In this paper, we study the change of the ADM mass of an ALE space along the Ricci flow. Thus we first show that the ALE property is preserved under the Ricci flow. Then, we show that the mass is invariant under the flow in dimension three (similar results hold in higher dimension with more assumptions). A consequence of this result is the following. Let (M, g) be an ALE manifold of dimension n...

Following work of Ecker (Comm Anal Geom 15:1025–1061, 2007), we consider a weighted Gibbons-Hawking-York functional on a Riemannian manifold-withboundary. We compute its variational properties and its time derivative under Perelman’s modified Ricci flow. The answer has a boundary term which involves an extension of Hamilton’s differential Harnack expression for the mean curvature flow in Euclid...

We derive the entropy formula for the linear heat equation on general Riemannian manifolds and prove that it is monotone non-increasing on manifolds with nonnegative Ricci curvature. As applications, we study the relation between the value of entropy and the volume of balls of various scales. The results are simpler version, without Ricci flow, of Perelman’s recent results on volume non-collaps...

1 Introduction The uniformalization of Kähler manifold has been an important topic in geometry for long. The famous Frankel Conjecture states: Compact n-dimensional Kähler Manifolds of Positive Bisectional Curvature are Bi-holomorphic to P n C. [10] by using stable harmonic maps in the context of Kähler geometry. Actually, Mori proved the Harthshorne Conjecture: Every irreducible n-dimensional ...

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 a purely expository article on Riemannian Ricci flow with emphasis on dimension three. None of the results in this paper are due to the author. In view of Hamilton’s vast works, Yau’s influence on the field, and Perelman’s recent inspiring paper on Ricci flow, there may be more interest in the area. The author has only partial knowledge of this field but hopefully this paper will help n...

One of the most important properties of a geometric flow is whether it preserves the positivity of various notions of curvature. In the case of the Kähler-Ricci flow, the positivity of the curvature operator (Hamilton [7]), the positivity of the biholomorphic sectional curvature (Bando [1], Mok[8]), and the positivity of the scalar curvature (Hamilton [4]) are all preserved. However, whether th...

This paper studies the normalized Ricci flow on surfaces with conical singularities. It’s proved that the normalized Ricci flow has a solution for a short time for initial metrics with conical singularities. Moreover, the solution makes good geometric sense. For some simple surfaces of this kind, for example, the tear drop and the football, it’s shown that they admit Ricci soliton metric. MSC 2...

In this paper we prove a conjecture by Feldman-IlmanenKnopf in [14] that the gradient shrinking soliton metric they constructed on the tautological line bundle over CP is the uniform limit of blowups of a type I Ricci flow singularity on a closed manifold. We use this result to show that limits of blow-ups of Ricci flow singularities on closed four dimensional manifolds do not necessarily have ...

We consider Ricci flow invariant cones C in the space of curvature operators lying between nonnegative Ricci curvature and nonnegative curvature operator. Assuming some mild control on the scalar curvature of the Ricci flow, we show that if a solution to Ricci flow has its curvature operator which satsisfies R+ε I ∈ C at the initial time, then it satisfies R+Kε I ∈ C on some time interval depen...