arXiv research

Li-Yau inequality applied to curves in 2D space.

In this paper, we derive Li-Yau inequality for unbounded Laplacian on complete weighted graphs with the assumption of the curvature-dimension inequality $CDE'(n,K)$, which can be regarded as a notion of curvature on graphs. Furthermore, we obtain some applications of Li-Yau inequality, including Harnack inequality, hea…

Derives Li & Yau estimates for heat equations on manifolds.

We prove a global Li-Yau inequality for a general Markov semigroup under a curvature-dimension condition. This inequality is stronger than all classical Li-Yau type inequalities known to us. On a Riemannian manifold, it is equivalent to a new parabolic Harnack inequality, both in negative and positive curvature, giving…

We introduce a new version of a curvature-dimension inequality for non-negative curvature. We use this inequality to prove a logarithmic Li-Yau inequality on finite graphs. To formulate this inequality, we introduce a non-linear variant of the calculus of Bakry and Émery. In the case of manifolds, the new calculus and …

The paper establishes new inequalities for Finsler measure spaces.

Probability versions of Li-Yau inequalities for manifolds with boundary.

Establishes a Li-Yau type inequality for curves in any codimension.

Proves Li-Yau inequality for Helfrich functional, ensuring embeddedness in spherical cases.

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.

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

New inequalities generalize Li's theorem on mixed Hodge structures.

Global existence of Willmore flow with boundary via Li-Yau inequality.

Researchers prove inequalities for reaction-diffusion systems using a new curvature-dimension condition.

New inequalities derived for hyperbolic space via specific flows.

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 …

Paper proves inequality for Green function on Kähler manifolds.

New gradient estimates for heat equation on Riemannian manifolds.

Extends Choi-Wang inequality to Li-Xia affine connections.

Researchers prove isoperimetric inequality in specific steady Ricci solitons.

Estimates for eigenvalues on Riemannian manifolds using classical inequalities.

We establish a one-parameter family of Harnack inequalities connecting the constrained trace Li-Yau differential Harnack inequality for a nonlinear parabolic equation to the constrained trace Chow-Hamilton Harnack inequality for this nonlinear equation with respect to evolving metrics related to Ricci flow on a 2-dimen…

Derives matrix Harnack inequalities for semilinear heat equations on manifolds.

Let $(X,d,μ)$ be a $RCD^\ast(K, N)$ space with $K\in mathbb{R}$ and $N\in [1,\infty)$. Suppose that $(X,d)$ is connected, complete and separable, and $\supp μ=X$. We prove that the Li-Yau inequality for the heat flow holds true on $(X,d,μ)$ when $K\ge 0$. A Baudoin-Garofalo inequality and Harnack inequalities for the h…

Study complete manifolds with weighted Poincaré inequality and Ricci curvature bounds.

In this paper, we study Li-Yau gradient estimates for the solutions $u$ to the heat equation $\partial_tu=Δu$ on graphs under the curvature condition $CD(n,-K)$ introduced by Bauer et al. in \cite{BHLLMY}. As applications, we derive Harnack inequalities and heat kernel estimates on graphs. Also we present a type of Ham…

In New York J. Math. 17 (2011), 41--49, Li has obtained an analogue of the Jørgensen inequality in the infinite-dimensional Möbius group. We show that this inequality is strict.

The study examines elastic curves with self-intersections and their properties.

In this paper, we generalize the Cao-Yau's gradient estimate for the sum of squares of vector fields up to higher step under assumption of the generalized curvature-dimension inequality. With its applications, by deriving a curvature-dimension inequality, we are able to obtain the Li-Yau gradient estimate for the CR he…

The paper derives new gradient and Hessian estimates for nonlinear parabolic equations.

We prove inequalities for Laplace eigenvalues of Kaehler manifolds generalising to higher eigenvalues the classical inequality for the first Laplace eigenvalue due to Bourguignon, Li, and Yau in 1994. We also obtain similar inequalities for analytic varieties in Kaehler manifolds.

The article examines entropy-information inequalities for continuous-time Markov chains under curvature-dimension conditions.

The paper generalizes a Steklov eigenvalue inequality for substatic triples under non-negative Ricci curvature.

Derives gradient estimate for a specific nonlinear parabolic equation on Finsler manifolds.

In the first part of this paper, we get new Li-Yau type gradient estimates for positive solutions of heat equation on Riemmannian manifolds with $Ricci(M)\ge -k$, $k\in \mathbb R$. As applications, several parabolic Harnack inequalities are obtained and they lead to new estimates on heat kernels of manifolds with Ricci…

In this paper, by maximum principle and cutoff function, we investigate gradient estimates for positive solutions to two nonlinear parabolic equations under Ricci flow. The related Harnack inequalities are deduced. An result about positive solutions on closed manifolds under Ricci flow is abtained. As applications, gra…

In this paper, motivated by finding sharp Li-Yau type gradient estimate for positive solution of heat equations on complete Riemannian manifolds with negative Ricci curvature lower bound, we first introduce the notion of Li-Yau multiplier set and show that it can be computed by heat kernel of the manifold. Then, an opt…

Proves inequalities for hypersurfaces in the sphere, solving a long-standing problem.

Sharp Li-Yau equality proven for shrinking Ricci solitons without curvature assumptions.

Paper extends isoperimetric inequalities for non-starshaped hypersurfaces.

Derives gradient estimates for CR heat equation on pseudo-Hermitian manifolds.

We prove a linear trace Li-Yau-Hamilton inequality for the Kaehler-Ricci flow. We then use this sharp differential inequality to study the Liouville properties of the plurisubharmonic functions on complete Kaehler manifolds with nonnegative bisectional curvature.

Study preserves planar and graphical properties of curves under elastic flow.

Classifies pinned $p$-elasticae and finds unique optimality exponents.

We prove the Bochner-Weitzenböck formula for the (nonlinear) Laplacian on general Finsler manifolds and derive Li-Yau type gradient estimates as well as parabolic Harnack inequalities. Moreover, we deduce Bakry-Émery gradient estimates. All these estimates depend on lower bounds for the weighted flag Ricci tensor.

We prove a variant of the Davies-Gaffney-Grigor'yan Lemma for the continuous time heat kernel on graphs. We use it together with the Li-Yau inequality to obtain strong heat kernel estimates for graphs satisfying the exponential curvature dimension inequality.

Intra-household inequality continues to remain a neglected corner despite renewed focus on income and wealth inequality. Using the LIS micro data, we present evidence that this neglect is equivalent to ignoring up to a third of total inequality. For a wide range of countries and over four decades, we show that at least…

Improved lower bounds for poly-Laplacian eigenvalues in arbitrary dimensions.