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Ricci decomposition
About: Ricci decomposition is a research topic. Over the lifetime, 1972 publications have been published within this topic receiving 45295 citations.
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145 citations
TL;DR: In this article, it was shown that the Ricci flow converges to a metric with constant bisectional curvature if and only if the curvature of the initial metric is positive.
Abstract: In this paper, we prove that if M is a Kahler-Einstein surface with positive scalar curvature, if the initial metric has nonnegative sectional curvature, and the curvature is positive somewhere, then the Kahler-Ricci flow converges to a Kahler-Einstein metric with constant bisectional curvature. In a subsequent paper [7], we prove the same result for general Kahler-Einstein manifolds in all dimension. This gives an affirmative answer to a long standing problem in Kahler Ricci flow: On a compact Kahler-Einstein manifold, does the Kahler-Ricci flow converge to a Kahler-Einstein metric if the initial metric has a positive bisectional curvature? Our main method is to find a set of new functionals which are essentially decreasing under the Kahler Ricci flow while they have uniform lower bounds. This property gives the crucial estimate we need to tackle this problem.
145 citations
Posted Content•
TL;DR: In this paper, the second variation of Perelman's λ$ and ε$ functionals for the Ricci flow is presented, and the linear stability of examples is investigated.
Abstract: In this announcement, we exhibit the second variation of Perelman's $\lambda$ and $
u$ functionals for the Ricci flow, and investigate the linear stability of examples. We also define the "central density" of a shrinking Ricci soliton and compute its values for certain examples in dimension 4. Using these tools, one can sometimes predict or limit the formation of singularities in the Ricci flow. In particular, we show that certain Einstein manifolds are unstable for the Ricci flow in the sense that generic perturbations acquire higher entropy and thus can never return near the original metric.
143 citations
140 citations
Posted Content•
TL;DR: In this paper, a metric quasi-Einstein metric is defined, where the Ricci tensor is a constant multiple of the metric tensor, which is a generalization of the Einstein metric.
Abstract: We call a metric quasi-Einstein if the $m$-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, which contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasi-Einstein metrics and prove several rigidity results. We also give a splitting theorem for some K\"ahler quasi-Einstein metrics.
138 citations