The total second variation of Perelman's $\mathcal{W}$-functional

Details The total second variation of Perelman's $\mathcal{W}$-functional The total second variation of Perelman's $\mathcal{W}$-functional

2012-01-04

arxiv.org/abs/1201.0969v1

Mathematics

Differential Geometry

37 pages

Scientific paper

We show a very simple and general total second variation formula for Perelman's $\mathcal{W}$-functional at arbitrary points in the space of Riemannian metrics. Moreover we perform a study of the properties of the variations of K\"ahler structures. We deduce a quite simple and general total second variation formula for Perelman's $\mathcal{W}$-functional with respect to such variations. In this case the main therm in the formula depends strongly on the variation of the complex structure. We discover also convexity of Perelman's $\mathcal{W}$-functional along particular variations over points with non-negative Bakry-Emery-Ricci tensor.

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