arXiv research

Study on completeness of Sobolev metrics on manifold-valued curves.

Study noncommutative Sobolev inequalities using quantum state metrics.

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

Completeness of surface metrics established for Sobolev spaces.

Uniform Sobolev inequality for Kähler metrics with entropy bound.

Study on fractional Sobolev metrics on curves, proving completeness and geodesic properties.

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…

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…

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…

The paper shows how Sobolev maps affect currents in metric spaces.

We study completeness properties of Sobolev metrics on the space of immersed curves and on the shape space of unparametrized curves. We show that Sobolev metrics of order $n\geq 2$ are metrically complete on the space $\mathcal I^n(S^1,\mathbb R^d)$ of Sobolev immersions of the same regularity and that any two curves i…

Solves Yamabe problem for 3D metrics of Sobolev class $W^{2,q}$.

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

Frank and Lieb proved sharp Sobolev inequalities without rearrangements.

Solves Yamabe problem for Sobolev-class asymptotically hyperbolic manifolds.

The paper establishes Sobolev inequalities between Riemannian metrics and their distance functions.

Local smoothing of metrics with small curvature, removing Ricci curvature condition.

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

Compact embeddings for invariant functions in metric-measure spaces.

Study on $G_2$-structures manifold geometry.

Study of geometric analysis on asymmetric metric spaces, including heat flow and Sobolev spaces.

Researchers prove rigidity for log-Sobolev inequality on specific metric spaces.

New metrics on curve spaces improve shape analysis.

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

Lipschitz maps on metric surfaces are rigid if they preserve area.

Sobolev mappings preserve the Rumin complex on contact manifolds.

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…

Holomorphic solutions vary in Sobolev spaces for Beltrami equations.

The paper examines stability of the Sobolev inequality in metric spaces with curvature dimension conditions.

The geodesic distance vanishes on the group of compactly supported diffeomorphisms of a Riemannian manifold $M$ of bounded geometry, for the right invariant weak Riemannian metric which is induced by the Sobolev metric $H^s$ of order $0\le s<\tfrac12$ on the Lie algebra $\mathfrak X_c(M)$ of vector fields with compact …

Paper proposes new density estimators for high-dimensional data.

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…

We find maximal representatives within equivalence classes of metric spheres. For Ahlfors regular spheres these are uniquely characterized by satisfying the seemingly unrelated notions of Sobolev-to-Lipschitz property, or volume rigidity. We also apply our construction to solutions of the Plateau problem in metric spac…

Many procedures in science, engineering and medicine produce data in the form of geometric shapes. Mathematically, a shape can be modeled as an un-parameterized immersed sub-manifold, which is the notion of shape used here. Endowing shape space with a Riemannian metric opens up the world of Riemannian differential geom…

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…

Extends Lipschitz functions while preserving local constants.

Uniform volume estimate for Kähler metrics in big cohomology classes.

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

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…

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…

Compact metrics found with specific curvature properties on 3D surfaces.

The paper studies the diameter of diffeomorphism groups with Sobolev metrics.

For a closed symplectic manifold $(M,ω)$ with compatible Riemannian metric $g$ we study the Sobolev $H^1$ geometry of the group of all $H^s$ diffeomorphisms on $M$ which preserve the symplectic structure. We show that, for sufficiently large $s$, the $H^1$ metric admits globally defined geodesics and the corresponding …

Simplified proof of Morrey's lemma for metric spaces.

Study fine Pólya-Szegő inequalities in metric spaces with applications.

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…

New tensorization theorem for Sobolev spaces on product spaces.

Stability of Dehn functions proven for ultralimits of Sobolev maps.