arXiv research

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

The transition maps for a Sobolev $G$-bundle are not continuous in the critical dimension and thus the usual notion of topology does not make sense. In this work, we show that if such a bundle $P$ is equipped with a Sobolev connection $A$, then one can associate a topological isomorphism class to the pair $\left( P, A\…

The authors study the Hodge theory of the exterior differential operator $d$ acting on $q$-forms on a smoothly bounded domain in $\RR^{N+1}$, and on the half space $\rnp$. The novelty is that the topology used is not an $L^2$ topology but a Sobolev topology. This strikingly alters the problem as compared to the classic…

We construct a large class of pathological $n$-dimensional topological spheres in ${\mathbb R}^{n+1}$ by showing that for any Cantor set $C\subset {\mathbb R}^{n+1}$ there is a topological embedding $f:{\mathbb S}^n\to{\mathbb R}^{n+1}$ of the Sobolev class $W^{1,n}$ whose image contains the Cantor set $C$.

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

We concerns here with the continuity on the geometry of the second Riemannian L^p-Sobolev best constant B_0(p,g) associated to the AB program. Precisely, for 1 <= p <= 2, we prove that B_0(p,g) depends continuously on g in the C^2-topology. Moreover, this topology is sharp for p = 2. From this discussion, we deduce som…

We extend the validity of a Gromov's dimension comparison estimate for topological hypersurfaces to sufficiently large classes of rectifiable sets, arising from Sobolev mappings. Our tools are a suitably weak exterior differentiation for pullback differential forms and a new low rank property for Sobolev mappings.

In this paper we show that certain generalizations of the $C^r$-Whitney topology, which include the Hölder-Whitney and Sobolev-Whitney topologies on smooth manifolds, satisfy the Baire property, to wit, the countable intersection of open and dense sets is dense.

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

A new framework for Einstein-Hilbert action with topological variations.

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

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

We show for a certain class of operators $A$ and holomorphic functions $f$ that the functional calculus $A\mapsto f(A)$ is holomorphic. Using this result we are able to prove that fractional Laplacians $(1+Δ^g)^p$ depend real analytically on the metric $g$ in suitable Sobolev topologies. As an application we obtain loc…

New model for shape graph registration with partial matching constraints.

Proposes GST for efficient computation of generalized Sobolev transport on graph metrics.

A new metric for comparing probability measures on graphs, scalable and negative definite.

Study on sphere-valued maps, proving energy convergence and current limits.

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

Given a compact manifold $N^n$, an integer $k \in \mathbb{N}_*$ and an exponent $1 \le p < \infty$, we prove that the class $C^\infty(\overline{Q}^m; N^n)$ of smooth maps on the cube with values into $N^n$ is dense with respect to the strong topology in the Sobolev space $W^{k, p}(Q^m; N^n)$ when the homotopy group $π_…

Survey of Fermat principle in general relativity and beyond.

Uniform bounds prove connection between Kähler metrics and RCD spaces.

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

Study nonexistence and gradient estimates for solutions on manifolds with bounded Ricci curvature.

We study stability properties of $f$-minimal hypersurfaces isometrically immersed in weighted manifolds with non-negative Bakry-Emery Ricci curvature under volume growth conditions. Moreover, exploiting a weighted version of a finiteness result and the adaptation to this setting of Li-Tam theory, we investigate the top…

We draw elliptic regularity results for 4-manifolds with an elliptic system, without Sobolev constant control. Direct use of analysis is circumvented; the results come mainly through geometric and topological arguments. In contrast to our previous paper, which worked predominantly on the scale of the curvature radius, …

We study the Sobolev stability of the Positive Mass Theorem (PMT) and the Riemannian Penrose Inequality (RPI) in the case where a region of a sequence of manifolds $M^3_i$ can be foliated by a smooth solution of Inverse Mean Curvature Flow (IMCF) which is uniformly controlled for time $t \in [0,T]$. In particular, we c…

We study critical Riemannian 4-manifolds with a lower bound on Ricci curvature, but no a priori analytic constraints such as on Sobolev constants. We derive elliptic-type estimates for the local curvature radius, which itself controls sectional curvature. The primary method is construction of blow-ups of degenerating m…

Proves long-time Ricci flow existence and topological rigidity for pinched integral curvature manifolds.

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…

New Sobolev inequalities on Kähler manifolds with positive Ricci curvature.

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

In this paper we study some fourth order elliptic equation involving the critical Sobolev exponent, related to the prescription of a fourth order conformal invariant on the standard sphere. We use a topological method to prove the existence of at least a solution when the function to be prescribed is close to a constan…

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 …

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.

New Sobolev inequalities found for curved spaces.

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.

The paper improves CR Sobolev inequalities and classifies minimizers.

Sharp Sobolev and Michael-Simon inequalities on curved manifolds.

Completeness of surface metrics established for Sobolev spaces.

The paper proves inequalities on Finsler manifolds under Ricci curvature bounds.