arXiv research

Study on $G_2$-structures manifold geometry.

Study of $H^1$ geometry of symplectomorphisms on manifolds.

The paper extends log-Sobolev inequalities to matrix-valued settings using combinatorial methods.

Paper proves inequalities for forms on sub-Riemannian manifolds.

Poincaré and Sobolev inequalities for differential forms on Heisenberg balls are derived.

Study RKHS on manifolds, linking Sobolev and diffusion spaces.

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

Tian's conjectures solved in Kahler geometry, linking metrics and inequalities.

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

Local Sobolev inequality on Ricci flows with applications.

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

Functional calculus results lead to smooth metric dependence on Riemannian geometry.

The paper proves density of smooth functions in Sobolev spaces on certain manifolds.

Quantitative Sobolev extensions lead to Neumann heat kernel bounds.

Sharp stability of isometries on Heisenberg group proven.

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…

New heat semigroup characterizes Sobolev and BV spaces in Carnot groups.

Study shows Sobolev functions on non-compact manifolds can't be approximated by smooth compactly supported functions.

Let $\M$ be a complete, connected noncompact manifold with bounded geometry. Under a condition near infinity, we prove that the Log Sobolev functional (\ref{logfanhan}) has an extremal function decaying exponentially near infinity. We also prove that an extremal function may not exist if the condition is violated. This…

We study the Yamabe problem on open manifolds of bounded geometry and show that under suitable assumptions there exist Yamabe metrics, i.e. conformal metrics of constant scalar curvature. For that, we use weighted Sobolev embeddings.

Ancient Ricci flows with asymptotic solitons have uniform bounds and inequalities.

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 …

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

Positive mass theorem and Yamabe equation on CR manifolds

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…

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

Let (M,g) be a complete noncompact riemannian manifold with bounded geometry and parallel Ricci curvature. We show that some operators, "affine" relatively to the Ricci curvature, are locally invertible, in some classical Sobolev spaces, near the metric g.

The study proves conditions for nontrivial solutions on Riemannian 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…

Paper introduces SGA for barycenter optimization in optimal transport.

Unified shape spaces with preserved invariances and regular metrics.

We study some basic analytic questions related to differential operators on Lie manifolds, which are manifolds whose large scale geometry can be described by a a Lie algebra of vector fields on a compactification. We extend to Lie manifolds several classical results on Sobolev spaces, elliptic regularity, and mapping p…

Note removes degeneracy in Kähler geometry estimates.

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

Study Bernstein-Gelfand-Gelfand complexes on Lipschitz domains, computing cohomology and applying to elasticity models.

The paper extends inequalities for convex bodies to higher dimensions and various norms.

Geometric approach solves Euler equations with random forces.

This paper simplifies complex geometry for shape analysis.

The study establishes inequalities on path space for sub-Riemannian manifolds.

Study shows vanishing distance in fluid dynamics equations.

This paper solves Hilbert's fourth problem for constant curvature metrics.

The paper defines function spaces on manifolds with bounded or singular geometries.

Constructs distance-like functions on manifolds with controlled curvature to study Sobolev spaces.

We show that gradient shrinking, expanding or steady Ricci solitons have potentials leading to suitable reference probability measures on the manifold. For shrinking solitons, as well as expanding soltions with nonnegative Ricci curvature, these reference measures satisfy sharp logarithmic Sobolev inequalities with low…

Diffeologies unify infinite-dimensional geometry and PDEs, enhancing classical function spaces.

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…

New boundary operators for sixth-order GJMS operator on manifolds.

Derives sub-Riemannian Ricci curvature for various manifolds.