arXiv research

In this paper, we show that any compact K$ä$hler manifold homotopic to a compact Riemannian manifold with negative sectional curvature admits a K$ä$hler-Einstein metric of general type. Moreover, we prove that, on a compact symplectic manifold $X$ homotopic to a compact Riemannian manifold with negative sectional curva…

The paper studies Ricci curvature on Kähler-Ricci flow.

We describe and construct here pseudo-Hermitian structures $θ$ without torsion (i.e. with transversal symmetry) whose Webster-Ricci curvature tensor is a constant multiple of the exterior differential $dθ$. We call these structures pseudo-Hermitian Einstein and our result states that they all can be derived locally fro…

We obtain a necessary and sufficient condition of existence of a K{ä}hler-Einstein metric on a $G\times G$-equivariant Fano compactification of a complex connected reductive group $G$ in terms of the associated polytope. This condition is not equivalent to the vanishing of the Futaki invariant. The proof relies on the …

Given a convex body $K \subset \mathbb{R}^n$ with the barycenter at the origin we consider the corresponding K{ä}hler-Einstein equation $e^{-Φ} = \det D^2 Φ$. If $K$ is a simplex, then the Ricci tensor of the Hessian metric $D^2 Φ$ is constant and equals $\frac{n-1}{4(n+1)}$. We conjecture that the Ricci tensor of $D^2…

Continuity of complex Monge-Ampère potentials on Kähler manifolds.

We prove that a compact stratied space satises the Riemannian curvature-dimension condition RCD(K, N) if and only if its Ricci tensor is bounded below by K $\in$ R on the regular set, the cone angle along the stratum of codimension two is smaller than or equal to 2$π$ and its dimension is at most equal to N. This gives…

Constructs two types of Eguchi-Hanson metrics with negative scalar curvature.

Construct Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations.

New metrics found without topological restrictions.

Constructs metrics with negative constant scalar curvature.

We show the existence of complete negative Kähler-Einstein metric on Stein manifolds with negatively pinched holomorphic sectional curvature. We prove that any Kähler metrics on such manifolds can be deformed to the complete negative Kähler-Einstein metric using the normalized Kähler-Ricci flow.

Constructs metrics with negative curvature on specific manifold types.

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

Constructs conformal metrics with negative curvature on manifolds with boundary.

New Einstein metrics found on complex manifolds.

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

Proves existence and uniqueness of metrics with negative curvature and singularities on compact surfaces.

The Bergman metric on symmetrized bidisc has negative curvature properties.

Sharp inequality in spaces with non-negative Ricci curvature.

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…

The study proves the existence of complete Kähler metrics with negative holomorphic bisectional curvature in specific domains.

New Einstein metrics found in curved spaces.

The paper finds many negatively curved Kähler metrics 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.

Upper bounds on Laplacian eigenvalues on manifolds with non-negative curvature.

We consider a smooth closed surface $M$ of fixed genus $\geqslant 2$ with a Riemannian metric $g$ of negative curvature with fixed total area. The second author has shown that the topological entropy of geodesic flow for $g$ is greater than or equal to the topological entropy for the metric of constant negative curvatu…

The paper bends Riemannian manifolds to achieve metrics with negative Ricci curvature.

Classifies Einstein metrics on R^4 with Heisenberg symmetry, finding incomplete Ricci-flat metrics and two complete negative-curvature examples.

The paper proves conjectures and classifies metrics on 3D manifolds.

Study Kähler metrics on complex tori with almost non-negative scalar curvature.

We study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds. We construct, in particular, the first classes of manifolds for which these moduli spaces have non-trivial rational homotopy, homology and cohomology groups. We also show tha…

A version of the singular Yamabe problem in bounded domains yields complete conformal metrics with negative constant scalar curvatures. In this paper, we study whether these metrics have negative Ricci curvatures. Affirmatively, we prove that these metrics indeed have negative Ricci curvatures in bounded convex domains…

Study compares metrics from negative curvature and quasi-Fuchsian representations.

New proof shows no negative curvature Einstein metrics in specific dimensions.

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 …

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 finds the minimum number of negative eigenvalues for conformal Laplacian metrics.

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

Study on Einstein deformations of negative Kähler Einstein metrics.

Study on moduli spaces of negatively curved metrics on surfaces.

New metrics with non-negative scalar curvature are always Ricci-flat on certain surgeries.

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

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

The sectional curvature of the Weil-Petersson metric on Teichmuller space is known to be negative. We show that this Weil-Petersson sectional curvature is not pinched from above by any negative constants, i.e., there is no negative upper bound.

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

New solutions found with negative mass in general relativity.

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