arXiv research

Study on moduli spaces of negatively curved metrics on surfaces.

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…

We prove that a complete Kähler manifold with holomorphic curvature bounded between two negative constants admits a unique complete Kähler-Einstein metric. We also show this metric and the Kobayashi-Royden metric are both uniformly equivalent to the background Kähler metric. Furthermore, all three metrics are shown to …

Study shows tori metrics converging to flat under specific conditions.

We show uniqueness of Ricci flows starting at a surface of uniformly negative curvature, with the assumption that the flows become complete instantaneously. Together with the more general existence result proved in [10], this settles the issue of well-posedness in this class.

The study proves the non-existence of certain Kähler metrics with specific curvature properties.

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

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.

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 Kähler-Ricci flow converges to a negative Kähler-Einstein metric under certain conditions.

We show that on Kahler manifolds with negative first Chern class, the sequence of algebraic metrics introduced by H. Tsuji converges uniformly to the Kahler-Einstein metric. For algebraic surfaces of general type and orbifolds with isolated singularities, we prove a convergence result for a modified version of Tsuji's …

New stability estimate for metric rigidity in hyperbolic dynamics.

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.

The paper studies minimal surface entropy on hyperbolic 3-manifolds and compares it to the hyperbolic case.

We prove that the Ricci flow g(t) starting at any metric on the euclidean space that is invariant by a transitive nilpotent Lie group N, can be obtained by solving an ODE for a curve of nilpotent Lie brackets. By using that this ODE is the negative gradient flow of a homogeneous polynomial, we obtain that g(t) is type-…

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

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…

We prove that the Teichmüller space $\mathcal{T}^{<0}(M)$ of negatively curved metrics on a hyperbolic manifold $M$ has nontrivial $i$-th rational homotopy groups for some $i> \dim M$. Moreover, some elements of infinite order in $π_i B\mbox{Diff}(M)$ can be represented by bundles over $S^i$ with fiberwise negatively c…

We prove that if a closed, smooth, simply-connected 4-manifold with a circle action admits an almost non-negatively curved sequence of invariant Riemannian metrics, then it also admits a non-negatively curved Riemannian metric invariant with respect to the same action. The same is shown for torus actions of higher rank…

This article generalizes the work of Ballmann and Światkowski to the case of Reflexive Banach spaces and uniformly convex Busemann spaces, thus giving a new fixed point criterion for groups acting on simplicial complexes.

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

We show that the positive mass theorem holds for continuous Riemannian metrics that lie in the Sobolev space $W^{2, n/2}_{loc}$ for manifolds of dimension less than or equal to $7$ or spin-manifolds of any dimension. More generally, we give a (negative) lower bound on the ADM mass of metrics for which the scalar curvat…

In this paper we prove that for all $n=4k-2$, $k\ge2$ there exists a closed smooth complex hyperbolic manifold $M$ with real dimension $n$ having non-trivial $π_1(\mathcal{T}^{<0}(M))$. $\mathcal{T}^{<0}(M)$ denotes the Teichmüller space of all negatively curved Riemannian metrics on $M$, which is the topological quoti…

We use classical results in smoothing theory to extract information about the rational homotopy groups of the space of negatively curved metrics on a high dimensional manifold. It is also shown that smooth M-bundles over spheres equipped with fiberwise negatively curved metrics, represent elements of finite order in th…

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…

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…

Let $(M, g)$ be a complete non-compact Kähler manifold with non-negative and bounded holomorphic bisectional curvature. Extending our techniques developed in \cite{CT3}, we prove that the universal cover $\wt M$ of $M$ is biholomorphic to $\ce^n$ provided either that $(M, g)$ has average quadratic curvature decay, or $…

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

Study of singular metrics with negative scalar curvature on compact manifolds.

Constructs metrics with negative curvature on specific manifold types.

Uniform K-stability of Calabi-Yau fibrations linked to base curve stability.

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.

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.

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

We prove the existence of $C^{1,1}$ isometric immersions of several classes of metrics on surfaces $(\mathcal{M},g)$ into the three-dimensional Euclidean space $\mathbb{R}^3$, where the metrics $g$ have strictly negative curvature. These include the standard hyperbolic plane, generalised helicoid-type metrics and gener…

Develops active intervals for geodesics in Teichmüller space.

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 …

The notion of nonpositive curvature in Alexandrov's sense is extended to include p-uniformly convex Banach spaces. Infinite dimensional manifolds of semi-negative curvature with a p-uniformly convex tangent norm fall in this class on nonpositively curved spaces, and several well-known results, such as existence and uni…

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.

Using the Donaldson-Auroux theory, we construct complete intersections in complex projective manifolds, which are negatively curved in various ways. In particular, we prove the existence of compact simply connected Kahler manifolds with negative holomorphic bisectional curvature. We also construct hyperbolic hypersurfa…

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…