arXiv research

Let (M,g) be a compact Riemannian manifold of dimension n \geq 2. In this work we prove the validity of the optimal L^p-Riemannian Gagliardo-Nirenberg inequality for 1 < p \leq 2. Our proof relies strongly on a new distance lemma which. In particular, we extend L^p-Euclidean Gagliardo-Nirenberg inequalities due to Del …

The paper proves the concavity of entropy power for diffusion equations and applies it to new inequalities.

This note proves sharp affine Gagliardo-Nirenberg inequalities which are stronger than all known sharp Euclidean Gagliardo-Nirenberg inequalities and imply the affine $L^{p}-$Sobolev inequalities. The logarithmic version of affine $L^{p}-$Sobolev inequalities is verified. Moreover, An alternative proof of the affine Mo…

In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Gagliardo-Nirenberg inequality with the same exponent $n$ $(n\geq 2)$, then it has exactly the $n$-dimensional volume growth. Besides, two interesting applications have also been given. The one is that we show that if…

The study shows that close hypersurfaces have uniformly bounded inequalities.

The biharmonic flow and Willmore flow are studied in higher dimensions using geometric evolution equations.

In this paper, we study the CR Yamabe flow with zero CR Yamabe invariant. We use the CR Poincaré inequality and a Gagliardo-Nirenberg type interpolation inequality to show that this flow has long time solution and the solution converges to a contact form with flat pseudo-Hermitian scalar curvature exponentially.

Stability of branched immersions with energy constraints.

The paper proves conditions under which certain geometric structures are rigid.

We introduce a family of conformal invariants associated to a smooth metric measure space which generalize the relationship between the Yamabe constant and the best constant for the Sobolev inequality to the best constants for Gagliardo-Nirenberg-Sobolev inequalities $\|w\r|_q \leq C\|\nabla w\|_2^θ\|w\|_p^{1-θ}$. Thes…

Improved score matching methods for estimating score functions and Hessians without high dimensionality.

The paper proves properties of Renyi entropy power on Riemannian manifolds.

The paper proves Hardy inequalities on Finsler manifolds using superharmonicity.

The paper improves stability estimates for soap bubble theorem in curved domains.

Study establishes Pólya-Szegő inequalities on submanifolds with small total mean curvature.

The L 1-Sobolev inequality states that the L n/(n--1)-norm of a compactly supported function on Euclidean n-space is controlled by the L 1-norm of its gradient. The generalization to differential forms (due to Lanzani & Stein and Bourgain & Brezis) is recent, and states that a the L n/(n--1)-norm of a compactly support…

We describe and partially solve a natural Yamabe-type problem on smooth metric measure spaces which interpolates between the Yamabe problem and the problem of finding minimizers for Perelman's $ν$-entropy. This problem reduces in all dimensions on Euclidean space to the characterization of the minimizers of the family …

For a closed Riemannian manifold $(M^m,g)$ of constant positive scalar curvature and any other closed Riemannian manifold $(N^n,h)$, we show that the limit of the Yamabe constants of the Riemannian products $(M\times N,g+rh)$ as $r$ goes to infinity is equal to the Yamabe constant of $(M^m \times R^n, [g+g_E])$ and is …

We define a very general "parametric connect sum" construction which can be used to eliminate isolated conical singularities of Riemannian manifolds. We then show that various important analytic and elliptic estimates, formulated in terms of weighted Sobolev spaces, can be obtained independently of the parameters used …

The isoperimetric inequality and related inequalities are explored.

New proof of Willmore inequality using geometric divergence inequality.

Lorentz-Finsler geometry reveals new and old inequalities.

The paper derives new inequalities on manifolds and applies them to convex hypersurfaces.

The paper proves inequalities on Finsler manifolds under Ricci curvature bounds.

New inequality on sphere generalizes circle inequality.

Paper proves anisotropic Minkowski inequality and related inequalities.

Explains geometric inequalities for minimal hypersurfaces.

The paper finds new inequalities for convex polygons.

The study improves Bochner inequality on Finsler manifolds to derive important inequalities.

The paper proves various inequalities on gradient shrinking Ricci solitons.

Sharp inequality found on three-balls for fourth order Sobolev traces.

Sharp inequalities for star bodies in 2D space.

The paper develops inequalities for log-concave functions and related surface areas.

Study on functional inequalities on simple edge spaces.

Proves inequalities on curved spaces with positive curvature.

The paper derives inequalities on Finsler manifolds, influenced by their curvatures.

Paper refines Talagrand inequality on Euclidean spaces.

New inequalities for convex curves with multiple geometric factors.

Alternative proofs for various inequalities on Riemannian manifolds.

The paper establishes inequalities and gradient estimates for harmonic functions on Finsler measure spaces.

The paper proves a Harnack inequality for heat equations on Finsler metric measure manifolds.

The paper explores how information geometry impacts classical CR inequalities.

Extends Riemannian geometry inequalities with sharper estimates.

The paper proves inequalities for hypersurfaces in weighted manifolds.

The study establishes inequalities on path space for sub-Riemannian manifolds.

Extends Gromov's optimal systolic inequality to manifolds with specific cohomology properties.

Study inequalities on hyperbolic spaces and Riemannian manifolds using symmetrization and heat semigroup.

Paper connects Fenchel-Willmore and Sobolev inequalities for submanifolds in curved spaces.