesic network realizing the first width of a surface with non-negative sectional curvature and strictly convex boundary.
Let (M, g) be a compact Einstein manifold with non-empty boundary. We prove that Killing fields at the boundary extend to Killing fields of any (M, g) provided the boundary is weakly convex and a simple condition on the fundamental group holds. This gives a new proof of the classical infinitesimal rigidity of convex su…
The study finds at least two short, simple geodesic chords on a disk with convex boundary.
problem Existence of short, simple geodesic chords on a 2-disk with convex boundary.
method Proof of existence using Riemannian geometry and bounds on lengths.
result Existence of at least two short, simple orthogonal geodesic chords on a 2-disk with convex boundary.
The paper studies Ricci flow on manifolds with boundary, proving existence, uniqueness, and boundary conditions preservation.
problem Ricci flow on manifolds with boundary.
method Proving short-time existence and uniqueness of the solution, and showing boundary conditions preservation.
result The flow preserves natural boundary conditions under certain curvature conditions.
We prove the existence of free boundary minimal annuli inside suitably convex subsets of three-dimensional Riemannian manifolds with nonnegative Ricci curvature − including strictly convex domains of the Euclidean space R3.
No stable minimal submanifolds in certain conformal domains.
problem Stability of minimal submanifolds in conformal domains.
method Analyzing sectional curvatures and boundary convexity.
result No compact stable free boundary minimal submanifolds exist.
Lower bound found for Steklov eigenvalue on curved manifolds.
problem Finding bounds for Steklov eigenvalues on curved spaces.
method Established a new lower bound using geometric curvature conditions.
result Found a new lower bound for the first non-zero Steklov eigenvalue.
Unbounded convex domains have zero mean curvature on disconnected boundaries.
problem Understanding mean curvature in unbounded convex domains.
method Analyzing mean curvature on disconnected boundary components.
result Mean curvature is zero on disconnected boundary components of unbounded mean convex domains.
We observe after Bayle and Rosales that the Levy-Gromov isoperimetric inequality generalizes to convex manifolds with boundary.