We prove the existence and uniqueness of the weak Kähler-Ricci flow on projective varieties with log terminal singularities. It is also shown that the weak Kähler-Ricci flow can be uniquely continued through divisorial contractions and flips if they exist. We then propose an analytic version of the Minimal Model Program with Ricci flow.

The problem of finding Kähler-Einstein metrics on a compact Kähler manifold has been the subject of intense study over the last few decades. In his solution to Calabi’s conjecture, Yau [Ya1] proved the existence of a Kähler-Einstein metric on compact Kähler manifolds with vanishing or negative first Chern class. An alternative proof of Yau’s theorem is given by Cao [Ca] using the Kähler-Ricci f...

where Rαβ(x, t) denotes the Ricci curvature tensor of the metric gαβ(x, t). One of the main problems in differential geometry is to find canonical structure on manifolds. The Ricci flow introduced by Hamilton [8] is an useful tool to approach such problems. For examples, Hamilton [10] and Chow [7] used the convergence of the Ricci flow to characterize the complex structures on compact Riemann s...

We present a novel colon flattening algorithm using the discrete Ricci flow. The discrete Ricci flow is a powerful tool for designing Riemannian metrics on surfaces with arbitrary topologies by user-defined Gaussian curvatures. Moreover, the discrete Ricci flow deforms the Riemannian metric on the surface conformally and minimizes the global distortion, which means the local shape is well prese...

In this paper, we prove a general maximum principle for the time dependent Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on complete Riemannian manifolds. As an application we construct complete manifolds with bounded nonnegative sectional curvature of dimension greater than or equal to four such that the Ricci flow does not preserve the nonnegativity of the sectio...

Suppose M is a compact n-dimensional manifold, n ≥ 2, with a metric gij(x, t) that evolves by the Ricci flow ∂tgij = −2Rij in M × (0, T ). We will give a simple proof of a recent result of Perelman on the non-existence of shrinking breather without using the logarithmic Sobolev inequality. It is known that Ricci flow is a very powerful tool in understanding the geometry and structure of manifol...

In [LY] a differential Harnack inequality was proved for solutions to the heat equation on a Riemannian manifold. Inspired by this result, Hamilton first proved trace and matrix Harnack inequalities for the Ricci flow on compact surfaces [H0] and then vastly generalized his own result to all higher dimensions for complete solutions of the Ricci flow with nonnegative curvature operator [ H2]. So...

The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton [8]. Later on, De Turck [4] gave a simplified proof. In the later of 80's, Shi [20] generalized the local existence result to complete noncompact manifolds. However, the uniqueness of the solutions to the Ricci flow on co...

We study the evolution of anticanonical line bundles along the Kähler Ricci flow. We show that under some conditions, the convergence of Kähler Ricci flow is determined by the properties of the anticanonical divisors of M . As examples, the Kähler Ricci flow on M converges when M is a Fano surface and c 1 (M) = 1 or c 1 (M) = 3. Combined with the work in [CW1] and [CW2], this gives a Ricci flow...

Abstract. Let M be a compact Riemannian manifold and the metrics g = g(t) evolve by the Ricci flow. We prove the following result. The Sobolev imbedding by Aubin or Hebey, perturbed by a scalar curvature term and modulo sharpness of constants, holds uniformly for (M, g(t)) for all time if the Ricci flow exists for all time; and if the Ricci flow develops a singularity in finite time, then the s...