We investigate the properties of the combinatorial Ricci flow for surfaces, both forward and backward – existence, uniqueness and singularities formation. We show that the positive results that exist for the smooth Ricci flow also hold for the combinatorial one and that, moreover, the same results hold for a more general, metric notion of curvature. Furthermore, using the metric curvature appro...

We study stability of non-compact gradient Kähler-Ricci flow solitons with positive holomorphic bisectional curvature. Our main result is that any compactly supported perturbation and appropriately decaying perturbations of the Kähler potential of the soliton will converge to the original soliton under Kähler-Ricci flow as time tends to infinity. To obtain this result, we construct appropriate ...

This is an expository article based on the author’s lecture delivered at the conference Lie Theory and Its Applications in March 2011, UCSD. We discuss various notions of positivity and their relations with the study of the Ricci flow, including a proof of the assertion, due to Wolfson and the author, that the Ricci flow preserves the positivity of the complex sectional curvature. We discuss th...

We show that for any solution gij̄(t) to the Kähler-Ricci flow with positive bisectional curvature Rīijj̄(t) > 0 on a compact Kähler manifold M , the bisectional curvature has a uniform positive lower bound Rīijj̄(t) > C > 0. As a consequence, gij̄(t) converges exponentially fast in C ∞ to an KählerEinstein metric with positive bisectional curvature as t → ∞, provided we assume the Futaki-invariant...

We show that complete conformally flat manifolds of dimension n > 3 with nonnegative Ricci curvature enjoy nice rigidity properties: they are either flat, or locally isometric to a product of a sphere and a line, or are globally conformally equivalent to R n or a spherical spaceform Sn/Γ. This extends previous results due to Q.-M. Cheng and B.-L. Chen and X.-P. Zhu. In this note, we study compl...

This text is a presentation of the general context and results of [Oll07] and [Oll09], with comments on related work. The goal is to present a notion of Ricci curvature valid on arbitrary metric spaces, such as graphs, and to generalize a series of classical theorems in positive Ricci curvature, such as spectral gap estimates, concentration of measure or log-Sobolev inequalities. The necessary ...

In this paper, we present a new proof of the upper and lower bound estimates for the first Dirichlet eigenvalue λ1 (B (p, r)) of Laplacian operator for the manifold with Ricci curvature Rc ≥ −K, by using Li-Yau’s gradient estimate for the heat equation.

We study the Kähler-Ricci flow on Fano manifolds. We show that if the curvature is bounded along the flow and if the manifold is K-polystable and asymptotically Chow semistable, then the flow converges exponentially fast to a Kähler-Einstein metric.

In this work we consider compact Kähler manifolds with non-positive mixed curvature which is a “convex combination” of Ricci and holomorphic sectional curvature. We show that in case, the canonical line bundle nef. Moreover, if negative at some point, then manifold projective being big If addition negative, ample. As an application, answer question Ni [Comm. Pure Appl. Math. 74 (2021), pp. 1100...