Alternative proof of Michael-Simon-Sobolev inequality using optimal transport.
problem Proving the Michael-Simon-Sobolev inequality for submanifolds of codimension 2.
method Optimal transport techniques.
result Sharpness of the inequality for submanifolds of codimension 2.
Optimizes transport on submanifolds for curvature inequalities.
problem Proving Michael-Simon-Sobolev inequalities in manifolds with intermediate Ricci curvature bounds.
method Generalizes optimal transport theory to submanifolds and applies to curvature inequalities.
result Proves a variant of the Michael-Simon-Sobolev inequality in manifolds with nonnegative intermediate Ricci curvatures.
Proves inequalities for tensor fields on submanifolds using ABP method.
problem Proving Michael-Simon inequalities for tensor fields.
method Alexandrov-Bakelman-Pucci (ABP) method
result Proved Michael-Simon inequalities for tensor fields.
Proves inequality for tensor fields on curved spaces.
problem Generalizing inequality for tensor fields on curved spaces.
method Alexandrov-Bakelman-Pucci (ABP) method
result Proves Michael-Simon-Sobolev inequality for tensor fields.
Michael-Simon inequality proven for anisotropic energies close to area.
problem Proving Michael-Simon inequality for anisotropic integrands close to area.
method New functional inequality for vector fields on the plane, simplifying Almgren's proof.
result Michael-Simon inequality holds for convex anisotropic integrands close to 1.
The paper proves Michael-Simon inequalities in hyperbolic space using novel curvature flows.
problem Proving the sharp Michael-Simon inequality for mean curvature in hyperbolic space.
method Developed new locally constrained curvature flows for proving the inequality.
result Sharp Michael-Simon inequalities for mean and k-th mean curvatures in starshaped hypersurfaces in hyperbolic space.
This paper continues the study of Alexandrov-Fenchel inequalities for quermassintegrals for k-convex domains. It focuses on the application to the Michael-Simon type inequalities for k-curvature operators. The proof uses optimal transport maps as a tool to relate curvature quantities defined on the boundary of a do…
Sharp Sobolev and Michael-Simon inequalities on curved manifolds.
problem Proving inequalities on manifolds with nonnegative curvature.
method Analyzing asymptotic volume ratio and curvature properties.
result Sharp Sobolev and Michael-Simon inequalities established.
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.
Develop an ABP approach to Sobolev and Michael-Simon inequalities beyond Euclidean volume growth.
problem Developing an ABP approach to Sobolev and Michael-Simon inequalities under volume noncollapsing assumptions.
method Using a refinement of Brendle's contact-set argument to derive lower bounds for the volumes of geodesic balls.
result A Michael-Simon type inequality for immersed submanifolds with nonnegative sectional curvature and volume noncollapsing.
Log Sobolev and Michael Simon inequalities for tensor fields on curved manifolds.
problem Establishing inequalities for tensor fields on curved manifolds.
method Applying the ABP method to symmetric tensor fields on manifolds with nonnegative sectional curvature.
result Log Sobolev and Michael Simon inequalities for tensor fields.
New inequalities derived for hyperbolic space via specific flows.
problem Sharp inequalities for mean and k-th mean curvatures in hyperbolic space.
method Locally constrained inverse curvature flow by Brendle, Guan, and Li.
result Established and verified new sharp inequalities for hyperbolic space.
Proves new Sobolev inequalities for submanifolds in manifolds with nonnegative intermediate Ricci curvature.
problem Proving Sobolev inequalities for submanifolds in manifolds with nonnegative intermediate Ricci curvature.
method Using the Alexandrov-Bakelman-Pucci method to prove Michael-Simon type inequalities.
result Extends existing inequalities to the k-Ricci curvature setting and provides isoperimetric inequalities. Study examines surfaces with bounded fractional mean curvature, proving control over local parametrization.
problem Understanding surfaces with bounded fractional mean curvature.
method Investigates bounded L^p-norm of fractional mean curvature, proving control over local parametrization.
result Proves control over local parametrization, leading to lower Ahlfors-regularity, weak Michael-Simon type inequality, and stability application.
The study proves inequalities on curved spaces without global curvature bounds.
problem Proving inequalities on manifolds with non-negative curvature outside compact sets.
method ABP method localized to regions of non-negative curvature, spectral properties of manifolds.
result Validated isoperimetric and Michael-Simon inequalities on manifolds with asymptotically non-negative curvature.
The paper proves new inequalities and flow properties for hypersurfaces.
problem Investigating new inequalities and flow properties for hypersurfaces.
method New locally constrained mean curvature flow and mean curvature type flow.
result Proves new sharp Michael-Simon inequalities and flow properties.
We prove a sharp logarithmic Sobolev inequality which holds for submanifolds in Euclidean space of arbitrary dimension and codimension. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature.
Completeness of surface metrics established for Sobolev spaces.
problem Ensuring completeness of reparametrization-invariant Sobolev metrics on surface spaces.
method Recasting completeness criteria for infinite-dimensional Riemannian manifolds and applying geometric estimates based on the Michael--Simon--Sobolev inequality.
result Established metric and geodesic completeness for specific Sobolev metrics on immersed surfaces, validating Mumford's conjecture.
The biharmonic flow and Willmore flow are studied in higher dimensions using geometric evolution equations.
problem Prove global existence of the Willmore flow in higher dimensions.
method Apply Michael-Simon-Sobolev inequality and Gagliardo-Nirenberg inequalities to establish local energy estimates and maximal existence time.
result Global existence of the Willmore flow in higher dimensions is proven.
New Sobolev inequality found for mean convex spacelike submanifolds in Minkowski space.
problem Finding a Sobolev inequality for mean convex spacelike submanifolds in Minkowski space.
method Applying the ABP estimate method to spacelike submanifolds in Rn,1. result Obtained a Sobolev inequality without a mean curvature term for mean convex hypersurfaces.
Paper doubles Hessian estimates for special Lagrangian equation with constraints.
problem Estimating Hessian for special Lagrangian equation under general phase constraints.
method Doubling argument, Alexandrov-type theorems.
result Established Hessian estimates for special Lagrangian equation.
Proves Hessian estimates for special Lagrangian equation with new proofs.
problem Interior Hessian estimates for special Lagrangian equation
method Doubling proofs, higher codimension analogue of previous methods
result Higher codimension analogue of gradient estimate for minimal hypersurfaces
We derive several mean value formulae on manifolds, generalizing the classical one for harmonic functions on Euclidean spaces as well as later results of Schoen-Yau, Michael-Simon, etc, on curved Riemannian manifolds. For the heat equation a mean value theorem with respect to `heat spheres' is proved for heat equation …
Sharp inequality for submanifolds in curved spaces.
problem Proving a logarithmic Sobolev inequality for submanifolds.
method Analyzes submanifolds in manifolds with nonnegative sectional curvature.
result Sharp inequality including mean curvature term.
Some of the most known integral inequalities are the Sobolev, Hardy and Rellich inequalities in Euclidean spaces. In the context of submanifolds, the Sobolev inequality was proved by Michael-Simon and Hoffman-Spruck. Since then, a sort of applications to the submanifold theory has been derived from those inequalities. …
In this paper we discuss a simple relation, which was previously missed, between the high co-dimensional isoperimetric problem of finding a filling with small volume to a given cycle, and extinction estimates for singular, high co-dimensional, mean curvature flow. The utility of this viewpoint is first exemplified by t…
Two-dimensional Lagrangian mean curvature equation solved with new inequality.
problem Solving the two-dimensional Lagrangian mean curvature equation.
method Using Warren-Yuan's super isoperimetric inequality and simplified approach.
result Derivation of a modified Hessian bound for solutions.
In this note we generalize an extension theorem in [5] and [9] of the mean curvature flow to the H^{k} mean curvature flow under some extra conditions. The main difficult problem in proving the extension theorem is to find a suitable version of Michael-Simon inequality for the H^{k} mean curvature flow, and to do a sui…
The paper proves Lp-Sobolev inequalities for minimal submanifolds.
problem Proving Lp-Sobolev inequalities for minimal submanifolds. method Optimal mass transport theory on Euclidean submanifolds.
result Asymptotically sharp and codimension-free Sobolev constant for p≥2. 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.
Sharp Lp-logarithmic-Sobolev inequalities on submanifolds with applications to hypercontractivity.
problem Developing inequalities on submanifolds of Euclidean space.
method Optimal mass transport theory on submanifolds, sharpness analysis.
result Sharp inequalities and equality conditions for submanifolds.