arXiv research

Quantitative Sobolev extensions lead to Neumann heat kernel bounds.

Research characterizes traces of Sobolev mappings into manifolds, identifying analytical obstructions and proving surjectivity under certain conditions.

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

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…

Completeness of surface metrics established for Sobolev spaces.

Global invertibility proven for orientation-preserving maps without homeomorphic extension.

In this article we prove completeness results for Sobolev metrics with nonconstant coefficients on the space of immersed curves and on the space of unparametrized curves. We provide necessary as well as sufficient conditions for the coefficients of the Riemannian metric for the metric to be metrically complete and we c…

Extends Lipschitz functions while preserving local constants.

Extends mapping results to non-compact Riemannian manifolds with positive reach.

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…

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 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 …

Existence and rigidity results for lifts in Carnot groups.

Compact embeddings for invariant functions in metric-measure spaces.

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

Let $(X, P)$ be a toric variety. In this note, we show that the $C^0$-norm of the Calabi flow $\varphi(t)$ on $X$ is uniformly bounded in $[0, T)$ if the Sobolev constant of $\varphi(t)$ is uniformly bounded in $[0, T)$. We also show that if $(X, P)$ is uniform $K$-stable, then the modified Calabi flow converges expone…

Sharp log-Sobolev inequalities proved for ${\sf CD}(0,N)$ spaces.

The paper extends Sobolev spaces to Finsler manifolds and shows density of smooth functions.

Paper relaxes the Lipschitz constraint in WGANs to improve performance.

Study geodesics in constrained curve spaces, including elastic curves and concentric circles.

By means of a general gluing and conformal-deformation construction, we prove that any smooth, metrically complete Riemannian manifold with smooth boundary can be realized as a closed domain into a smooth, geodesically complete Riemannan manifold without boundary. Applications to Sobolev spaces, Nash embedding and loca…

In this article we study Sobolev metrics of order one on diffeomorphism groups on the real line. We prove that the space $\operatorname{Diff}_{1}(\mathbb R)$ equipped with the homogenous Sobolev metric of order one is a flat space in the sense of Riemannian geometry, as it is isometric to an open subset of a mapping sp…

We formulate the notion of minimax estimation under storage or communication constraints, and prove an extension to Pinsker's theorem for nonparametric estimation over Sobolev ellipsoids. Placing limits on the number of bits used to encode any estimator, we give tight lower and upper bounds on the excess risk due to qu…

Study shows rates for Laplacian-eigenmap methods in nonparametric regression.

Let $(M,F)$ be a $C^\infty$ Finsler manifold, $p\geq 1$ a real number, $k$ a positive integer and $H_k^p (M)$ a certain Sobolev space determined by a Finsler structure $F$. Here, it is shown that the set of all real $C^{\infty}$ functions with compact support on $M$ is dense in the Sobolev space $H_1^p (M)$. This resul…

Develops Poisson structures on weak Sobolev loop spaces for integrable systems.

We extend the well-known result that any $f \in W^{1,n}(Ω,\mathbb{R}^n)$, $Ω\subset \mathbb{R}^n$ with strictly positive Jacobian is actually continuous: it is also true for fractional Sobolev spaces $W^{s,\frac{n}{s}}(Ω)$ for any $s \geq \frac{n}{n+1}$, where the sign condition on the Jacobian is understood in a distr…

The paper introduces new boundary operators and proves higher order CR Sobolev trace inequalities for Siegel domain and complex ball.

Study infinite circle patterns in the Weil-Petersson class using discrete harmonic functions.

We study the structure of abelian extensions of the group $L_qG$ of $q$-differentiable loops (in the Sobolev sense), generalizing from the case of central extension of the smooth loop group. This is motivated by the aim of understanding the problems with current algebras in higher dimensions. Highest weight modules are…

Study shows limitations of Lie bracket commutation for nonsmooth vector fields.

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

Paper introduces new fractional Dirac operator and Q-curvature.

Proposes a new generalization bound for Bayesian deep nets without strict assumptions.

New Sobolev inequalities on Kähler manifolds with positive Ricci curvature.

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

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

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

Study partial derivatives on non-smooth metric measure structures.

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

The paper constructs Levi flat structures using structure sheaves and differential complexes.

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 …

Proves Sobolev inequality on manifolds with specific curvature properties.

In this paper we study a sharp Hardy-Littlewood-Sobolev (HLS) type inequality with Riesz potential on bounded smooth domains. We obtain the inequality for a general bounded domain $Ω$ and show that if the extension constant for $Ω$ is strictly larger than the extension constant for the unit ball $B_1$ then extremal fun…

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

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

New Sobolev inequalities found for curved spaces.

We consider maps into Riemannian manifolds of non-positive curvature and start developing a systematic PDE theory. We control the Sobolev $H^{2,2}$-norm of such a map in terms of its energy, the $L^2$-norm of its tension field and a topological term depending on the homotopy class. We also solve a Dirchlet problem with…