Proves a higher rank rigidity theorem for convex real projective manifolds.
problem No specific problem stated; focuses on proving a theorem.
method Analogue of Ballmann and Burns-Spatzier's higher rank rigidity theorem.
result Proves a higher rank rigidity theorem for convex real projective manifolds.
The paper proves a flat torus theorem for certain groups acting on convex domains.
problem Establishing a flat torus theorem for specific groups acting on convex domains.
method Analyzing discrete groups in mPGLd(R) acting convex co-compactly on a properly convex domain. result An analogue of the flat torus theorem for mCAT(0) spaces is proven for these groups. Convexity theorem for Hamiltonian actions on conformal symplectic manifolds.
problem Establishing convexity in conformal symplectic geometry.
method Proving a convexity theorem for moment maps under Lee type actions.
result Analog of Kirwan's convexity theorem in conformal symplectic geometry.
Paper improves proof of theorem on convex 2-disk homeomorphisms.
problem Space of SL homeomorphisms of a convex 2-disk. method Improved proof of main lemma.
result Major improvement in understanding homeomorphisms of convex 2-disk.
The study proves curvature rigidity for convex polytopes.
problem Proving curvature rigidity for convex polytopes.
method Using Fredholm theory for Dirac operators and a theorem of Fefferman and Phong.
result Scalar curvature rigidity theorem for convex polytopes proved.
Combination theorems for convex projective geometry subgroups.
problem Understanding discrete subgroups in convex projective geometry.
method General combination theorems for discrete subgroups preserving properly convex open subsets.
result Free products of convex cocompact subgroups are convex cocompact.
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.
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. The paper proves a rigidity theorem for non-compact convex sets in hyperbolic 3-space.
problem Determining a closed convex set in hyperbolic 3-space by its boundary metric.
method Pogorelov's rigidity theorem, Hausdorff measure, and complex analysis techniques.
result The intrinsic path metric on the boundary determines a closed convex set up to isometry under certain conditions.
New geometric proof of convex function differentiability and approximation.
problem Second-order differentiability of convex functions and their approximations.
method Elementary geometric approach to prove classical and recent results.
result New proofs of Lusin approximation of convex functions and bodies by C1,1 functions. The study proves Liouville theorems on curved manifolds with convex boundaries.
problem Proving Liouville theorems on manifolds with nonnegative curvature and strictly convex boundary.
method Analyzing smooth compact Riemannian manifolds with nonnegative sectional curvature and strictly convex boundary.
result Derives Liouville theorems and verifies a conjecture about eigenvalues and inequalities.
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.
This note contains some observations on abelian convexity theorems. Convexity along an orbit is established in a very general setting using Kempf-Ness functions. This is applied to give short proofs of the Atiyah-Guillemin-Sternberg theorem and of abelian convexity for the gradient map in the case of a real analytic su…
Entropy rigidity proven for 3D and higher convex projective manifolds.
problem Entropy rigidity for strictly convex projective manifolds.
method Uses techniques from Besson, Courtois, and Gallot's entropy rigidity theorem.
result Uniform lower bounds on volume for finite volume strictly convex projective manifolds in dimensions ≥ 3.
Minimal graph theorem proven for convex domains.
problem Characterizing minimal graphs over convex domains.
method Analyzing minimal surface equation solutions on convex domains.
result Minimal graphs over convex domains are linear.
The study proves a discrete Blaschke theorem for convex polygons in 2-dimensional space forms.
problem Investigating curvature and circumradius constraints for convex polygons in 2-space forms.
method Defining curvature at each vertex and proving a Blaschke-type theorem.
result The circumradius of a convex polygon satisfies a specific inequality related to its vertex curvatures.
Geometrically, Kostant's Convexity Theorem is extended to submetries with a fat section.
problem Extending Kostant's Convexity Theorem to a broader class of representations.
method Introducing a new concept of 'fat section' and proving the theorem for submetries with this property.
result Kostant's Convexity Theorem is partially extended to submetries with a fat section.
We investigate special lcs and twisted Hamiltonian torus actions on strict lcs manifolds and characterize them geometrically in terms of the minimal presentation. We prove a convexity theorem for the corresponding twisted moment map, establishing thus an analog of the symplectic convexity theorem of Atiyah and Guillemi…
Paper proves flexibility of specific relations using convex integration.
problem Holonomic approximation theorem in differential topology.
method Proves the holonomic approximation theorem for first order jets using convex integration.
result Relation is open and ample, leading to flexibility of the theorem.
Paper constructs L2 estimates for flat vector bundles and generalizes Prékopa's theorem.
problem Constructing L2 estimates for flat vector bundles. method Using Hörmander's L2-estimate for the operator d on a flat vector bundle over a p-convex Riemannian manifold. result Generalizes Prékopa's theorem in convex analysis.
New theorem proves convex bodies with specific curvature measures are rescaled Wulff shapes.
problem Characterizing convex bodies based on anisotropic curvature measures.
method Analyzing k-th anisotropic curvature measures and their relation to anisotropic perimeter.
result Arbitrary convex bodies with specific curvature measures are rescaled Wulff shapes.
We introduce and study a new class of $\eps$-convex bodies (extending the class of convex bodies) in metric and normed linear spaces. We analyze relations between characteristic properties of convex bodies, demonstrate how $\eps$-convex bodies connect with some classical results of Convex Geometry, as Helly theorem, an…
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. Ancient Lagrangian flows get limited convex solutions.
problem Controlling convex solutions of Lagrangian flows at antiquity.
method Proving a Liouville type theorem with quadratic growth restrictions.
result Ancient convex solutions are unique.
The paper proves a theorem linking convex body centroids and category theory.
problem Understanding centroids of sections of convex bodies.
method Lusternik-Schnirelmann category theory.
result At least n hyperplanes exist such that the center of mass of their intersection with a convex body lies on the boundary of the convex body.
Proves hard Lefschetz theorem and Hodge-Riemann relations for convex valuations.
problem Proving properties of convex valuations analogous to Kähler manifolds.
method Elliptic operator theory and perturbation theory applied to unbounded operators on a Hilbert space.
result Establishes hard Lefschetz theorem and Hodge-Riemann relations for convex bodies.
Proof of Gromov's theorem on convex polytopes with acute angles.
problem Gromov's conjecture on extremal scalar curvature of convex polytopes.
method Smoothing construction using Dirac operator techniques.
result Detailed proof of Gromov's theorem.
We show that the cone associated with a moment map for an action of a torus on a contact compact connected manifold is a convex polyhedral cone and that the moment map has connected fibers provided the dimension of the torus is bigger than 2 and that no orbit is tangent to the contact distribution. This may be consider…
Generalized Blaschke rolling theorem for curved spaces.
problem Extending classical theorem to curved spaces.
method Generalization to Riemannian manifolds with bounded curvature.
result Sharp results in arbitrary dimensions, new even in constant curvature spaces.
Paper extends Schur's theorem to spherical curves via monotonicity.
problem Comparing chord lengths of convex and spherical curves.
method Monotonicity and expansion module approach.
result Schur's Theorem extended to spherical curves.
Proves Riemannian positive mass theorem with singularities.
problem Proves Riemannian positive mass theorem for specific types of singular manifolds.
method Uses initial data sets with a second fundamental form to transfer convexity between different singularity components.
result Proves the theorem for manifolds with some mean-concave components and others mean-convex.
The Blaschke rolling disk theorem is extended to non-convex domains.
problem Classical inclusion principle for non-convex domains.
method Geometric conditions based on curvature, algorithm for decomposition.
result Necessary and sufficient conditions for rolling disks in non-convex domains.
The paper extends Strassen's theorem to include biased martingales for American options.
problem Existence of martingales for arbitrage-free prices of American options.
method Derives an extension of Strassen's theorem linking biased martingales to strengthened convex order.
result Characterizes the strengthened convex order through integrals with respect to compensated Poisson processes.
The paper proves a Liouville theorem for special Lagrangian equations with convexity conditions.
problem Proving Liouville theorems for special Lagrangian equations with specific conditions.
method Using Neumann-Poincaré inequality, mean value inequality for superharmonic functions, and geometric measure theory.
result Derives global and interior Hessian estimates for solutions of special Lagrangian equations.
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.
The paper extends flow theory with free boundaries, proving key bounds and theorems.
problem Mean convex mean curvature flow with free boundary conditions.
method Triple-approximation scheme combining maximum principle and various theorems.
result A priori bound on the ratio of second fundamental form to mean curvature.
We extend to the framework of locally L0-convex modules some results from classical convex analysis. Namely, randomized versions of Mazur lemma and Krein-Smulian theorem under mild stability properties are provided.
Simplifies Wulff theorem for crystalline shapes using Minkowski Theory.
problem Proving the Wulff theorem for crystalline integrands.
method Direct approach using Minkowski Theory to exploit convex properties.
result Simpler proof of the Wulff theorem for crystalline shapes.
In the last 15 years, White and Huisken-Sinestrari developed a far-reaching structure theory for the mean curvature flow of mean convex hypersurfaces. Their papers provide a package of estimates and structural results that yield a precise description of singularities and of high curvature regions in a mean convex flow.…
Non-compact convex sets in hyperbolic 3-space are rigid under isometries.
problem Rigidity of non-compact convex sets in hyperbolic 3-space
method Proving rigidity using Pogorelov's theorem and properties of locally convex surfaces
result Any intrinsic isometry between the boundaries of two non-compact closed convex subsets extends to a global isometry of the ambient space
We prove that the torsion of any closed space curve which bounds a simply connected locally convex surface vanishes at least 4 times. This answers a question of Rosenberg related to a problem of Yau on characterizing the boundary of positively curved disks in Euclidean space. Furthermore, our result generalizes the 4 v…
The paper proves a theorem about splitting manifolds with specific curvature properties.
problem Understanding the structure of manifolds with nonnegative Ricci curvature and mean-convex boundaries.
method Proving a splitting theorem for manifolds with spectral nonnegative Ricci curvature and mean-convex boundary.
result The manifold is either isometric to a closed manifold with nonnegative Ricci curvature or has no interior ends.
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.
The main point of this paper is to prove the following useful result: If the almost everywhere 2-jet of a locally quasi-convex function u satisfies a degenerate elliptic constraint F, then u is F-subharmonic, i.e., u is a viscosity F-subsolution. This AE Theorem makes otherwise difficult results transparent. Some insta…
The paper defines quasi-convex subsets in spaces with lower curvature bound.
problem Understanding the geometry of spaces with lower curvature bound.
method Introducing and exploring quasi-convex subsets in Alexandrov spaces.
result Quasi-convex subsets are a fundamental concept for comparing Riemannian and Alexandrov spaces.
Theorem proves congruence for compact submanifolds in a sphere.
problem Understanding submanifolds in a sphere with specific embedding properties.
method Used a Reilly type formula for space forms.
result Proved a congruence theorem for compact embedded hypersurfaces.
The main goal of the paper is to prove the sandwich theorem for geodesic convex functions in a complete Riemannian manifold. Then by using this theorem we have proved an inequality in a manifold with bounded sectional curvature. Finally, we have shown that the gradient of a convex function is orthogonal to the tangent …
We prove several geometric theorems using tools from the theory of convex optimization. In the Riemannian setting, we prove the max flow-min cut theorem for boundary regions, applied recently to develop a "bit-thread" interpretation of holographic entanglement entropies. We also prove various properties of the max flow…