arXiv research

For non-compact manifolds with boundary we prove that bounded geometry defined by coordinate-free curvature bounds is equivalent to bounded geometry defined using bounds on the metric tensor in geodesic coordinates. We produce a nice atlas with subordinate partition of unity on manifolds with boundary of bounded geomet…

We study collapsed manifolds with Ricci bounded covering geometry i.e., Ricci curvature is bounded below and the Riemannian universal cover is non-collapsed or consists of uniform Reifenberg points. Via Ricci flows' techniques, we partially extend the nilpotent structural results of Cheeger-Fukaya-Gromov, on collapsed …

3-manifolds with positive scalar curvature and bounded geometry are contractible.

Quantizes symplectic manifolds with bounded geometry using Berezin-Toeplitz method.

Generalizes tools for studying collapsed manifolds to new geometry.

We prove a persistence result for noncompact normally hyperbolic invariant manifolds in Riemannian manifolds of bounded geometry. The bounded geometry of the ambient manifold is a crucial assumption in order to control the uniformity of all estimates throughout the proof.

In this paper, we investigate analytical and geometric properties of certain non-compact boundary-manifolds, namely manifolds of bounded geometry. One result are strong Bochner type vanishing results for the L^2-cohomology of these manifolds: if e.g. a manifold admits a metric of bounded geometry which outside a compac…

The paper establishes lower bounds on injectivity radius and constructs metrics with bounded geometry.

Optimization rates improved for manifolds with bounded geometry.

The study connects polyhedral manifolds to Riemannian ones with geometric bounds.

The study limits the number of specific foliations with bounded geometry.

The paper extends Yamabe flow results to non-compact manifolds with bounded geometry.

We prove a persistence result for noncompact normally hyperbolic invariant manifolds in the setting of Riemannian manifolds of bounded geometry. Bounded geometry of the ambient manifold is a crucial assumption required to control the uniformity of all estimates throughout the proof. The $C^{k,α}$-smoothness result is o…

The paper proves wellposedness of flows on manifolds with bounded geometry.

We characterize functions which are growth types of Riemannian manifolds of bounded geometry.

In this paper, we systematically investigate the geometry and topology of manifolds with integral radial curvature bounds, and obtain many interesting and important conclusions.

A surgery classification theory is introduced for manifolds of bounded geometry up to quasi-isometry. The Borel conjecture for this theory is proven for flat Euclidean space.

Paper proves inequalities for forms on sub-Riemannian manifolds.

The paper proves rigidity results for Einstein manifolds with specific geometric constraints.

Study collapsing geometry with Ricci curvature, proving Kähler metrics and Killing structures.

Study on the geometry of limit spaces of manifolds with boundary.

Generalizes fully augmented links to doubled 3-manifolds with geometric bounds.

We prove that simply connected open Riemannian manifolds of bounded geometry, linear growth and sublinear filling growth (e.g. finite filling area) are simply connected at infinity.

Uniformly proves index invariance for signature operators on manifolds.

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.

We prove well-posedness and regularity results for elliptic boundary value problems on certain domains with a smooth set of singular points. Our class of domains contains the class of domains with isolated oscillating conical singularities, and hence they generalize the classical results of Kondratiev on domains with c…

In this paper we develop the geometry of bounded Fréchet manifolds. We prove that a bounded Fréchet tangent bundle admits a vector bundle structure. But the second order tangent bundle $T^2M$ of a bounded Fréchet manifold $M$, becomes a vector bundle over $M$ if and only if $M$ is endowed with a linear connection. As a…

Local Hardy spaces defined for Riemannian manifolds with bounded geometry.

We revisit Spakula's uniform K-homology, construct the external product for it and use this to deduce homotopy invariance of uniform K-homology. We define uniform K-theory and on manifolds of bounded geometry we give an interpretation of it via vector bundles of bounded geometry. We further construct a cap product with…

Ricci flow stability on manifolds with bounded geometry ensures convergence to hyperbolic metrics.

The main theorem states that any complete connected Riemannian manifold of bounded geometry can be isometrically realized as a leaf with trivial holonomy in a compact Riemannian foliated space.

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

Develops Lefschetz theory for noncompact manifolds.

If Pi: M -> B is an onto smooth maximal rank map between complete Riemannian manifolds M and B with bounded geometry, we prove sufficient conditions for M to be roughly isometric to the Riemannian product FxB, where F is a fiber of M.

We prove that the moduli space of complete Riemannian metrics of bounded geometry and uniformly positive scalar curvature on an orientable 3-manifold is path-connected. This generalizes the main result of the fourth author [Mar12] in the compact case. The proof uses Ricci flow with surgery as well as arguments involvin…

The study proves the existence and properties of isoperimetric clusters in Riemannian manifolds with bounded geometry.

The paper extends a theorem to manifolds with local Ricci bounded covering geometry.

Liouville theorems extended to graphs with bounded geometry.

The study extends calibrated geometry to smooth maps and finds energy bounds.

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.

Study compares manifolds with boundary under weighted Ricci curvature bounds.

We show that a Kleinian surface group, or hyperbolic 3-manifold with a cusp-preserving homotopy-equivalence to a surface, has bounded geometry if and only if there is an upper bound on an associated collection of coefficients that depend only on its end invariants. Bounded geometry is a positive lower bound on the leng…

Let $M$ be a Riemannian manifold with a smooth boundary. The main question we address in this article is: "When is the Laplace-Beltrami operator $Δ\colon H^{k+1}(M)\cap H^1_0(M) \to H^{k-1}(M)$, $k\in \mathbb{N}_0$, invertible?" We consider also the case of mixed boundary conditions. The study of this main question lea…

The study extends convergence theorems for Ricci-limit spaces with bounded curvature.

The Serre-Swan theorem in differential geometry establishes an equivalence between the category of smooth vector bundles over a smooth compact manifold and the category of finitely generated projective modules over the unital ring of smooth functions. This theorem is here generalized to manifolds of bounded geometry. I…

Classifies holomorphic parabolic geometries on complex manifolds.

We study the large-scale geometry of 3-manifolds with nontrivial 2-dimensional bounded cohomology, with a view to proving a weak version of the geometrization conjecture for such manifolds.

Geometrically refines Cramér-Rao bound using extrinsic manifold curvature.