arXiv research

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

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

Unified study of Riemannian and sub-Riemannian geometries with synthetic Ricci curvature bounds.

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 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.

Study of Lagrangian submanifolds with Riemannian bounds and their metric properties.

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

Study of submanifolds in symplectic and contact manifolds using Hausdorff metrics.

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 geometry on a slope of a mountain is the geometry of a Finsler metric, called here the {\it slope metric}. We study the existence of globally defined slope metrics on surfaces of revolution as well as the geodesic's behavior. A comparison between Finslerian and Riemannian areas of a bounded region is also studied.

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

In this paper we continue our study on the canonical metrics on the Teichmüller and the moduli space of Riemman surfaces. We first prove the equivalence of the Bergman metric and the Carathéodory metric to the Kähler-Einstein metric, solving another old conjecture of Yau. We then prove that the Ricci curvature of the p…

The paper reconstructs Lorentzian spacetimes from causal sets.

Geodesics found in a metric space of m-subharmonic functions.

New surfaces defy traditional pants decompositions.

The study constructs a Lorentzian length space and explores its properties and relationships with metric and causal geometry.

Introduces bounded scale measure and generalizes property A.

Quantum complexity lowerbound proved using differential geometry.

The aim of this article is to present the category of bounded Frechet manifolds in respect to which we will review the geometry of Frechet manifolds with a stronger accent on its metric aspect. An inverse function theorem in the sense of Nash and Moser in this category is proved, and some applications to Riemannian geo…

Study bounds on curvature for special Finsler metrics.

In this paper, we define and study the Weil-Petersson geometry. Under the framework of the Weil-Petersson geometry, we study the Weil-Petersson metric and the Hodge metric. Among the other results, we represent the Hodge metric in terms of the Weil-Petersson metric and the Ricci curvature of the Weil-Petersson metric f…

Extends construction of Kähler-Einstein metrics to noncompact manifolds.

The paper bounds higher Steklov eigenvalues of graphs on surfaces.

Proper holomorphic isometries between Bergman domains are biholomorphisms.

The Fisher-Rao geometry is applied to elliptical distributions for optimization and classification.

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…

Estimates Kähler metric diameters with entropy bound alone.

Study on ratio of intrinsic to extrinsic metrics and its relation to surface area.

The norm of Cartan torsion plays an important role for studying of immersion theory in Finsler geometry. Indeed, Finsler manifold with unbounded Cartan torsion can not be isometrically imbedded into any Minkowski space. In this paper, we find two subclasses of (?, ?)-metrics which have bounded Cartan torsion. Then, we …

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…

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 proves complex geometry results for manifolds of the form X × R², answering a 1994 conjecture.

Quiver varieties' geometry at infinity studied using Nakajima metric.

Upper bounds on Einstein metrics on homogeneous spaces.

New uniformity definition on noncompact manifolds without metrics.

The study defines a canonical nilpotent structure for certain collapsed manifolds.

In this paper, we prove results concerning the large scale geometry of connected, simply connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics. Precisely, we prove that there do not exist quasi-isometric embeddings of such a nilpotent Lie group into either a CAT(0) metric space or an…

It is shown that the Hilbert geometry $(D,h_D)$ associated to a bounded convex domain $D\subset \mathbb{E}^n$ is isometric to a normed vector space $(V,||\cdot ||)$ if and only if $D$ is an open $n$-simplex. One further result on the asymptotic geometry of Hilbert's metric is obtained with corollaries for the behavior …

A geometric framework for metrics of maximal acceleration which is applicable to large proper accelerations is discussed, including a theory of connections associated with the geometry of maximal acceleration. In such a framework it is shown that the uniform bound on the proper maximal acceleration implies an uniform b…

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

Study Finsler metric measure manifolds' concentration properties.

We partially confirm an old conjecture of Donaldson that if there exists a cscK metrics in a given Kähler class, then there is no degenerated geodesic ray which is tamed by a bounded ambient geometry unless it parallels to a holomorphic line consists of cscK metrics only. We also prove that for simple test configuratio…

The paper connects Bergman geometry with information geometry.

Compact RCD spaces derived from singular Kahler metrics on 3D projective varieties.

We investigate the geometry of a normal bundle equipped with a $(p,q)$-metric, i.e., Riemannian metric of Cheeger-Gromoll type, to the submanifold of a Riemannian manifold. We derive all natural object as the Levi-Civita connection, curvature tensor, sectional and scalar curvature. We prove that under some natural cond…

We calculate an upper bound for the second nonzero eigenvalue of the scalar Laplacian, $λ_{2}$, for toric Kähler-Einstein metrics in terms of the polytope data. We give some detailed examples in complex dimensions 1, 2 and 3. We also discuss extensions of this method to other geometries.

The paper proves conditions for positive scalar curvature and Yamabe constant on noncompact cylinders.

We relate $L^{q,p}$-cohomology of bounded geometry Riemannian manifolds to a purely metric space notion of $\ell^{q,p}$-cohomology, packing cohomology. This implies quasi-isometry invariance of $L^{q,p}$-cohomology together with its multiplicative structure. The result partially extends to the Rumin $L^{q,p}$-cohomolog…