Concrete distribution properties examined on simplex.
problem Properties of Concrete distribution on simplex.
method Reflection and location-scale transformation of uniform distribution; explicit parameterization to Poincaré half-space.
result Fisher information and information metric are hyperbolic space; Fisher-Rao geodesic distance computed.
The paper classifies energy-minimizing sets in specific domains.
problem Classifying volume-constraint local energy-minimizing sets.
method Proved a Poincaré-type inequality for stable sets.
result Relative boundary of energy-minimizing sets is smooth.
We investigate the possibility of improving the p p p -Poincaré inequality ∥ ∇ H N u ∥ p ≥ Λ p ∥ u ∥ p \|\nabla_{\mathbb{H}^N} u\|_p \ge Λ_p \|u\|_p ∥ ∇ H N u ∥ p ≥ Λ p ∥ u ∥ p on the hyperbolic space, where p > 2 p>2 p > 2 and Λ p : = [ ( N − 1 ) / p ] p Λ_p:=[(N-1)/p]^{p} Λ p := [( N − 1 ) / p ] p is the best constant for which such inequality holds. We prove several different, and independent, improved inequalities, one of which is …
Fundamental solutions of Dirac type operators are introduced for a class of conformally flat manifolds. This class consists of manifolds obtained by factoring out the upper half-space of R n \mathbb{R}^n R n by arithmetic subgroups of generalized modular groups. Basic properties of these fundamental solutions are presented t…
Explicit formulas for fractional GJMS operators on hyperbolic spaces and inequalities proved.
problem Proving explicit formulas and inequalities for fractional operators on hyperbolic spaces.
method Scattering theory on hyperbolic space, Helgason-Fourier analysis, and special function analysis.
result Sharp constants in fractional Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities coincide with Euclidean space constants.
Researchers prove rigidity of convex polytopes in hyperbolic space using spinor techniques.
problem Proving rigidity of convex polytopes in hyperbolic space.
method Spinor techniques and recent smoothing constructions of Brendle-Wang.
result Scalar curvature rigidity for parabolic convex polytopes in hyperbolic space.
Paper classifies critical points in half-space with new distance function.
problem Classifying critical points in half-space with capillary CMC hypersurfaces.
method New shifted distance function for capillary problem in half-space.
result Proves Alexandrov-type theorem for singular capillary CMC hypersurfaces.
We study, on a weighted Riemannian manifold of Ric N ≥ K > 0 _{N} \geq K > 0 N ≥ K > 0 for N < − 1 N < -1 N < − 1 , when equality holds in the isoperimetric inequality. Our main theorem asserts that such a manifold is necessarily isometric to the warped product R × cosh ( K / ( 1 − N ) t ) Σ n − 1 \mathbb{R} \times_{\cosh(\sqrt{K/(1-N)}t)} Σ^{n-1} R × c o s h ( K / ( 1 − N ) t ) Σ n − 1 of hyperbolic nature, where Σ n − 1 Σ^{n-1} Σ n − 1 i…
Study extends eigenvalue formulas to weighted manifolds and proves global rigidity theorems.
problem Eigenvalue formulas and rigidity theorems for weighted manifolds.
method Extends variational formulae to weighted manifolds, proving global rigidity theorems.
result Global rigidity theorems for critical domains in Gaussian half-space.
Paper studies stability of curved surfaces in a half-space.
problem Stability of anisotropic capillary hypersurfaces in a half-space.
method Analyzes weak stability and proves Bernstein-type theorems.
result Compact hypersurfaces are stable if and only if they are a truncated Wulff shape.
Developed a half-space model for pseudo-hyperbolic space.
problem Modeling pseudo-hyperbolic space for any dimensions.
method Created an isometric embedding of pseudo-hyperbolic space into a half-space.
result Geodesics, totally geodesic submanifolds, horospheres, and isometry group are described in the half-space model.
Paper introduces capillary Schwarz symmetrization in half-space.
problem Capillary problems in half-space.
method Introduces a special anisotropic gauge to transform capillary symmetrization to convex symmetrization.
result Capillary Schwarz symmetrization in half-space established.
The paper establishes inequalities for p p p -capacitary functions in flat half-spaces.
problem Understanding p p p -capacitary functions in asymptotically flat half-spaces. method Establishes monotone quantities and mass-capacity inequalities.
result Sharp inequalities attain equality on a Schwarzschild half-space.
Study connects weighted isoperimetric problems to nonlocal elliptic operator extensions.
problem Sharp inequalities for weighted Poisson integrals and their extremizers.
method Formulates variational problem on conformal metric measure space.
result Sharp inequalities are linked to variational problem on CCE manifolds.
Anisotropic minimal graphs over half-spaces are flat.
problem Characterizing minimal graphs over half-spaces.
method Maximum principle and fully nonlinear PDE theory.
result Anisotropic minimal graphs over half-spaces are flat.
Study proves no minimal surfaces can be contained in certain half-spaces or cones.
problem Prohibiting minimal surfaces from certain geometric configurations.
method Analyzes weighted minimal surfaces in R 3 \mathbb{R}^3 R 3 with height-dependent weights. result No proper surfaces can be contained in specific half-spaces or cones.
Study on minimal surfaces with free boundary in a half-space, improving index estimates.
problem Non-existence of index two embedded minimal surfaces with free boundary in a half-space.
method Improved estimates of Neumann and Dirichlet indices, simplified proof of lower bounds.
result Answered Ambrozio et al.'s question and provided new lower bounds.
Paper proves flatness of anisotropic minimal graphs in half-spaces.
problem Anisotropic minimal graphs with free boundaries in half-spaces.
method Proves flatness using linear growth conditions.
result Anisotropic minimal graphs in half-spaces are flat if they have at most one-sided linear growth.
The study proves properties of capillary graphs in half-spaces.
problem Characterizing capillary minimal graphs in half-spaces.
method Analyzing tangent cones and regular set properties.
result Capillary minimal graphs in low dimensions or specific cone conditions are linear.
Paper solves inequalities for capillary hypersurfaces in half-spaces.
problem Finding inequalities for convex capillary hypersurfaces in half-spaces.
method Introduced quermassintegrals and constructed a new locally constrained curvature flow to prove convergence to spherical caps.
result Obtained Alexandrov-Fenchel inequalities for convex capillary hypersurfaces.
New theorem for nonlocal minimal surfaces in any dimension.
problem Classical half-space theorems for minimal surfaces.
method Established a new half-space theorem.
result Result holds in any dimension.
This study proves the Half Space Property for RCD(0,N) and RCD(K,N) spaces.
problem Proving the Half Space Property for RCD(K,N) spaces.
method Analyzing locally perimeter minimizing sets and extending Green's functions results.
result The Half Space Property holds for RCD(K,N) spaces under specific conditions.
We propose a simple proof of the vertical half-space theorem for Heisenberg space.
New Minkowski inequality for capillary surfaces in half-space.
problem Establishing a new Minkowski inequality for capillary surfaces.
method Flow of inverse mean curvature type for capillary hypersurfaces in a half-space.
result Derive a new Minkowski-type inequality for star-shaped and mean convex capillary hypersurfaces.
Paper extends Wente's result to anisotropic capillary surfaces in half-spaces.
problem Extending Wente's result to anisotropic capillary surfaces.
method New Heintze-Karcher inequality and Minkowski formula.
result Anisotropic capillary hypersurfaces in half-spaces are Wulff shapes.
Proposes NUV priors for half-space and box constraints.
problem Adding constraints to linear Gaussian models without computational cost.
method Introduces NUV representations for half-space and box constraints.
result Adds constraints to linear Gaussian models without affecting computational tractability.
Paper analyzes iterative learning for concept classes and learns half-spaces.
problem Learning concept classes efficiently with iterative learners.
method Analyzes various settings of iterative learning and provides a constructive algorithm for half-spaces.
result Constructive iterative algorithm for learning half-spaces from informant.
The paper classifies solutions to a Liouville equation on a half-space with a specific boundary condition.
problem Classifying solutions to a Liouville equation with a nonlinear Neumann boundary condition.
method Analyzing the n n n -Laplacian Liouville equation on the half-space R + n \mathbb{R}^{n}_{+} R + n with positive nonlinear Neumann boundary condition. result The classification of solutions extends previous results for n = 2 n=2 n = 2 and p = n p=n p = n . We show that one can obtain logarithmic improvements of L 2 L^2 L 2 geodesic restriction estimates for eigenfunctions on 3-dimensional compact Riemannian manifolds with constant negative curvature. We obtain a ( log λ ) − 1 2 (\logλ)^{-\frac12} ( log λ ) − 2 1 gain for the L 2 L^2 L 2 -restriction bounds, which improves the corresponding bounds of Burq, Gérard …
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.
Inverse metric matrices on Siegel-Jacobi spaces are calculated for Berezin quantization.
problem Calculating inverse metric matrices on Siegel-Jacobi spaces.
method Inversion of metric matrices on X n J {\mathcal{X}}^J_n X n J and i l d e X n J ilde{\mathcal{X}}^J_n i l d e X n J . result Explicit calculations of inverse metric matrices for n = 2 n=2 n = 2 . 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.
For n ≥ 2 , n\geq2, n ≥ 2 , we obtain Liouville type theorems for minimal surface equations in half space R + n \mathbf R^n_+ R + n with affine Dirichlet boundary value or constant Neumann boundary value.
Solves capillary L p L_p L p -Christoffel-Minkowski problem in half-space.
problem Capillary surfaces in half-space geometry.
method Non-collapsing estimate for height and capillary support function.
result Extends Christoffel-Minkowski existence result.
Study flows for capillary Minkowski problems in half-spaces.
problem Existence and behavior of capillary Minkowski problems.
method Anisotropic capillary Gauss curvature flows.
result Long-time existence and asymptotic behavior of flows.
Solves Christoffel-Minkowski problem for capillary convex bodies in Euclidean half-space.
problem Finding capillary convex bodies with prescribed k k k -th capillary area measure. method Solving a Hessian-type equation with Robin boundary condition.
result Existence and uniqueness of a smooth solution under natural conditions.
A 3D catenoid in 4D space is a minimal hypersurface that cannot be extended to a higher-dimensional half-space.
problem Extending the half-space theorem to higher dimensions in R 4 \mathbb{R}^4 R 4 . method Analyzing the topological constraints on minimal hypersurfaces in R 4 \mathbb{R}^4 R 4 . result A complete, properly embedded minimal hypersurface in a slab in R 4 \mathbb{R}^4 R 4 must be a hyperplane. New proofs and inequalities for capillarity problems quantify asymmetries.
problem Quantifying asymmetries in capillarity functionals.
method ABP-type technique, symmetrization, selection-type argument.
result Sharp quantitative inequalities for asymmetries in capillarity problems.
A new screening test for Lasso improves solution speed.
problem Improving the efficiency of Lasso solution methods.
method Safe region with dome geometry based on dual cutting half-spaces.
result The new screening test leads to significant acceleration.
Develops a method to prove Penrose inequality for half-spaces.
problem Proving the Riemannian Penrose inequality for asymptotically flat half-spaces.
method Doubling procedure for asymptotically flat half-spaces with non-negative scalar curvature and mean-convex boundary.
result Obtains the Penrose-type inequality for dimensions 3 to 7.
Study calculates mass of special polyhedra in hyperbolic space.
problem Evaluating mass in hyperbolic geometry.
method Used upper half space model and special polyhedra.
result Evaluated mass functional on polyhedra.
Study classifies solutions to specific equations on half-space and ball.
problem Classifying nonnegative solutions to Q Q Q -flat and constant T T T -curvature equations. method Introduced a biharmonic Poisson kernel and derived its explicit representation formula.
result Established classification theorems for solutions on R + n + 1 \mathbb{R}_+^{n+1} R + n + 1 and B n + 1 \mathbb{B}^{n+1} B n + 1 . The paper proves nonexistence results for translating solitons in r-mean curvature flow.
problem Proving nonexistence of translating solitons in r-mean curvature flow.
method Establishing nonexistence results under suitable growth conditions on curvature and second fundamental form.
result Properly immersed translating solitons cannot be confined to certain half-spaces.
New geometry based on Siegel upper half-space with volume formula.
problem Developing a new 3D geometry based on Siegel upper half-space.
method Constructing a geometry fibered over Siegel upper half-space and providing a volume formula.
result Volume of Siegel-Seifert closed manifolds is the fiber circle length times base manifold's Euler characteristic.
The paper constructs homotopically non-trivial spheres in complexified spaces.
problem Embedding spheres in complexified spaces defined by hyperplane arrangements.
method Introducing locally consistent systems of half-spaces, embedding a sphere, and computing twisted intersection numbers.
result The constructed sphere is homotopically non-trivial if the half-space system is globally consistent.
We study a half-space problem related to graphs in H 2 × R \mathbb{H}^2\times\mathbb{R} H 2 × R , where H 2 \mathbb{H}^2 H 2 is the hyperbolic plane, having constant mean curvature H H H defined over unbounded domains in H 2 \mathbb{H}^2 H 2 .
Paper extends capillary convex body results to anisotropic setting with Alexandrov-Fenchel inequalities.
problem Extending capillary convex body results to anisotropic setting.
method Developed theory for anisotropic capillary convex bodies in half-space and established Alexandrov-Fenchel inequality for mixed volumes.
result Established a general Alexandrov-Fenchel inequality for mixed volumes of anisotropic capillary convex bodies, weakening and extending previous results.
The paper explores properties of 1-surfaces in hyperbolic space and proves strong half-space theorems.
problem Investigating properties of 1-surfaces in Riemannian manifolds with specific curvature bounds.
method Analyzes the intersection problem for 1-surfaces with bounded curvature in a complete Riemannian three-manifold with Ricci curvature bounded from below.
result Strong half-space theorems for complete 1-surfaces in hyperbolic space with bounded curvature.