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.
Explains geometric inequalities for minimal hypersurfaces.
problem Geometric inequalities for minimal hypersurfaces.
method Expository discussion of known inequalities.
result Discussion of classical inequalities for minimal hypersurfaces.
New inequalities for convex curves with multiple geometric factors.
problem Establishing inequalities for convex curves with multiple geometric factors.
method Parametric isoperimetric-type inequalities for closed convex curves with parameter conditions and equality conditions.
result Derived new inequalities and improved versions of existing inequalities.
The paper proves geometric inequalities in sphere using locally constrained flows.
problem Deriving geometric inequalities in sphere.
method Established the longtime existence and convergence of a locally constrained flow.
result Proved new families of three-term geometric inequalities in sphere.
Sharp geometric inequalities for free boundary hypersurfaces in balls.
problem Understanding geometric properties of free boundary hypersurfaces in balls.
method Proving a family of sharp geometric inequalities.
result Family of sharp geometric inequalities for free boundary hypersurfaces in balls.
The paper derives inequalities on Finsler manifolds, influenced by their curvatures.
problem Deriving inequalities on Finsler manifolds.
method Local and global geometric inequalities on Riemannian and Finsler manifolds.
result Generalized Caffarelli-Kohn-Nirenberg and Hardy type inequalities on Finsler manifolds.
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.
The paper finds inequalities in Grassmannian geometry.
problem Understanding geometric properties of Grassmannians.
method Analyzes inequalities for elements in Grassmannians.
result Law of Cosines and geodesic triangle inequalities.
Study on electrostatic systems with boundary, proving new geometric inequalities.
problem Electrostatic systems with boundary in higher dimensions.
method Investigation of electrostatic systems on compact manifolds with boundary, establishing new geometric properties.
result Proved sharp boundary estimates and isoperimetric-type inequalities for electrostatic manifolds.
The paper establishes sharp geometric inequalities for hypersurfaces in warped product manifolds.
problem Geometric inequalities involving three distinct quantities in warped product manifolds.
method Two families of inequalities comparing three geometric quantities in space forms or warped product manifolds.
result Generalizes and extends previous results on Weinstock-type inequalities and Steklov/Wentzell eigenvalues.
Geometric inequalities for static convex domains in hyperbolic space proved.
problem Proving geometric inequalities for static convex domains in hyperbolic space.
method Using static convexity of flow hypersurfaces, new inequalities are derived.
result New family of geometric inequalities for static convex domains in hyperbolic space.
The paper extends geometric inequalities from Euclidean space to Riemannian manifolds.
problem Proving geometric inequalities on smooth oriented Riemannian manifolds.
method Introducing symmetric decreasing rearrangement inequalities and testing their applicability to Riemannian manifolds.
result Smooth co-area formula and re-formulated geometric inequalities on Riemannian manifolds.
This paper explains a technique for proving geometric inequalities.
problem Proving various geometric inequalities in different contexts.
method Unified framework based on Alexandrov-Bakelman-Pucci technique.
result Unified approach to proving geometric inequalities.
The ABP method is used to prove geometric inequalities for submanifolds and tensors.
problem Establishing geometric inequalities for submanifolds and tensors.
method Application of the Alexandrov-Bakelman-Pucci (ABP) method.
result Logarithmic Sobolev inequality and Sobolev-type inequality for submanifolds and tensors.
As we showed in [3], a geometric inequality can be regarded as an optimization problem. In this paper we find another proof for a Chen's inequality,regarding the Ricci curvature [2] and we improve this inequality in the Lagrangian case.
The paper proves geometric inequalities for pinched convex hypersurfaces in de Sitter space.
problem Geometric inequalities for convex hypersurfaces in de Sitter space.
method Locally constrained flows with initial compact spacelike hypersurfaces pinched in de Sitter space.
result Established geometric inequalities related to quermassintegrals and weighted curvature integrals.
The paper proves various inequalities on gradient shrinking Ricci solitons.
problem Understanding geometric inequalities on gradient shrinking Ricci solitons.
method Proving multiple inequalities equivalent on complete gradient shrinking Ricci solitons.
result Various inequalities (Sobolev, logarithmic Sobolev, Schrödinger, etc.) are equivalent on gradient shrinking Ricci solitons.
Through the study of some elliptic and parabolic fully nonlinear PDEs, we establish conformal versions of quermassintegral inequality, the Sobolev inequality and the Moser-Trudinger inequality for the geometric quantities associated to the Schouten tensor on locally conformally flat manifolds.
A generalization of the affine-geometric Wirtinger inequality for curves to hypersurfaces is given.
New weighted geometric inequalities for hypersurfaces in R^n proved.
problem Proving new weighted geometric inequalities for hypersurfaces in R^n.
method Proof of a family of sharp weighted inequalities involving weighted k-th mean curvature integral and quermassintegrals.
result Generalization and new proof of Wei and Zhou's result without relying on earlier results.
Two geometric inequalities are established for Einstein totally real submanifolds in a complex space form. As immediate applications of these inequalities, some non-existence results are obtained.
The paper extends geometric inequalities for nearly spherical sets in various space forms.
problem Investigating weighted inequalities for nearly spherical sets in space forms.
method Generalizing and extending inequalities for nearly spherical sets in C1 and W2,∞ settings, with convex weight functions. result Quantitative stability estimates for weighted inequalities in Rn+1 and Hn+1. Adapts Stein's method for geometric inequalities, addressing boundary terms.
problem Geometric inequalities and their stability under constraints.
method Uses elliptic PDE with oblique boundary condition to handle boundary terms.
result Stability results for various geometric inequalities with respect to a new distance.
The study shows that close hypersurfaces have uniformly bounded inequalities.
problem Bounding inequalities for close hypersurfaces.
method Analyzing families of smooth hypersurfaces close to a fixed one.
result Uniformly bounded constants in Sobolev, Gagliardo-Nirenberg, and geometric Calderón-Zygmund inequalities.
Sharp inequality found for hypersurfaces in curved spaces.
problem Establishing geometric inequalities for hypersurfaces in curved spaces.
method Standard comparison methods in Riemannian Geometry.
result Sharp geometric inequality for closed hypersurfaces in manifolds with asymptotically nonnegative curvature.
The paper proves a geometric capacitary inequality for sub-static manifolds with harmonic potentials.
problem Proving a geometric capacitary inequality for sub-static manifolds with harmonic potentials.
method Introducing a one-parameter family of functions that are monotone along the level-set flow of the potential, up to the optimal threshold.
result Proves a geometric capacitary inequality where the capacity of the horizon plays the same role as the ADM mass in the celebrated Riemannian Penrose Inequality.
We prove three optimal conformal geometric inequalities of Blatter type on the Klein bottle. These inequalities provide conformal lower bounds of the volume and involve lengths of homotopy classes of curves that are candidates to realize the systole.
We give a geometric interpretation of the linear trace Harnack inequality for the Ricci flow.
Study geometric inequalities and boundary estimates for Einstein-type manifolds with boundary.
problem Investigate geometric properties of Einstein-type manifolds with boundary.
method Investigate geometric inequalities and establish boundary estimates.
result Established boundary estimates in terms of eigenvalues and Brown-York mass.
This review explores Ricci soliton inequalities in Riemannian geometry.
problem Understanding geometric and analytic characteristics of Riemannian manifolds.
method Comprehensive study of Ricci soliton inequalities, summarizing historical evolution and current developments.
result Complex interactions between curvature conditions and geometric inequalities.
We generalize Brendle's geometric inequality considered in \cite{B} to static manifolds. The inequality bounds the integral of inverse mean curvature of an embedded mean-convex hypersurface by geometric data of the horizon. As a consequence, we obtain a reverse Penrose inequality on static asymptotically locally hyperb…
Extends gradient estimates for heat equation under Finsler geometric flows.
problem Global gradient estimates for positive solutions to heat equation.
method General compact Finsler CD(−K,N) geometric flow. result Derives Harnack inequality for positive solutions.
The study constructs geometrically decomposable aspherical 4-manifolds with non-zero signature and explores their properties.
problem Characterizing geometrically decomposable aspherical 4-manifolds with non-zero signature.
method Constructing examples and proving inequalities for geometrically decomposable aspherical 4-manifolds.
result All geometrically decomposable aspherical 4-manifolds with non-zero signature satisfy the inequality \( \chi \geq 3|σ| \).
The study improves Wintgen inequalities for submanifolds in specific geometric spaces.
problem Improving Wintgen inequalities for submanifolds in various geometric spaces.
method Analyzing submanifolds in conformally flat manifolds and deriving inequalities for different types of spaces.
result Derived inequalities for submanifolds in various geometric spaces, including Riemannian manifolds of quasi-constant curvature and warped products.
In this article, we propose the notion of the general p-affine capacity and prove some basic properties for the general p-affine capacity, such as affine invariance and monotonicity. The newly proposed general p-affine capacity is compared with several classical geometric quantities, e.g., the volume, the p-var…
We construct transformations which take asymptotically AdS hyperbolic initial data into asymptotically flat initial data, and which preserve relevant physical quantities. This is used to derive geometric inequalities in the asymptotically AdS hyperbolic setting from counterparts in the asymptotically flat realm, whenev…
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.
In this paper, we use the inverse curvature flow to prove a sharp geometric inequality on star-shaped and two-convex hypersurface in hyperbolic space.
Proves Hodge-Riemann relations for mixed valuations and strengthens geometric inequalities.
problem Geometric inequalities and mixed Hodge-Riemann relations for translation-invariant valuations.
method Proves mixed Hodge-Riemann relations for various convex bodies and their mixed volumes.
result Strengthened geometric inequalities for lower dimensional convex bodies.
The paper derives Harnack inequalities for evolving Riemannian manifolds without dimensionality restrictions.
problem Deriving Harnack inequalities for geometric flows with evolving metrics.
method Probabilistic representation of conjugate semigroups and supercontractivity.
result Established dimension-free Harnack inequalities for geometric flows.
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 is a survey article on recent progress of comparison geometry and geometric analysis on Finsler manifolds of weighted Ricci curvature bounded below. Our purpose is two-fold: Give a concise and geometric review on the birth of weighted Ricci curvature and its applications; Explain recent results from a nonlinear an…
The paper proves geometric inequalities for hypersurfaces in weighted manifolds.
problem Geometric inequalities for hypersurfaces in weighted manifolds.
method Noncompact smooth metric measure spaces with nonnegative Bakry-Émery Ricci curvature.
result Sharp geometric inequalities for the boundary of open sets in weighted manifolds.
Disproves Fedotov's conjecture on higher-order Shephard inequalities.
problem Fedotov's conjecture on higher-order Shephard inequalities.
method Using Hodge-Riemann relations for simple convex polytopes.
result Fedotov's conjecture is disproved.
We give a geometric interpretation of Hamilton's matrix Harnack inequality for the Ricci flow as the curvature of a connection on space-time.
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 …
Both analytic and geometric forms of an optimal monotone principle for Lp-integral of the Green function of a simply-connected planar domain Ω with rectifiable simple curve as boundary are established through a sharp one-dimensional power integral estimate of Riemann-Stieltjes type and the Huber analytic and geome…
New method uses entropy dissipation to prove isoperimetric inequalities.
problem Proving isoperimetric inequalities in geometric settings.
method Information-theoretic approach based on entropy dissipation under heat flow.
result New proof of Euclidean isoperimetric inequality with sharp constant.