arXiv research

Explains geometric inequalities for minimal hypersurfaces.

Survey of methods for solving smooth stochastic variational inequalities.

The paper extends Busemann's inequalities to complex and quaternionic spaces.

Modified Bakry-Émery criterion inequality for Tsallis entropy monotonicity.

The paper extends intersection theory for b-divisors, proving monotonicity and volume inequalities.

Study noncommutative Sobolev inequalities using quantum state metrics.

The paper proves inequalities for submanifolds in Riemannian manifolds.

New proof of log-Brunn-Minkowski inequality for zonoids and convex bodies.

The paper derives inequalities for $p$-capacitary functions in 3-manifolds with nonnegative scalar curvature.

Improved algorithms for convex-concave min-max optimization and monotone variational inequalities.

Develops a dynamical method to prove the sharp Berezin-Li-Yau inequality.

The paper studies frequency monotonicity for solutions of nonlinear equations under Ricci flow.

The paper establishes inequalities for $p$-capacitary functions in flat half-spaces.

Derives monotonic quantities for $p$-harmonic functions on manifolds.

New deficit functions link elliptic and parabolic inequalities, proving log Sobolev.

The paper studies gradient estimates and monotonicity of parabolic frequency for solutions to the Laplacian G_2 flow.

Derive monotone quantities for harmonic functions on asymptotically flat 3-manifolds with nonnegative scalar curvature.

This paper shows how to learn variational inequalities fast with strong monotonicity.

In this paper we first prove some linear isoperimetric inequalities for submanifolds in the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds. Moreover, the equality is attained. Next, we prove some monotonicity formulas for submanifolds with bounded mean curvature vector in warped product manifolds and, as cons…

Derives new monotone quantities for p-harmonic functions on asymptotically flat 3-manifolds.

Alternative neural network training using monotone variational inequality.

The paper proves the monotonicity of a modified Perelman's W-entropy for mean curvature flow.

New proof of Positive Mass Theorem using Green's function and monotonicity formula.

The paper proves a geometric capacitary inequality for sub-static manifolds with harmonic potentials.

Local Sobolev inequality on Ricci flows with applications.

Two simple proofs of the triangle inequality for the Jaccard distance in terms of nonnegative, monotone, submodular functions are given and discussed.

New algorithms reduce variance in solving complex mathematical problems.

The paper proves a Minkowski inequality on specific Riemannian manifolds.

The extragradient method fails for hypomonotone variational inequalities.

We prove Harnack inequalities for hypersurfaces flowing on the unit sphere by $p$-powers of a strictly monotone, 1-homogeneous, convex, curvature function $f$, $0<p\leq 1.$ If $f$ is the mean curvature, we obtain stronger Harnack inequalities.

Sharp Gaussian isoperimetry proven along Ricci flow.

Improved analysis of extragradient methods for structured VIPs.

We give a proof to the Li-Yau-Hamilton type inequality claimed by Perelman on the fundamental solution to the conjugate heat equation. The rest of the paper is devoted to improving the known differential inequalities of Li-Yau-Hamilton type via monotonicity formulae.

A new method solves variational inequality problems with multiple constraints without needing optimal Lagrange multipliers.

We give a family of monotone quantities along smooth solutions to the inverse curvature flows in Euclidean spaces. We also derive a related geometric inequality for closed hypersurfaces with positive k-th mean curvature.

We generalize an entropy calculation of Perelman to the case of domains evolving inside a Ricciflow solution. In the case of Euclidean space as ambient manifold an interesting relation with Harnack inequalities emerges.

The paper defines new geometric concepts on Riemannian manifolds and applies them to optimization problems.

The paper proves an inequality for symmetric polynomials under a fixed point measure.

New algorithms avoid non-monotonic risk curves in statistical learning.

We find a monotone quantity along the inverse mean curvature flow and use it to prove an Alexandrov-Fenchel-type inequality for strictly convex hypersurfaces in the $n$-dimensional sphere, $n \geq 3$.

In this paper, under the generalized curvature-dimension inequality recently introduced by F. Baudoin and N. Garofalo, we obtain differential Harnack inequalities for the positive solutions to the Schödinger equation associated to subelliptic operator with potential. As applications of the differential Harnack inequali…

Simon type monotonicity formulas for the Willmore functional $\int | \mathbf{H} |^2$ in the hyperbolic space $\mathbb{H}^n$ and $\mathbb{S}^n$ are obtained. The formula gives a lower bound of $\int_Σ | \mathbf{H} |^2$ where $Σ^2$ is any closed surface in $\mathbb{H}^n$.

Both analytic and geometric forms of an optimal monotone principle for $L^p$-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…

The purpose of this work is to study some monotone functionals of the heat kernel on a complete Riemannian manifold with nonnegative Ricci curvature. In particular, we show that on these manifolds, the gradient estimate of Li and Yau, the gradient estimate of Ni, the monotonicity of the Perelman's entropy and the volum…

We prove a generalization of the Li-Yau estimate for a board class of second order linear parabolic equations. As a consequence, we obtain a new Cheeger-Yau inequality and a new Harnack inequality for these equations. We also prove a Hamilton-Li-Yau estimate, which is a matrix version of the Li-Yau estimate, for these …

We apply Gromov's ham sandwich method to get (1) domain monotonicity (up to a multiplicative constant factor); (2) reverse domain monotonicity (up to a multiplicative constant factor); and (3) universal inequalities for Neumann eigenvalues of the Laplacian on bounded convex domains in a Euclidean space.

We prove a sharp inequality for hypersurfaces in the n-dimensional Anti-deSitter-Schwarzschild manifold for general n greater or equal to 3. This inequality generalizes the classical Minkowski inequality for surfaces in the three dimensional Euclidean space, and has a natural interpretation in terms of the Penrose ineq…

Harmonic map flow's singularity properties proven with Lojasiewicz inequalities.