arXiv research

New method associates topological classes to Sobolev bundles in critical dimensions.

Develops topological concepts for Morrey-Sobolev bundles in high dimensions.

Unified framework for Sobolev spaces on vector bundles, including explicit integration by parts.

Study compactness and continuity in Sobolev wave front set spaces for smooth vector bundles.

A new statistical model uses Orlicz-Sobolev spaces with Gaussian weight.

We prove the smoothness of the L^2-analytic torsion form on some fiber bundles with non-compact fibers of positive Novikov-Shubin invariant. We do so by generalizing the arguments of Azzali-Goette-Schick to an appropriate Sobolev space, and proving that the Novikov-Shubin invariant remains positive in the Sobolev setti…

Study shows smooth holomorphic structures can be approximated from weak connections.

The paper derives inequalities and formulas for generalized Ricci flow.

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

The paper defines and analyzes $abla$-Sobolev spaces and operators on manifolds.

New Fredholm criteria for pseudodifferential operators on manifolds.

The loop space $L\mathbb{P}_1$ of the Riemann sphere consisting of all $C^k$ or Sobolev $W^{k,p}$ maps from the circle $S^1$ to $\mathbb{P}_1$ is an infinite dimensional complex manifold. The loop group $LPGL(2,\mathbb{C})$ acts on $L\mathbb{P}_1$ . We prove that the group of $LPGL(2,\mathbb{C})$ invariant holomorphic …

Kähler-Ricci flow on Kähler manifolds converges to negative Kodaira dimension

By adapting some ideas of M. Ledoux \cite{ledoux2}, \cite{ledoux-stflour} and \cite{Led} to a sub-Riemannian framework we study Sobolev, Poincaré and isoperimetric inequalities associated to subelliptic diffusion operators that satisfy the generalized curvature dimension inequality that was introduced by F. Baudoin and…

We study geometric variational problems for a class of effective models in quantum field theory known as Faddeev-Skyrme models. Mathematically one considers minimizing an energy functional on homotopy classes of maps from closed 3-manifolds into homogeneous spaces of compact Lie groups. The energy minimizers known as H…

We consider the problem of extending functions φ:\to S^n to functions u:B^{n+1}\to S^n for n=2,3. We assume φto belong to the critical space W^{1,n} and we construct a W^{1,(n+1,\infty)}-controlled extension u. The Lorentz-Sobolev space W^{1,(n+1,\infty)} is optimal for such controlled extension. Then we use such resul…

We study fractional Sobolev and Besov spaces on noncompact Riemannian manifolds with bounded geometry. Usually, these spaces are defined via geodesic normal coordinates which, depending on the problem at hand, may often not be the best choice. We consider a more general definition subject to different local coordinates…

The study shows how to regularize weakly harmonic maps using Sobolev norms and Coulomb frames.

Proves sub-Riemannian manifolds are infinitesimally Hilbertian with arbitrary measures.

The paper extends von Neumann's theory to normed modules and shows how they can be represented.

Uniform bounds prove connection between Kähler metrics and RCD spaces.

This monograph develops the theory of covariant Schrödinger semigroups acting on sections of vector bundles over noncompact Riemannian manifolds from scratch. Contents: I. Sobolev spaces on vector bundles II. Smooth heat kernels on vector bundles III. Basis differential operators in Riemannian manifolds IV. Some specif…

The paper develops quantitative estimates for holomorphic sections over bounded domains.

Let $X$ be a compact connected orientable CR manifold with the action of a connected compact Lie group $G$. Under natural pseudoconvexity assumptions we show that the CR Guillemin-Strernberg map is Fredholm at the level of Sobolev spaces of CR functions. As application we study this map for holomorphic line bundles whi…

We consider complete Riemannian manifolds with a controlled growth of the covariant derivatives of Ricci curvatures up to order $k-2$ and a controlled decay of the injectivity radii. On such manifolds we construct distance-like functions with a control on covariant derivatives up to order $k$. Alternatively, the assump…

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…

The paper explores variational problems on Riemannian manifolds with special foliations, proving existence results.

New Sobolev inequalities on Kähler manifolds with positive Ricci curvature.

Extends Coulomb gauge existence to non-associative gauge theory.

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

Novel algorithm speeds up computation of Sobolev IPM for graph-based probability measures.

We propose a new Integral Probability Metric (IPM) between distributions: the Sobolev IPM. The Sobolev IPM compares the mean discrepancy of two distributions for functions (critic) restricted to a Sobolev ball defined with respect to a dominant measure $μ$. We show that the Sobolev IPM compares two distributions in hig…

Explains optimal functional inequalities, focusing on Sobolev and fractional Sobolev.

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

In this paper we study the problem of deriving further Sobolev inequalities from a given Sobolev inequality. We use several different methods, including Bessel potentials and Riesz transforms. We apply the results to the Ricci flow to extend the author's results on the $W^{1,2}$ Sobolev inequality along the Ricci flow …

Frank and Lieb proved sharp Sobolev inequalities without rearrangements.

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

Proves Sobolev inequality on manifolds with specific curvature properties.

Alternative proof of Michael-Simon-Sobolev inequality using optimal transport.

Study shows how close functions are to optimal in Riemannian manifolds.

Narasimhan and Ramadas showed that the restricted holonomy group of the Coulomb connection is dense in the connected component of the identity of the gauge group when one considers the product principal bundle $S^3 \times SU(2)\to S^3$. Instead of a base manifold S^3, we consider here a base manifold of dimension $n\ge…

New Sobolev inequalities found for curved spaces.

A Riemannian metric on a manifold M induces a family of Riemannian metrics on the loop space LM depending on a Sobolev space parameter s. We compute the connection forms of these metrics and the higher symbols of their curvature forms, which take values in pseudodifferential operators. These calculations are used in a …

Study noncommutative Sobolev inequalities using quantum state metrics.

The paper examines stability of Sobolev inequalities on manifolds with Ricci curvature bounds.

Study continuity of complex Sobolev functions, with applications to Kaehler metrics.

Improved CR Sobolev inequalities on CR sphere established.

Generalizes Sobolev IPM for graph-based measures using Orlicz geometric structure.