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.
Universal inequalities found for Laplacian eigenvalues on convex domains.
problem Finding bounds for Laplacian eigenvalues on convex domains.
method Established two universal inequalities.
result Found new bounds for Laplacian eigenvalues.
Magnitude of Euclidean domains predicts Willmore energy in odd dimensions.
problem Magnitude function of compact domains in odd dimensions.
method Asymptotic expansion of magnitude function at infinity.
result Magnitude function determines Willmore energy of boundary in odd dimensions.
Study bounds derivatives of solutions to a specific equation on domains.
problem Bounding second derivatives of solutions to the σk-Yamabe equation. method Proves local pointwise second derivative estimates for positive W2,p solutions. result Establishes bounds for derivatives of solutions to the σk-Yamabe equation. Universal inequalities for Laplacian eigenvalues on convex domains.
problem Eigenvalue distribution of the Laplacian on convex domains.
method Established two universal inequalities.
result Two new inequalities for Laplacian eigenvalues.
The paper defines flexible domains for minimal surfaces in Euclidean spaces and explores their properties.
problem Understanding the flexibility of domains in Euclidean spaces for minimal surfaces.
method Investigates the concept of flexibility in terms of minimal surfaces contained in domains.
result Defines flexible domains and shows how they can be approximated by minimal immersions.
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.
The paper examines geometric properties of domains for the p-Laplacian in Euclidean and hyperbolic spaces.
problem Exploring geometric properties of unbounded extremal domains for the p-Laplacian operator.
method Analyzing properties in Euclidean and hyperbolic spaces, proving constraints on domains and their asymptotic boundaries.
result Extremal domains in two dimensions must be balls, and in hyperbolic space, they have specific geometric constraints.
Study classifies graphs in Euclidean and non-Euclidean spaces with specific curvature conditions.
problem Classifying graphs with prescribed curvature in various spaces.
method Proves rigidity and classification results for graphs in Riemannian manifolds, focusing on R2 and R3. result Provides general splitting theorems for graphs in these settings.
Study optimizes perimeter in convex domains with anisotropic constraints.
problem Optimizing perimeter in convex domains with anisotropic constraints.
method Analytical properties, topological features, and geometric measure theory results.
result Sharp isoperimetric inequalities and existence of minimizers.
New upper bound for Neumann Laplacian eigenvalues on convex domains.
problem Bounding Neumann eigenvalues on convex domains.
method Deriving a new upper bound for eigenvalues.
result Universal inequalities for Neumann eigenvalues derived from the upper bound.
Quasiregular curves are Hölder continuous and have higher integrability.
problem Understanding the Hölder continuity and integrability of quasiregular curves.
method Analyzing the Hölder continuity and integrability of curves defined by a K-quasiregular function with respect to a covector ω. result Quasiregular curves are (1/K)(∥ωVert/∣ω∣ℓ1)-Hölder continuous and have higher integrability. We compute the whole spectrum of the Dirichlet-to-Neumann operator acting on differential p-forms on the unit Euclidean ball. Then, we prove a new upper bound for its first eigenvalue on a domain Ω in Euclidean space in terms of the isoperimetric ratio ${\rm Vol}(\bdΩ)/{\rm Vol}(Ω)$.
Radial graphs with constant mean curvature found in Euclidean space.
problem Existence of hypersurfaces with constant mean curvature.
method Radial graphs over domains of the unit sphere, Dirichlet problem.
result Existence of hypersurfaces with positive constant mean curvature.
We study the existence or not of harmonic diffeomorphisms between certain domains in the Euclidean 2-sphere. In particular, we show harmonic diffeomorphisms from circular domains in the complex plane onto finitely punctured spheres, with at least two punctures. This result follows from a general existence theorem for m…
In this paper we study the maximal stable domains on minimal catenoids in Euclidean and hyperbolic spaces and in H2×R. We in particular investigate whether half-vertical catenoids are maximal stable domains (\emph{Lindelöf's property}). We also consider stable domains on catenoid-cousins in hyperbolic space. …
In this paper we prove that a flat free-boundary minimal n-disk, n≥3, in the unit Euclidean ball Bn+1 is the unique compact free boundary minimal hypersurface in the unit Euclidean ball which the squared norm of the second fundamental form is less than either 4n2 or 4∣x∣2(n−2)2. Mor…
The paper proves new inequalities in hyperbolic space using Euclidean methods.
problem Proving weighted isoperimetric inequalities in hyperbolic space.
method Using isoperimetric inequality with log-convex density in Euclidean space.
result Removed horo-convex assumption and proved new inequalities for star-shaped domains.
Proves uniqueness of capillary disks in 3D domains.
problem Proving uniqueness of capillary disks in three-dimensional domains modeled by elliptic PDEs.
method Using elliptic PDEs and properties of surfaces in 3D domains, generalizing Nitsche's and Hopf's theorems.
result Generalizes Nitsche's result for capillary constant mean curvature disks in the Euclidean ball.
In this paper, we prove some isoperimetric bounds for lower order eigenvalues of the Wentzell-Laplace operator on bounded domains of a Euclidean space or a Hadamard manifold, of the Laplacian on closed hypersurfaces of a Euclidean space or a Hadamard manifold, and of a biharmonic Steklov problem on bounded domains of a…
The ball maximizes the first biharmonic Steklov eigenvalue.
problem Maximizing the first biharmonic Steklov eigenvalue for bounded domains.
method Comparing domains with fixed measure to find the maximum eigenvalue.
result The ball maximizes the first positive biharmonic Steklov eigenvalue.
Investigates properties of a pseudometric on domains in Euclidean space, linking it to hyperbolic geometry.
problem Defines and analyzes a pseudometric on domains in Rn to understand their hyperbolic properties. method Introduces a pseudometric based on conformal harmonic discs and studies its properties and conditions for hyperbolicity.
result Characterizes domains as hyperbolic based on their geometric properties and provides sufficient conditions for hyperbolicity.
We show that the eigenvalues of the intrinsic Dirac operator on the boundary of a Euclidean domain can be obtained as the limits of eigenvalues of Euclidean Dirac operators, either in the domain with a MIT-bag type boundary condition or in the whole space, with a suitably chosen zero order mass term.
In this paper, we use a weighted isoperimetric inequality to give a lower bound on the first Dirichlet eigenvalue of the Laplacian on a bounded domain inside a Euclidean cone. Our bound is sharp, in that only sectors realize it. This result generalizes a lower bound of Payne and Weinberger in two dimensions.
Maximal distortion between geodesic and Euclidean diameters in polygonal domains is studied.
problem Maximal ratio of geodesic to Euclidean diameters in polygonal domains with holes.
method Analyzes convex polygons with holes, using geometric triangulations as a comparison.
result The supremum of the ratio is between Ω(h1/3) and O(h1/2) for convex polygons. We show that if a bounded domain in complex Euclidean space with C1,1 boundary covers a compact manifold, then the domain is biholomorphic to the unit ball.
Characterizes kernel of linearization for minimal surfaces problem
problem Characterizing kernel of linearization for minimal surfaces problem
method Show kernel consists of potential fields and TT fields
result In whole-space Euclidean decomposition, kernel consists of potential fields and TT fields
The paper examines Euclidean domains with nearly maximal Yamabe quotients.
problem Understanding domains with nearly maximal Yamabe quotients in Euclidean space.
method Analyzes the properties of domains in R3 with nearly maximal Yamabe quotients, proving conditions for equality and comparing to quasi-conformal maps. result Domains with nearly maximal Yamabe quotients are diffeomorphic to balls and are close to a ball in a metric space sense.
Sharp inequality outside ball proved using Neumann method.
problem Anisotropic isoperimetric inequality for domains outside an Euclidean ball.
method Applied ABP method to Neumann boundary value problem.
result Proved sharp anisotropic isoperimetric inequality.
We construct open domains in Euclidean 3-space which do not admit complete properly immersed minimal surfaces with an annular end. These domains can not be smooth by a recent result of Martin and Morales
New families of non-tiling domains satisfy Pólya's conjecture.
problem Finding non-tiling domains that satisfy Pólya's conjecture.
method Analyzing partitioning and eigenvalue orders of domains.
result Existence of families of non-tiling domains satisfying Pólya's conjecture.
Study on shape optimization for specific eigenvalue problems on domains.
problem Shape optimization of eigenvalue problems for fourth order Steklov.
method Asymptotic expansion and sharp upper bound derivation.
result Derivation of eigenvalue spectra and shape optimization conclusions.
The paper proves rigidity theorems on holomorphic isometries into homogeneous domains.
problem Characterizing and comparing holomorphic isometries into homogeneous bounded domains.
method Two rigidity theorems on holomorphic isometries into homogeneous bounded domains.
result The flat (definite or indefinite) complex Euclidean space is not a relative of a homogeneous bounded domain.
We investigate the remainder in the asymptotic formula for the number of integer points in a family of bounded domains in the Euclidean space, which remain unchanged along some linear subspace and expand in the directions, orthogonal to this subspace. We prove some estimates for the remainder, imposing additional assum…
Maximal surfaces in Lorentz-Minkowski space have conjugate graphs.
problem Characterizing maximal surfaces in Lorentz-Minkowski space.
method Three proofs showing correspondence to minimal surfaces in Euclidean space.
result Conjugate surface of a maximal graph over a convex domain is also a graph.
We present a boundary version of a theorem about solenoidal unit vector fields with minimum energy on a spherical domain of an odd dimensional Euclidean sphere.
We present in this paper a \boundary version" for theorems about minimality of volume and energy functionals on a spherical domain of threedimensional Euclidean sphere.
In this article we compute the best Sobolev constants for various Hardy-Sobolev inequalities with sharp Hardy term. This is carried out in three different environments: interior point singularity in Euclidean space, interior point singularity in hyperbolic space and boundary point singularity in Euclidean domains.
New GL-GP models learn covariance respecting domain geometry.
problem Suboptimal results from nonparametric regression on restricted domains.
method Graph Laplacian based Gaussian Processes (GL-GPs) with Nyström extension.
result Performance gains in various applications.
Stability is a key aspect of data analysis. In many applications, the natural notion of stability is geometric, as illustrated for example in computer vision. Scattering transforms construct deep convolutional representations which are certified stable to input deformations. This stability to deformations can be interp…
We study polyharmonic (k-harmonic) maps between Riemannian manifolds with finite j-energies (j=1, cdots, 2k-2). We show if the domain is complete and the target is the Euclidean space, then such a map is harmonic.
A simple proof shows standard billiard for certain convex domains.
problem Characterizing billiards in convex domains that are both projective and Minkowski.
method Direct simple proof in C1-smoothness, semi-local and local versions proved. result Standard Euclidean billiard in an appropriate structure.
This paper autoformalizes Euclidean geometry using LLMs and theorem provers.
problem Challenges in formalizing Euclidean geometry due to reliance on diagrams.
method Combines neuro-symbolic framework, SMT solvers, and LLMs to fill in diagrammatic gaps.
result Demonstrates the capability and limitations of LLMs on autoformalizing geometry problems.
We apply Gromov's ham sandwich method to get (1) domain monotonicity (up to a multiplicative constant factor); (2) reverse domain monotonicity (up to a multiplicative constant factor); and (3) universal inequalities for Neumann eigenvalues of the Laplacian on bounded convex domains in a Euclidean space.
UNOT solves optimal transport problems efficiently using neural networks.
problem Computational expense in solving optimal transport problems.
method UNOT (Universal Neural Optimal Transport) uses Fourier Neural Operators to predict OT distances and plans accurately and efficiently.
result UNOT achieves up to 7.4x speedup over the Sinkhorn algorithm while maintaining accuracy.
In this Note we establish a relation between sections in globally generated holomorphic vector bundles on Kähler manifolds, isotropic with respect to a non-degenerate quadratic form, and totally geodesic foliations on Euclidean open domains. We find a geometric condition for a totally geodesic foliation to originate in…
The goal of this article is to show that five explicitly given transformations, a rotation, two screw Heisenberg rotations, a vertical translation and an involution generate the Euclidean Picard modular groups with coefficient in the Euclidean ring of integers of a quadratic imaginary number field. We also obtain the r…
The paper estimates eigenvalues for specific differential operators on curved spaces.
problem Estimating eigenvalues for a class of elliptic differential operators on Riemannian manifolds.
method Analyzes eigenvalue estimates for a broader class of elliptic differential operators in divergence form.
result Provides eigenvalue estimates for Gaussian shrinking solitons and specific domains.