Solves relative isoperimetric problem on polygonal domains, focusing on corners.
problem Relative isoperimetric problem on polygonal domains in R2. method Developed techniques for polygonal domains, with special attention to corners.
result Solved the relative isoperimetric problem for a square with a square corner removed.
Direct proof of Neumann isoperimetric inequality for convex domains.
problem Neumann isoperimetric inequality on convex domains.
method Direct proof using Riemannian manifold properties.
result New proof of Neumann isoperimetric inequality.
Proves isoperimetric inequality for spacelike domains in generalized Robertson-Walker spaces.
problem Proving the isoperimetric inequality for spacelike domains in generalized Robertson-Walker spaces.
method Locally constrained mean curvature flow
result Proves the isoperimetric inequality
Solves equality case in isoperimetric inequality for non-convex domains.
problem Equality case in relative isoperimetric inequality outside convex sets.
method Analyzes non-convex domains to settle the equality case.
result Solves the equality case for relative isoperimetric inequality outside arbitrary convex sets.
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 a reverse isoperimetric inequality in curved spaces.
problem Finding the minimum area of a region with a boundary of bounded curvature.
method Analyzes domains homeomorphic to a disc in Alexandrov spaces with curvature constraints.
result Establishes a reverse isoperimetric inequality for specific geometric regions.
Paper finds inequalities for convex domains in hyperbolic space.
problem Finding inequalities for convex domains in hyperbolic space.
method Introducing hyperbolic ellipsoids and using orthogonal projection to establish inequalities.
result Affine isoperimetric inequalities for static convex domains in hyperbolic space characterized by hyperbolic ellipsoids.
Sharp isoperimetric inequalities for Neumann eigenvalues in symmetric spaces.
problem Finding bounds for eigenvalues of Neumann Laplacian on domains in symmetric spaces.
method Proving sharp inequalities for eigenvalues in compact and noncompact rank-1 symmetric spaces.
result Generalization of previous results for hyperbolic space and symmetric spaces.
Paper proves isoperimetric inequality for Minkowski spacetime.
problem Maximizing volume of domain of dependence for finite lightcone.
method Lorentz polarisation to study variational problem.
result Maximal volume achieved by spacelike hyperplane truncated finite lightcone.
The paper shows how almost isoperimetric domains are close to spheres.
problem Understanding the geometry of almost isoperimetric domains.
method Analyzing finite perimeter subsets with small isoperimetric deficit and applying integral curvature bounds.
result Finite perimeter subsets with small isoperimetric deficit are close to spheres up to a small measure.
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.
The paper proves Gromov hyperbolicity of certain metrics using isoperimetric inequalities.
problem Investigating Gromov hyperbolicity of specific metrics.
method Using isoperimetric inequalities to characterize Gromov hyperbolicity.
result Characterization of domains where these metrics are Gromov hyperbolic.
Study preserves volume in warped spaces, solves isoperimetric problem.
problem Isoperimetric problem in warped product spaces.
method Normalized hypersurface flow in warped product spaces.
result Solves isoperimetric problem for specific domains.
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.
Sharp inequalities on Riemannian manifolds for domain areas and volumes.
problem Finding sharp isoperimetric inequalities for domains on Riemannian manifolds.
method Generalized convexity, cut distance, mean curvature, extrinsic radius, Hausdorff measure.
result Geodesic balls maximize area-to-volume ratios under certain curvature conditions.
Study proves inequality for eigenvalues in symmetric spaces.
problem Proving an inequality for Steklov eigenvalues in symmetric spaces.
method Analyzes eigenvalues on bounded domains in noncompact rank-1 symmetric spaces.
result Extends previous results to non-Euclidean spaces.
The paper finds bounds for eigenvalues of various operators on domains and hypersurfaces.
problem Finding bounds for eigenvalues of specific operators on domains and hypersurfaces.
method Proving isoperimetric bounds for eigenvalues of the Wentzell-Laplace operator, Laplacian, and biharmonic Steklov problem.
result Interesting rigidity results can be obtained with sharp bounds.
Proves inequality for Steklov eigenvalues in hyperbolic space.
problem Finding bounds for Steklov eigenvalues in hyperbolic geometry.
method Proves isoperimetric inequality for harmonic mean of eigenvalues.
result Establishes inequality for hyperbolic Steklov eigenvalues.
New proof of isoperimetric inequality using Steiner's formula.
problem Proving the isoperimetric inequality in the plane.
method Direct proof using Steiner's formula, bypassing domain existence.
result Establishes the isoperimetric inequality directly.
Study on Neumann eigenvalues controlled by domain isoperimetric ratio.
problem Control the number of Neumann eigenvalues no greater than the first Dirichlet eigenvalue.
method Combination of analytical and numerical results, related to Yau's conjecture.
result Neumann eigenvalues are controlled by the isoperimetric ratio of the domain.
Study the isoperimetric problem in Riemannian manifolds with non-trivial conformal vector fields.
problem Isoperimetric problem in Riemannian manifolds with non-trivial conformal vector fields.
method Introduced a mean curvature type flow to study the isoperimetric problem.
result Established the isoperimetric inequality for star-shaped hypersurfaces in such manifolds.
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 show that any star-shaped convex hypersurface with constant Weingarten curvature in the deSitter-Schwarzschild manifold is a sphere of symmetry. Moreover, we study an isoperimetric problem for bounded domains in the doubled Schwarzschild manifold. We prove the existence of an isoperimetric surface for any value of t…
We prove that a plane domain which is almost isoperimetric (with respect to the L1 metric) is close to a square whose sides are parallel to the coordinates axis. Closeness is measured either by L∞ Haussdorf distance or Fraenkel asymmetry. In the first case, we determine the extremal domains.
Sharp HLS inequality on bounded domains with applications to curvature and isoperimetric constants.
problem Sharp Hardy-Littlewood-Sobolev inequality on bounded domains.
method Extension operator and suitable test functions.
result Existence of extremal functions and abstract domains with zero scalar curvature and larger isoperimetric constant.
Sharp inequalities and symmetries on Riemannian surfaces quantified.
problem Understanding symmetries and asymmetries in Riemannian surfaces.
method Introducing scattering energy to measure asymmetry and proving isoperimetric inequalities.
result Sharp quantitative isoperimetric inequalities and domains with vanishing scattering energy characterized.
Quantifies nearly spherical subsets in complex ball geometry.
problem Isoperimetric inequality for nearly spherical domains in Bergman ball.
method Proves a quantitative isoperimetric inequality for nearly spherical subsets of Bergman ball.
result First result on isoperimetric phenomenon in Bergman ball.
The study confirms two cases of the convex body isoperimetric conjecture in the plane.
problem The least perimeter to enclose a given area inside a unit disk is greater than inside any other convex set.
method Examined symmetric domains and perturbations of the unit disk.
result Two cases of the convex body isoperimetric conjecture are confirmed.
In this paper we prove that isoperimetric sets in three-dimensional homogeneous spaces diffeomorphic to R3 are topological balls. We also prove that in three-dimensional homogeneous spheres isopermetric sets are either two-spheres or symmetric genus-one tori. We then apply our first result to the three-dime…
Study sharp upper bounds for Aharonov-Bohm eigenvalues on surfaces.
problem Finding sharp upper bounds for eigenvalues of magnetic Laplacian.
method Isoperimetric inequalities and bounds in terms of Gaussian curvature.
result Maximal first eigenvalue for geodesic disk on simply connected surfaces.
We establish a new symmetrization procedure for the isoperimetric problem in symmetric spaces of noncompact type. This symmetrization generalizes the well known Steiner symmetrization in euclidean space. In contrast to the classical construction the symmetrized domain is obtained by solving a nonlinear elliptic equatio…
Paper proves a new isoperimetric inequality for Steklov eigenvalues.
problem Finding a new isoperimetric inequality for Steklov eigenvalues.
method Proving a Brock-type inequality under specific conditions.
result Extension of Brock's classical result to Witten-Laplacian.
Paper finds eigenvalue bounds for hyperbolic space domains.
problem Finding eigenvalue bounds for Robin Laplacian in hyperbolic space.
method Lower and upper bounds derived for eigenvalues.
result Geodesic ball maximizes eigenvalue in negative boundary parameter case.
Let N be a complete Riemannian manifold of dimension n+1 whose Riemannian metric g is conformally equivalent to a metric with non-negative Ricci curvature. The normalized Steklov eigenvalues of a bounded domain in N are bounded above in terms of the isoperimetric ratio of the domain. Consequently, the normalized Steklo…
Paper derives inequalities for eigenvalues of Witten-Laplacian under fixed volume constraint.
problem Eigenvalue inequalities of Witten-Laplacian on bounded domains.
method Rearrangement technique and trial functions under fixed weighted volume constraint.
result Several isoperimetric inequalities for eigenvalues of Witten-Laplacian.
In this paper we provide an extension to the Jellett-Minkowski's formula for immersed submanifolds into ambient manifolds which possesses a pole and radial curvatures bounded from above or below by the radial sectional curvatures of a rotationally symmetric model space. Using this Jellett-Minkowski's generalized formul…
Geodesic disks maximize the first non-trivial Neumann eigenvalue on spheres.
problem Maximizing the first non-trivial Neumann eigenvalue on spheres.
method Proving maximizers are geodesic disks.
result Geodesic disks maximize the first non-trivial Neumann eigenvalue.
Solves isoperimetric problem for curl operator on compact 3-manifolds.
problem Isoperimetric problem for the curl operator on compact 3-manifolds.
method Analyzes eigenvalues and optimal domains of the curl operator.
result Optimal lower bounds for eigenvalues of curl operator are always attained.
The main purpose of this paper is to prove a sharp Sobolev inequality in an exterior of a convex bounded domain. There are two ingredients in the proof: One is the observation of some new isoperimetric inequalities with partial free boundary, and the other is an integral inequality (due to Duff [9]) for any nonnegative…
In this article, by following the method in \cite{PT}, combining Willmore energy with isoperimetric inequalities, we construct two examples of singularities under mean curvature flow in H3. More precisely, there exists a torus, which must develop a singularity under MCF before the volume it encloses decreas…
In this paper, we deals with isoperimetric-type inequalities for closed convex curves in the Euclidean plane R^2. We derive a family of parametric inequalities involving the following geometric functionals associated to a given convex curve with a simple Fourier series proof: length, area of the region included by the …
Sharp inequalities for curved surfaces and cones.
problem Optimizing areas in nonpositively curved spaces.
method Proving inequalities for disks and triangles in cones.
result Minimal area properties for specific shapes in cones.
} In this article, we put forward a Neumann eigenvalue problem for the bi-harmonic operator Δ2 on a bounded smooth domain $\Om$ in the Euclidean n-space Rn (n≥2) and then prove that the corresponding first non-zero eigenvalue $Υ_1(\Om)$ admits the isoperimetric inequality of Szegö-Weinberger type: $Υ_…
The paper extends inequalities to closed Riemannian manifolds.
problem Extending inequalities from Euclidean domains to Riemannian manifolds.
method Proving explicit bounds using geometric quantities.
result Explicit bounds on Riemannian manifolds are derived.
Let (M,g) be a n-dimensional Riemannian manifold and Ω be any compact connected domain in M. We study the problem of finding the {\em maxima} of the functional E(Ω) (known as {\em torsional rigidity} associated to Ω) among all domains of prescribed volume v. Our results show tha…
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.
Study sharp inequalities for perimeter functionals in capillarity and convex cones.
problem Quantitative isoperimetric inequalities for perimeter functionals in capillarity and convex cones.
method Derivation of Fuglede-type estimates and application of selection principle.
result Sharp quantitative isoperimetric inequalities in strong and barycentric forms.
Proves a generalized isoperimetric inequality for spheres in dimensions 4 and above.
problem Proving a generalized isoperimetric inequality for spheres in dimensions 4 and above.
method Reduced to a theorem about thick embeddings of graphs, proved using Kolmogorov-Barzdin theorem and max-flow min-cut theorem. Counterexample in dimension 3 uses coarea inequality and winding number computation.
result A generalized isoperimetric inequality for spheres in dimensions 4 and above.