arXiv research

We apply our abstract gradient inequalities developed by the authors in arXiv:1510.03817 to prove Lojasiewicz--Simon gradient inequalities for the harmonic map energy function using Sobolev spaces which impose minimal regularity requirements on maps between closed, Riemannian manifolds. Our Lojasiewicz--Simon gradient …

In this sequel to arXiv:1510.03817, we apply our abstract Lojasiewicz-Simon gradient inequality to prove Lojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions using Sobolev spaces which impose minimal regularity requirements on pairs of connections and sections. The Lojasiewicz-Simon gradient …

We prove several abstract versions of the Lojasiewicz-Simon gradient inequality for an analytic functional on a Banach space that generalize previous abstract versions of this inequality, weakening their hypotheses and, in particular, the well-known infinite-dimensional version of the gradient inequality due to Lojasie…

Uniqueness of nondegenerate blowups for planar networks shown.

For any compact Lie group $G$ and closed, smooth Riemannian manifold $(X,g)$ of dimension $d\geq 2$, we extend a result due to Uhlenbeck (1985) that gives existence of a flat connection on a principal $G$-bundle over $X$ supporting a connection with $L^p$-small curvature, when $p>d/2$, to the case of a connection with …

The elastic flow of curves converges smoothly to a critical point.

The article analyzes the stability of a curve shortening flow for planar networks.

In this note, we prove an ${L^{\frac{n}{2}}}$-energy gap result for Yang-Mills connections on a principal $G$-bundle over a compact manifold without using Lojasiewicz-Simon gradient inequality (arXiv:1502.00668).

The paper proves smooth convergence of evolving hypersurfaces to critical points.

The Willmore flow preserves surface volume, leading to convergence to a sphere.

The paper proves stability and convergence of minimal networks under curvature motion.

In this monograph, we develop results on global existence and convergence of solutions to abstract gradient flows on Banach spaces for a potential function that obeys the Lojasiewicz-Simon gradient inequality. We prove a Lojasiewicz-Simon gradient inequality for the Yang-Mills energy functional over closed, smooth Riem…

Study curves evolving by gradient flow of elastic energy, proving existence, smoothing, and convergence.

It is a consequence of the Morse-Bott Lemma on Banach spaces that a smooth Morse-Bott function on an open neighborhood of a critical point in a Banach space obeys a Lojasiewicz gradient inequality with the optimal exponent one half. In this article we prove converses for analytic functions on Banach spaces: If the Loja…

We study the Riemannian quantiative isoperimetric inequality. We show that direct analogue of the Euclidean quantitative isoperimetric inequality is--in general--false on a closed Riemannian manifold. In spite of this, we show that the inequality is true generically. Moreover, we show that a modified (but sharp) versio…

In this article, we introduce a new method (based on Perelman's lambda-functional) to study the stability of compact Ricci-flat metrics. Under the assumption that all infinitesimal Ricci-flat deformations are integrable we prove: (A) a Ricci-flat metric is a local maximizer of lambda in a C^2,alpha-sense iff its Lichne…

The paper studies the convergence of elastic flows of curves into manifolds, proving smooth convergence under certain conditions.

Study proves Lojasiewicz inequalities for harmonic maps near simple bubble trees.

The paper proves a Lojasiewicz inequality for maps from the 2-sphere to itself.

New functional proves mass positivity for ALE metrics.

In this sequel to [arXiv:1412.4114], we prove an $L^{d/2}$ energy gap result for Yang-Mills connections on principal $G$-bundles, $P$, over arbitrary, closed, Riemannian, smooth manifolds of dimension $d\geq 2$. We apply our version of the Lojasiewicz-Simon gradient inequality [arXiv:1409.1525, arXiv:1510.03815] to rem…

We characterize the rate of convergence of a converging volume-normalized Yamabe flow in terms of Morse theoretic properties of the limiting metric. If the limiting metric is an integrable critical point for the Yamabe functional (for example, this holds when the critical point is non-degenerate), then we show that the…

The paper proves various inequalities on gradient shrinking Ricci solitons.

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

Derives inequality for optimal transport on manifolds.

The paper derives inequalities and formulas for generalized Ricci flow.

Extends gradient estimates for heat equation under Finsler geometric flows.

Using the curvature-dimension inequality proved in Part~I, we look at consequences of this inequality in terms of the interaction between the sub-Riemannian geometry and the heat semigroup $P_t$ corresponding to the sub-Laplacian. We give bounds for the gradient, entropy, a Poincaré inequality and a Li-Yau type inequal…

This paper proves AdaGrad and Adam converge linearly under PL inequality.

We study some equivalent properties of the curvature-dimension conditions $CD(n,K)$ inequality on infinite, but locally finite graph. These equivalences are gradient estimate, Poincaré type inequalities and reverse Poincaré inequalities. And we also obtain one equivalent property of gradient estimate for a new notion o…

By using a coupling method, an explicit log-Harnack inequality with local geometry quantities is established for (sub-Markovian) diffusion semigroups on a Riemannian manifold (possibly with boundary). This inequality as well as the consequent $L^2$-gradient inequality, are proved to be equivalent to the pointwise curva…

New algorithm samples superlinearly growing log-gradient distributions.

The paper proves inequalities for hypersurfaces in weighted manifolds.

This paper explores geometric and analytic aspects of Lojasiewicz inequalities on vector bundles.

The paper proves a reverse isoperimetric inequality and applies it to analyze surface flows.

Study on stability of free boundary Willmore problem using new gradient inequality.

The study establishes inequalities for functions on manifolds using Green function estimates.

New Harnack inequality for heat equation on compact manifolds.

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

Gradient flow of elastic energy converges to elastica.

The paper proves a spin manifold's 4D quasi-Einstein satisfies Hitchin-Thorpe inequality.

For sub-Riemannian manifolds with a chosen complement, we first establish the derivative formula and integration by parts formula on path space with respect to a natural gradient operator. By using these formulae, we then show that upper and lower bounds of the horizontal Ricci curvature correspond to functional inequa…

Paper proves Harnack inequality for $f$-mean curvature flow.

New gradient estimates for heat equation on Riemannian manifolds.

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…

Unified framework for analyzing neural networks trained by gradient descent.

The paper proves gradient estimates for a weighted p-Laplacian equation on Riemannian manifolds.

Proves error bounds for PGD, extending log-Sobolev and Talagrand inequalities.