New Heintze-Karcher inequality helps understand droplet shapes.
problem Characterize the shape of droplets inside smooth containers.
method Obtained a new form of the Heintze-Karcher inequality for mean convex hypersurfaces with boundary on curved substrates.
result New mathematical inequality aids in understanding droplet shapes.
The paper extends Heintze-Karcher inequalities to fractional Q-curvature.
problem Extending Heintze-Karcher inequalities to fractional Q-curvature.
method Generalization of Heintze-Karcher inequalities to fractional Q-curvature on conformally compact Einstein manifolds.
result Rigidity theorems for specific values of γ.
Study proves rigidity for Heintze-Karcher inequality in substatic manifolds.
problem Characterizing equality cases in geometric inequalities.
method Rigidity statement and application to warped product settings.
result Fully removes assumption (H4) in Brendle's characterization.
Paper proves inequalities in sub-static warped product manifolds.
problem Proving inequalities in sub-static warped product manifolds.
method Proved Heintze-Karcher type inequalities involving shifted mean curvature.
result Uniqueness results for hypersurfaces satisfying curvature equations.
Optimal inequality for free boundary hypersurfaces in convex domains.
problem Proving an optimal Heintze-Karcher inequality for free boundary hypersurfaces.
method Analyzing anisotropic free boundary hypersurfaces in convex domains.
result Optimal Heintze-Karcher-type inequality achieved for anisotropic free boundary Wulff shapes.
Paper proves inequality for capillary hypersurfaces in a wedge.
problem Proving a best version of Heintze-Karcher inequality for capillary hypersurfaces.
method Utilized Heintze-Karcher method and modified parallel hypersurfaces.
result Classified capillary constant mean curvature hypersurfaces hitting the edge in a wedge.
In this paper, we prove a generalization of Reilly's formula in \cite{Reilly}. We apply such general Reilly's formula to give alternative proofs of the Alexandrov's Theorem and the Heintze-Karcher inequality in the hemisphere and in the hyperbolic space. Moreover, we use the general Reilly's formula to prove a new Hein…
Paper proves inequality linking capillary surfaces to Finsler geometry.
problem Proving a Heintze-Karcher inequality for capillary hypersurfaces.
method Introduced a Finsler metric for geodesic flow and studied the hypersurface properties.
result Established a new inequality relating capillary surfaces to Finsler geometry.
Paper proves inequality for capillary hypersurfaces with new proof.
problem Proving a Heintze-Karcher type inequality for hypersurfaces with capillary boundary.
method Using a mixed boundary value problem in Reilly type formula to establish the inequality.
result New proof of Alexandrov type theorem for capillary hypersurfaces.
The paper shows how to make certain sets on a sphere smooth and flat.
problem Understanding the smoothness of level-sets of distance functions on spheres.
method Isometric embedding into Rn+2, and analysis on codimension-2 graphs. result Level-sets of distance functions on spheres are C1,1-rectifiable. Proves new inequality for hyperbolic space hypersurfaces.
problem Finding inequalities for hypersurfaces in hyperbolic space.
method Proves a Heintze-Karcher type inequality for shifted mean convex hypersurfaces.
result Proves Alexandrov type theorem and uniqueness result for hypersurfaces.
Study geometric inequalities for quasi-Einstein manifolds using new formulas.
problem Investigate geometric inequalities on quasi-Einstein manifolds.
method Use generalized Reilly's formulas and establish new boundary estimates and isoperimetric inequalities.
result Present a Heintze-Karcher type inequality for compact quasi-Einstein manifolds.
In this note we prove the Heintze-Karcher inequality in the context of essentially non-branching metric measure spaces satisfying a lower Ricci curvature bound in the sense of Lott-Sturm-Villani. The proof is based on the the needle decomposition technique for metric measure spaces introduced by Cavalletti-Mondino. Mor…
Study p-Laplacian equation on Riemannian manifolds with positive Ricci curvature.
problem Overdetermined problem for p-Laplacian equation on compact Riemannian manifolds.
method Introduced a new P-function related to the first nonzero eigenvalue for p-Laplacian, derived integral identities, and applied them to achieve inequalities and the Soap Bubble Theorem.
result Achieved the Heintze-Karcher type inequality and the Soap Bubble Theorem.
The paper studies constant mean curvature hypersurfaces in Finsler manifolds.
problem Understanding geometric properties of hypersurfaces in Finsler manifolds.
method Using volume preserving variation and homothetic navigation.
result Deduced a Heintze-Karcher type inequality and proved an Alexandrov type theorem.
The paper solves Serrin-type problems on Riemannian manifolds using new inequalities and identities.
problem Solving Serrin-type problems in Riemannian manifolds.
method Using a Heintze-Karcher inequality, a Soap Bubble result, and a new Pohozaev identity.
result New results on Serrin-type problems in Riemannian manifolds, including rigidity theorems.
The paper proves rigidity results for capillary hypersurfaces in hyperbolic space.
problem Understanding the rigidity of capillary hypersurfaces in hyperbolic space.
method Proving a Heintze-Karcher type inequality and applying it to Alexandrov type theorems.
result Rigidity results for capillary hypersurfaces, including totally umbilical and totally geodesic cases.
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.
Using spinorial techniques, we prove, for a class of pseudo-hyperbolic ambient manifolds, a Heintze-Karcher type inequality. We then use this inequality to show an Alexandrov type theorem in such spaces.
New comparison theorem for submanifolds with geometric inequalities.
problem Geometric inequalities for submanifolds in ambient spaces.
method Explicit Jacobian determinant formula for normal exponential map.
result Establishes new comparison theorem related to Heintze-Karcher's.
Study anisotropic capillary surfaces in a wedge using generalized Minkowski norms.
problem Understanding capillary surfaces with anisotropic forces.
method Generalized Minkowski norm on the unit sphere, new Minkowski formulae, Heintze-Karcher inequality.
result Proved an Alexandrov-type theorem in the anisotropic setting.
Study on Serrin's problem in convex cones with rigidity results and geometric inequalities.
problem Serrin's overdetermined problem in convex cones of Riemannian manifolds.
method Rigidity results, soap bubble theorem, Heintze-Karcher inequality, drift Laplacian analysis.
result Characterization of intersections of geodesic balls with cones in Riemannian manifolds.
The paper proves stability of Wulff shapes using anisotropic curvature functionals.
problem Stability of Wulff shapes under anisotropic curvature.
method Estimates distance to Wulff shape using Lp-norm of traceless F-Hessian of a foliating function. result Quantitative stability results for anisotropic inequalities and problems.
Study proves only origin-centered spheres solve certain curvature problems.
problem Proving uniqueness of solutions to curvature problems.
method Using the Heintze-Karcher inequality, the study proves the uniqueness of smooth, strictly convex solutions to a class of Minkowski type problems.
result Only origin-centered spheres solve isotropic and Lp-Gaussian-Minkowski problems. Study flow on de Sitter space for convex hypersurfaces.
problem Behavior of locally constrained inverse curvature flow in de Sitter space.
method Analyze flow on de Sitter space with specific initial conditions and inequalities.
result Derive Alexandrov-Fenchel type inequalities.
Paper solves a mixed boundary value problem in space forms with umbilical boundaries.
problem Solving a partially overdetermined mixed boundary value problem in space forms.
method Generalizing previous results to domains with partial umbilical boundaries.
result A partially overdetermined problem in a domain with partial umbilical boundary admits a solution if and only if the rest part of the boundary is also part of an umbilical hypersurface.
Paper derives a Reilly type integral formula and applies it to inequalities and eigenvalue problems.
problem Developing a new integral formula and its applications in geometric inequalities and eigenvalue problems.
method Derives a Reilly type integral formula associated with the φ-Laplacian and applies it to inequalities and eigenvalue problems. result Obtains Heintze-Karcher and Minkowski type inequalities, and eigenvalue relationships.
In this article, we will use the harmonic mean curvature flow to prove a new class of Alexandrov-Fenchel type inequalities for strictly convex hypersurfaces in hyperbolic space in terms of total curvature, which is the integral of Gaussian curvature on the hypersurface. We will also use the harmonic mean curvature flow…
We consider closed orientable hypersurfaces in a wide class of warped product manifolds, which include space forms, deSitter-Schwarzschild and Reissner-Nordström manifolds. By using a new integral formula or Brendle's Heintze-Karcher type inequality, we present some new characterizations of umbilic hypersurfaces. These…
Study on p-Laplace equation in convex cones, proving rigidity under specific conditions.
problem Overdetermined problem for p-Laplace equation in convex cones. method Established properties of capacitary potential, used P-function, isoperimetric inequality, and Heintze-Karcher inequality. result Rigidity result under orthogonal intersection assumption.
In this paper we prove that any immersed stable capillary hypersurfaces in a ball in space forms are totally umbilical. This solves completely a long-standing open problem. In the proof one of crucial ingredients is a new Minkowski type formula. We also prove a Heintze-Karcher-Ros type inequality for hypersurfaces in a…
The paper proves reverse inequalities in various geometric settings using curvature radius data.
problem Proving reverse Alexandrov-Fenchel inequalities in different geometric settings.
method Using curvature radius data and associated evolute or focal maps.
result Sharp reverse Alexandrov-Fenchel estimates and inequalities in smooth convex curves and hypersurfaces.
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.
This paper extends IMCF theory to Heisenberg group, solving Penrose inequality.
problem Extending inverse mean curvature flow theory to Heisenberg group.
method Developed a sub-Riemannian theory for Heisenberg group, introduced a flow preserving H-perimeter.
result Established a Minkowski-type formula in Heisenberg group, proving Heintze-Karcher inequality.
In this article, we first establish the main tool - an integral formula for Riemannian manifolds with multiple boundary components (or without boundary). This formula generalizes Reilly's original formula from \cite{Re2} and the recent result from \cite{QX}. It provides a robust tool for sub-static manifolds regardless…
New integral estimates on substatic manifolds improve Alexandrov Theorem.
problem Improving integral estimates on substatic manifolds.
method Introducing a new vector field with nonnegative divergence.
result Generalization and improvement of integral estimates leading to Alexandrov Theorem.
Stability results for geometric equations in warped product spaces.
problem Geometric partial differential equations in warped product spaces.
method Stability theorem development for level sets of functions.
result Quantitative stability theorems for Serrin's problem and Alexandroff's theorem.
The paper derives new inequalities for non-convex domains and flows.
problem Inequalities for non-convex domains and flows.
method Inverse curvature flow and Alexandrov-Fenchel-type inequalities.
result New inequalities for non-convex domains and flows.
We study the isoperimetric, functional and concentration properties of n-dimensional weighted Riemannian manifolds satisfying the Curvature-Dimension condition, when the generalized dimension N is negative, and more generally, is in the range N∈(−∞,1), extending the scope from the traditional range $N \i…
New static black hole uniqueness theorems for negative cosmological constant.
problem Uniqueness of static black holes in asymptotically locally hyperbolic spaces.
method Inequality relating surface gravity and topology, rigidity of Kottler black holes, monotone quantities under IMCF, regularity theorem for IMCF.
result Static black holes are uniquely determined by their geometry and topology.
New optimal surfaces found in Heisenberg group defy Euclidean sphere optimality.
problem Optimizing mean curvature in Heisenberg group sub-Riemannian setting.
method Developed variational theory, established first and second variation formulas, introduced new critical surfaces.
result Identified and characterized a new family of rotationally invariant critical surfaces, the Pansu-Minkowski spheres.
In this paper we prove an area comparison result for certain totally geodesic surfaces in 3-manifolds with a lower bound on the scalar curvature. This result is a variant of a comparison theorem of Heintze-Karcher for minimal hypersurfaces in manifolds of nonnegative Ricci curvature. Our assumptions on the ambient mani…
Refines d'Alembertian for signed Lorentz distance functions in metric measure spacetimes.
problem Exact representation and bounds of d'Alembertian for signed Lorentz distance functions.
method Metric geometry techniques, localization, Sobolev calculus.
result Distributional d'Alembertian is a signed measure with integration by parts formula.
The paper studies curvature measures and volume-preserving flows on convex bodies.
problem Characterizing and understanding convex bodies through anisotropic curvature measures.
method Developed anisotropic curvature measures, used Minkowski formulas and Heintze-Karcher inequalities, and analyzed volume-preserving flows.
result Characterized Wulff shapes via anisotropic curvature measures and proved convergence of volume-preserving flows.
The isoperimetric inequality and related inequalities are explored.
problem Proving the isoperimetric inequality and related inequalities.
method Discussing classical and recent proofs.
result Various proofs of the isoperimetric inequality and Sobolev inequality.
New proof of Willmore inequality using geometric divergence inequality.
problem Proving the Willmore inequality for bounded domains.
method Using a parametric geometric inequality derived from a divergence form geometric differential inequality.
result New proofs of quantitative Willmore-type and weighted Minkowski inequalities.
Lorentz-Finsler geometry reveals new and old inequalities.
problem Finding new inequalities using Lorentz-Finsler geometry.
method Applying reverse Cauchy-Schwarz and reverse triangle inequalities in Lorentz-Finsler geometry.
result Proved new and refined inequalities, including refinements of Aczél's inequality.
The paper derives new inequalities on manifolds and applies them to convex hypersurfaces.
problem Deriving new inequalities on manifolds and convex hypersurfaces.
method Using Fourier theory and geometric implications of Poincare-type inequalities.
result Sharp Minkowski-type inequalities, including stability and Alexandrov-Fenchel inequalities.