arXiv research

Paper shows regions close to negatively curved metrics are minimal fillings and rigid.

Study shows nontrivial homotopy groups for negatively curved metrics on hyperbolic manifolds.

New Einstein metrics found on complex manifolds.

We show that the space of negatively curved metrics of a closed negatively curved Riemannian $n$-manifold, $n\geq 10$, is highly non-connected.

The Teichmüller space of negatively curved metrics on complex hyperbolic manifolds is not contractible.

We show that the space of nonpositively curved metrics of a negatively curved manifold is highly non connected.

New Einstein metrics found in curved spaces.

The paper finds many negatively curved Kähler metrics on complex manifolds.

We study the moduli space of negatively curved metrics of a hyperbolic manifold.

We study the Teichmüller space of negatively curved metrics on a high dimensional manifold, with applications to bundles with negatively curved fibers.

Study shows non-negative curvature on 4-manifolds with torus symmetry.

Constructs negatively curved complete intersections in complex manifolds.

The paper extends Heintze-Kobayashi-Wolf theory to negatively curved homogeneous Finsler manifolds.

Proves existence of smooth isometric immersions for certain curved surfaces.

This is a survey on known results and open problems about Smooth and PL-Rigidity Problem for negatively curved locally symmetric spaces. We also review some developments about studying the basic topological properties of the space of negatively curved Riemannian metrics and the Teichmuller space of negatively curved me…

Study on curved metrics on manifolds using smoothing theory.

Study counts minimal surfaces in curved 3D spaces, finding hyperbolic space minimizes area.

In this paper we prove that for all $n=4k-2$, $k\ge2$ there exists closed $n$-dimensional Riemannian manifolds $M$ with negative sectional curvature that do not have the homotopy type of a locally symmetric space, such that $π_{1}(\mathcal{T}^{<0}(M))$ is non-trivial. $\mathcal{T}^{<0}(M)$ denotes the Teichmüller space…

The study finds sparse sets that uniquely determine metrics on negatively curved manifolds.

The classic 2pi-Theorem of Gromov and Thurston constructs a negatively curved metric on certain 3-manifolds obtained by Dehn filling. By Geometrization, any such manifold admits a hyperbolic metric. We outline a program using cross curvature flow to construct a smooth one-parameter family of metrics between the "2pi-me…

The paper proves conditions for Kähler manifolds with negative curvature.

Let $(X, g_0)$ be a simply connected, complete, negatively curved Riemannian manifold. We prove local and infinitesimal rigidity results for compactly supported deformations of the metric $g_0$. For any negatively curved metric $g$ equal to $g_0$ outside a compact, the identity map of $X$ induces a natural boundary map…

Examines properties of negatively curved 3-manifolds and their embeddings, introduces Cross Curvature Flow.

New theorem shows certain curved surfaces are uniquely identified by their geodesic lengths.

Constructs metrics with negative curvature on specific manifold types.

Study on moduli spaces of negatively curved metrics on surfaces.

In this paper we announce the following result: ``Every manifold of dimension $\ge3$ admits a complete negatively Ricci curved metric.'' Furthermore we describe some sharper results and sketch proofs.

The paper constructs foliations of minimal surfaces in negatively curved 3-manifolds.

We prove that the space of complete, finite volume, pinched negatively curved Riemannian metrics on a smooth high-dimensional manifold is either empty or it is highly non-connected, provided their behavior at infinity is similar.

In this paper, we prove a global rigidity theorem for negatively curved Finsler metrics on a compact manifold of dimension n>2. We show that for such a Finsler manifold, if the flag curvature is a scalar function on the tangent bundle, then the Finsler metric is of Randers type. We also study the case when the Finsler …

Geodesics on curved surfaces can't fill certain configurations.

Constructs a convex Finsler metric on vector bundles under specific conditions.

This paper extends FP's method to complex hyperbolic branched covers to find Einstein metrics.

As the first step in the direction of the Hopf conjecture on the non-existence of metrics with positive sectional curvature on $S^2 \times S^2$ D.Gromoll and K.Tapp in [GT] suggested the following (Weak Hopf) conjecture (on the rigidity of non-negatively curved metrics on $S^2 \times R^3$): "The boundary $S^2\times S^2…

Study proves uniqueness of corrugated negatively curved immersions in differential geometry.

The paper studies Kähler-Einstein metrics with singularities and their limits.

Under mild assumptions on a group G, we prove that the class of complete Riemannian n-manifolds of uniformly bounded negative sectional curvatures and with the fundamental groups isomorphic to G breaks into finitely many tangential homotopy types. It follows that many aspherical manifolds do not admit complete negative…

For a smooth manifold $M$ we define the Teichmüller space $\cT(M)$ of all Riemannian metrics on $M$ and the Teichmüller space $\cT^ε(M)$ of $ε$-pinched negatively curved metrics on $M$, where $0\leqε\leq\infty$. We prove that if $M$ is hyperbolic the natural inclusion $\cT^ε(M)\hookrightarrow\cT(M)$ is, in general, not…

New gaps found in metric curvature.

Study on 4-manifolds restricts negatively curved metrics and fundamental groups.

Study on moduli spaces of non-negative curvature metrics on manifolds.

New Teichmüller space for negatively curved surfaces defined.

In this paper we study the evolution of almost non-negatively curved (possibly singular) three dimensional metric spaces by Ricci flow. The non-negatively curved metric spaces which we consider arise as limits of smooth Riemannian manifolds (M_i,g_i), i \in N, whose Ricci curvature is not less than -c^2(i), where c^2(i…

Ricci flow deforms metrics with positive curvature to include negative curvature.

Negative curvature proven in Sasaki manifold space completion.

The study examines the flexibility of entropies for negatively curved surfaces.

In this paper, we introduce the flag-wise positively curved condition for Finsler spaces (the (FP) Condition), which means that in each tangent plane, we can find a flag pole in this plane such that the corresponding flag has positive flag curvature. Applying the Killing navigation technique, we find a list of compact …

Extends construction of Kähler-Einstein metrics to noncompact manifolds.