The study shows how strictly convex domains in Euclidean spaces are rigid.
problem Understanding the rigidity of strictly convex domains in Euclidean spaces.
method Proved a rigidity theorem for smooth strictly convex domains in Euclidean spaces.
result Smooth strictly convex domains in Euclidean spaces are rigid.
New group preserves convex shape without repeating patterns.
problem No virtually abelian unipotent group preserves strictly convex domains.
method Examined Heisenberg group acting on convex shape.
result First example of non-virtually abelian unipotent group preserving strictly convex domain.
Study volume expansion on convex domains using Blaschke metric.
problem Volume expansion of Blaschke metric on strictly convex domains.
method Expressed logarithmic coefficient L as integrals of affine invariants over the boundary and formulated intrinsic geometry as conformal Codazzi structure.
result L is a global conformal invariant of the boundary.
In this article, we study convex affine domains which can cover a compact affine manifold. For this purpose, we first show that every strictly convex quasi-homogeneous projective domain has at least C1 boundary and it is an ellipsoid if its boundary is twice differentiable. And then we show that an n-dimensional par…
We consider the existence problem for `Steiner networks' (trivalent graphs with 120 degree angles at each junction) in strictly convex domains, with `Neumann' boundary conditions (orthogonal intersection with the domain boundary.) For each of the three possible combinatorial possibilities, sufficient conditions on the …
Uniform convexity in divisible domains leads to hyperbolic geometry.
problem Understanding the geometry of divisible convex sets in Finsler manifolds.
method Proving β-uniform convexity of a specific Finsler metric. result A strictly convex divisible domain induces a β-uniformly convex Finsler metric. Researchers develop Q-curvature for convex hypersurfaces using ambient metrics.
problem Calculating Q-curvature for convex hypersurfaces in projective manifolds.
method Using the ambient metric, they construct GJMS operators and relate Q-curvature to the logarithmic coefficient in volume expansion.
result Derived first and second variation formulas for strictly convex domains.
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.
Consider a strictly convex bounded regular domain C of R3. For any arbitrary finite topological type we find a compact Riemann surface M, an open domain M⊂M with the fixed topological type, and a conformal complete proper minimal immersion X:M→C which can be extended to a conti…
Sharp inradius estimates for convex domains in 2D Alexandrov spaces.
problem Estimating radii of inscribed balls in convex domains.
method Sharp lower bounds on inradius derived for strictly convex domains in 2D Alexandrov spaces.
result Characterization of the case when inradius bounds are attained.
A complex surface's Teichmüller space can't be locally convex.
problem Proving Teichmüller space's non-convexity locally.
method Analyzing properties of Teichmüller spaces and convex domains.
result Teichmüller space cannot be locally strictly convex.
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 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.
New convex domains in hyperbolic space can have lower fundamental gap than constant potentials.
problem Finding convex domains with lower fundamental gap than constant potentials.
method Constructing specific convex domains and potentials with controlled eigenfunctions.
result Fundamental gap of −Δ+V can be strictly smaller than −Δ for convex domains. Paper shows non-convexity in solutions to Hessian equations.
problem Non-convexity of solutions to k-Hessian equations in exterior domains. method Examples and new proof for quasiconvexity of harmonic functions.
result Solutions to k-Hessian equations are not quasiconvex in exterior domains. The notion of topological degree is studied for mappings from the boundary of a relatively compact strictly pseudo-convex domain in a Stein manifold into a manifold in terms of index theory of Toeplitz operators on the Hardy space. The index formalism of non-commutative geometry is used to derive analytic integral form…
In this paper, we study entire spacelike translating solitons in Minkowski space. By constructing convex spacelike solutions to (1.3) in bounded convex domains, we obtain many entire smooth convex strictly spacelike translating solitons by prescribing boundary data at infinity.
The paper classifies vertices in planar polygons formed by convex domains.
problem Classifying vertices in planar polygons formed by convex domains.
method Analyzing polygons formed by homothets and translates of a convex domain.
result The number of singular boundary points in a C-polygon is between n and 2(n−1)+m for a strictly convex domain with m singular boundary points. In view of A. Andreotti and H. Grauert's vanishing theorem for q-complete domains in C^n, (Théorème de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193--259,) we re-prove a vanishing result by J.-P. Sha, (p-convex Riemannian manifolds, Invent. Math. 83 (1986), no. 3, 437--447,)…
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.
New proof for 2D lens rigidity from billiard trajectories.
problem Determining obstacles in 2D space from billiard trajectories.
method Separate proof for 2D case, using billiard trajectories.
result Proves rigidity for 2D strictly convex bodies.
The paper studies complex Finsler metrics and their equivalence to the Kobayashi metric.
problem Investigating properties and equivalence of complex Finsler metrics.
method Using curvature properties of Bergman metrics and Schwarz lemma, the paper analyzes complex Finsler metrics and their equivalence to the Kobayashi metric.
result Uniform equivalences of the Kobayashi metric and Carathéodory metric on bounded strongly convex domains with smooth boundaries are proven.
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 N be a complete, simply-connected surface of constant curvature κ\leq 0. Moreover, suppose that Ωand \tildeΩ are strictly convex domains in N with the same area. We show that there exists an area-preserving diffeomorphism from Ωto \tildeΩ whose graph is a minimal submanifold of N \times N.
We consider strictly convex hypersurfaces which are evolving by the non-parametric logarithmic Gauss curvature flow subject to a Neumann boundary condition. Solutions are shown to converge smoothly to hypersurfaces moving by translation. In particular, for bounded domains we prove that convex functions with prescribed …
The study pinches conditions for constant mean curvature surfaces in convex 3-manifolds.
problem Understanding the topology and geometry of constant mean curvature surfaces with free boundaries in convex 3-manifolds.
method Provided pinching conditions on the traceless second fundamental form to guarantee surface topology.
result The surface is either a disk, annulus, spherical cap, or Delaunay surface under certain conditions.
New examples of hyperbolic manifolds with convex projective structures.
problem Finding new examples of hyperbolic manifolds with convex projective structures.
method Generalized Dehn filling and real projective structures.
result Infinitely many hyperbolic manifolds with cusps admit properly convex real projective structures.
In this paper, we compute the derivatives of the line segment energy for a symmetric tensor field and apply them to obtain slightly more general log-concavity estimates for positive solutions of heat equations and first eigenfunctions on bounded strictly convex domains.
New method constructs asymptotic convex hypersurfaces via equidistant hyperplanes.
problem Constructing asymptotic convex hypersurfaces in hyperbolic space.
method Approximating hypersurface by geodesic graphs over equidistant hyperplanes.
result Existence of complete, strictly locally convex hypersurfaces with prescribed asymptotic boundary.
The paper finds convex hypersurfaces with specific curvature properties.
problem Finding convex hypersurfaces with prescribed Hessian curvatures and Gauss images.
method Used novel C2 boundary estimates based on orthogonal invariance and infinitesimal rotations. result Proved existence of strictly convex graphic hypersurfaces with prescribed k-Hessian curvatures. Proves Gromov's conjecture and answers Stoker's polyhedron conjecture.
problem Gromov's flat corner domination conjecture and Stoker's conjecture for convex polyhedra.
method Same techniques applied to prove conjectures.
result Proves Gromov's conjecture and answers Stoker's polyhedron conjecture.
Study on curvature equation in Heisenberg group with convex boundary.
problem Existence of solutions to prescribed mean curvature equation in sub-Finsler Heisenberg group.
method Finsler approximation scheme to prove existence of Lipschitz solutions.
result Existence of a Lipschitz solution for the Dirichlet problem.
Sharp lower bound for curvature in Kähler manifolds.
problem Curvature bounds for real hypersurfaces in Kähler manifolds.
method Gauss equation for semi-isometric CR immersions.
result Proves $ rac12$-positivity of Tanaka-Webster scalar curvature.
Study shows surfaces in Lorentz manifold evolve by translation.
problem Investigating space-like graphs over compact convex domains in Lorentz manifold.
method Non-parametric mean curvature flow with contact angle boundary condition.
result Solutions converge to translation-only motion.
Let M,N be two smooth compact hypersurfaces of Rn which bound strictly convex domains equipped with two absolutely continuous measures μ and ν (with respect to the volume measures of M and N). We consider the optimal transportation from μ to ν for the quadratic cost. Let $(φ:m \to \mathbb{R},ψ…
The paper studies quasi-X-convex functions and their applications in optimization.
problem Optimization problems with quasi-X-convex functions. method Definition and study of X-convex, quasi-X-convex, and related functions. result Applications of quasi-X-convex functions in optimization problems. Study finds strictly convex surfaces with specific curvature and boundary in space forms.
problem Finding strictly locally convex hypersurfaces with prescribed curvature and boundary in space forms.
method Using C2 a priori estimates and degree theory arguments, the study establishes existence results. result Existence of strictly locally convex hypersurfaces with prescribed curvature and boundary in space forms.
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.
In this paper we prove that there exists a smooth classical solution to the HJB equation for a large class of constrained problems with utility functions that are not necessarily differentiable or strictly concave. The value function is smooth if admissible controls satisfy an integrability condition or if it is contin…
We find all extremal Lagrangian tori in symplectic unit balls and some toric domains.
problem Characterize extremal Lagrangian tori in symplectic manifolds.
method Analyzing symplectic area and using geometric properties of toric domains.
result Every extremal Lagrangian torus in the unit ball is on the boundary.
The paper proves geometric inequalities and their stabilities for curves in hyperbolic space.
problem Geometric inequalities and their stabilities for curves in hyperbolic space.
method Curve flow for shifted principal curvatures, Heintze-Karcher type inequality for h-convex curves.
result Geometric inequalities and their stabilities for curves in hyperbolic space.
Paper proves disks are holomorphic in Teichmüller spaces.
problem Understanding holomorphic disks in Teichmüller spaces.
method Analyzes totally-geodesic isometries and disk-rigid domains.
result Every isometry is holomorphic or anti-holomorphic.
Study of pseudometric properties on domains in Nagano spaces.
problem Characterize pseudometrics on domains in real-type Nagano spaces.
method Analyze Kobayashi-type pseudometrics on domains, proving properties and computing specific cases.
result The pseudometric is a genuine metric under certain conditions and has specific properties in higher rank.
Study shows bound on Uryson width for specific 3D manifolds.
problem Bounding Uryson width for 3D manifolds with non-negative Ricci curvature and strictly mean convex boundary.
method Proved existence of a Morse function with uniform diameter bounds on level sets.
result Upper bound on Uryson width for the specified 3D manifolds.
The study connects contact forms and Ruelle invariant in convex domains.
problem Understanding the relationship between contact forms and Ruelle invariant in convex domains.
method Using the extrinsic curvature and Ruelle invariant, the authors prove bounds and construct counterexamples.
result First examples of dynamically convex contact 3-spheres not strictly contactomorphic to convex boundaries.
Algorithm extends Euclidean cell decomposition to projective surfaces.
problem Computing Euclidean cell decomposition for non-hyperbolic surfaces.
method Generalised Weeks' algorithm to strictly convex projective surfaces.
result Algorithm successfully decomposes Euclidean cell structure for projective surfaces.
We prove that the Hilbert geometry of a convex domain in Rn has bounded local geometry, i.e., for a given radius, all balls are bilipschitz to a euclidean domain of Rn. As a consequence, if the Hilbert geometry is also Gromov hyperbolic, then the bottom of its spectrum is strictly positive. We…
This study of properly or strictly convex real projective manifolds introduces notions of parabolic, horosphere and cusp. Results include a Margulis lemma and in the strictly convex case a thick-thin decomposition. Finite volume cusps are shown to be projectively equivalent to cusps of hyperbolic manifolds. This is pro…