arXiv research

Study noncommutative Sobolev inequalities using quantum state metrics.

Develops local elliptic regularity for geometrically-natural operators with low regularity coefficients.

We study properties of Sobolev-type metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolev-type metrics with constant coefficients of order 2 and higher is globally well-posed for smooth initial data as well as initial data in certain Sobolev spaces. Thus the space of closed plane…

Study on $G_2$-structures manifold geometry.

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

Study on Brownian motion on discrete curve spaces, proving stochastic completeness.

We define a manifold $M$ where objects $c\in M$ are curves, which we parameterize as $c:S^1\to R^n$ ($n\ge 2$, $S^1$ is the circle). Given a curve $c$, we define the tangent space $T_cM$ of $M$ at $c$ including in it all deformations $h:S^1\to R^n$ of $c$. In this paper we study geometries on the manifold of curves, pr…

We study reparametrization invariant Sobolev metrics on spaces of regular curves. We discuss their completeness properties and the resulting usability for applications in shape analysis. In particular, we will argue, that the development of efficient numerical methods for higher order Sobolev type metrics is an extreme…

For general varifolds in Euclidean space, we prove an isoperimetric inequality, adapt the basic theory of generalised weakly differentiable functions, and obtain several Sobolev type inequalities. We thereby intend to facilitate the use of varifold theory in the study of diffused surfaces.

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

The paper proves inequalities for varifolds on Riemannian manifolds.

In this paper we prove a characterization of $p$-hyperbolic ends on complete Riemannian manifolds which carries a Sobolev type inequality.

A rather complete investigation of anisotropic Bessel potential, Besov, and Hölder spaces on cylinders over (possibly) noncompact Riemannian manifolds with boundary is carried out. The geometry of the underlying manifold near its 'ends' is determined by a singularity function which leads naturally to the study of weigh…

Proves inequality linking function deviation to gradient norm on compact manifolds.

In this paper, firstly, inspired by Natário's recent work \cite{Na}, we use the isoperimetric inequality to derive some Alexandrov-Fenchel type inequalities for closed convex hypersurfaces in the hyperbolic space $\H^{n+1}$ and in the sphere $\SS^{n+1}$. We also get the rigidity in the spherical case. Secondly, we use …

Study shows optimal rates for independence testing via U-statistic permutation tests.

The paper extends inequalities to twisted differential forms on Kähler manifolds.

We prove new Beckner-Sobolev type inequalities on compact Kähler manifolds with positive Ricci curvature. As an application, we obtain a diameter upper bound that improves the Bonnet-Myers bound.

The paper extends sphere theorems to higher-order mean curvature functions on specific hypersurfaces.

Motivated by the equation satisfied by the extremals of certain Hardy-Sobolev type inequalities, we show sharp $L^q$ regularity for finite energy solutions of p-laplace equations involving critical exponents and possible singularity on a sub-space of $\mathbb{R}^n$, which imply asymptotic behavior of the solutions at i…

The skew mean curvature flow(SMCF), which origins from the study of fluid dynamics, describes the evolution of a codimension two submanifold along its binormal direction. We study the basic properties of the SMCF and prove the existence of a short-time solution to the initial value problem of the SMCF of compact surfac…

We provide upper bounds of the expected Wasserstein distance between a probability measure and its empirical version, generalizing recent results for finite dimensional Euclidean spaces and bounded functional spaces. Such a generalization can cover Euclidean spaces with large dimensionality, with the optimal dependence…

Study Hardy identities and inequalities on Cartan-Hadamard manifolds.

The paper proves inequalities for twisted differential forms on manifolds.

We present a new approach for matching regular surfaces in a Riemannian setting. We use a Sobolev type metric on deformation vector fields which form the tangent bundle to the space of surfaces. In this article we compare our approach with the diffeomorphic matching framework. In the latter approach a deformation is pr…

Optimal geometric estimates for Kähler manifolds with bounded Nash entropy

New Sobolev inequalities on Kähler manifolds with positive Ricci curvature.

In this paper, we propose a method for estimating the Sobolev type embedding constant on a domain with minimally smooth boundary. We estimate the embedding constant by constructing an extension operator and computing its operator norm. We also present some examples of estimating the embedding constant for certain domai…

We consider the results of combining two approaches developed for the design of Riemannian metrics on curves and surfaces, namely parametrization-invariant metrics of the Sobolev type on spaces of immersions, and metrics derived through Riemannian submersions from right-invariant Sobolev metrics on groups of diffeomorp…

The paper finds new inequalities for Laplacian and biharmonic eigenvalues on manifolds.

Study improves Poincaré-Sobolev inequalities for differential forms.

Log Sobolev and Michael Simon inequalities for tensor fields on curved manifolds.

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 …

Study disproves conjecture about metric completion of curve spaces.

Completeness of surface metrics established for Sobolev spaces.

Characterizes complex Hessian equations for bounded energy functions.

When revisiting the Faber-Krahn inequality for the principal $p$-Laplacian eigenvalue of a bounded open set in $\mathbb R^n$ with smooth boundary, we simply rename it as the $p$-Faber-Krahn inequality and interestingly find that this inequality may be improved but also characterized through Maz'ya's capacity method, th…

The ABP method is used to prove geometric inequalities for submanifolds and tensors.

The paper derives inequalities and formulas for generalized Ricci flow.

Researchers prove inequalities for differential forms in Heisenberg groups, extending Euclidean results.

Second order Sobolev metrics are a useful tool in the shape analysis of curves. In this paper we combine these metrics with varifold-based inexact matching to explore a new strategy of computing geodesics between unparametrized curves. We describe the numerical method used for solving the inexact matching problem, appl…

Motivated by applications in the field of shape analysis, we study reparametrization invariant, fractional order Sobolev-type metrics on the space of smooth regular curves $\operatorname{Imm}(S^1,\mathbb{R}^d)$ and on its Sobolev completions $\mathcal{I}^{q}(S^1,\mathbb{R}^{d})$. We prove local well-posedness of the ge…

The paper proves gradient and comparison inequalities for RCD spaces.

We define abstract Sobolev type spaces on $\mathsf{L}^p$-scales, $p\in [1,\infty)$, on Hermitian vector bundles over possibly noncompact manifolds, which are induced by smooth measures and families $\mathfrak{P}$ of linear partial differential operators, and we prove the density of the corresponding smooth Sobolev sect…

Study on new Monge-Ampère functionals and their variational problems.

Metrics on shape space are used to describe deformations that take one shape to another, and to determine a distance between them. We study a family of metrics on the space of curves, that includes several recently proposed metrics, for which the metrics are characterised by mappings into vector spaces where geodesics …

Sharp comparison theorems are derived for all eigenvalues of the (weighted) Laplacian, for various classes of weighted-manifolds (i.e. Riemannian manifolds endowed with a smooth positive density). Examples include Euclidean space endowed with strongly log-concave and log-convex densities, extensions to $p$-exponential …

This manuscript served as lecture notes for a mini-course in the 2016 Southern California Geometric Analysis Seminar Winter School. The goal is to give a quick introduction to Kahler geometry by describing the recent resolution of Tian's three influential properness conjectures in joint work with T. Darvas. These resul…