25 nov 2009

Abstract.

Wednesday, November 25 17 in Sala Riunioni , Dipartimento di Matematica Prof. Robert Haslhofer (ETH Zurich) Will speak about:

Perelman's $\lambda$-functional and the stability of Ricci-flat metrics

Abstract: Hamilton's Ricci flow can be interpreted as gradient flow of Perelman's $\lambda$-functional on the space of metrics modulo diffeomorphisms. In particular, if compact Ricci-flat metrics are dynamically stable fixed points of the Ricci flow, then they locally maximize $\lambda$, and this in turn implies that their Lichnerowicz Laplacians have only nonpositive eigenvalues. In this talk, I will show that the converse implications are also true, provided the premoduli space of Ricci-flat metrics is a manifold of prescribed dimension. To prove the local maxima result, I use the Ebin-Palais slice theorem and estimate the error term in the Taylor expansion of $\lambda$ coming from the third variation. To show dynamical stability, I prove a Lojasiewicz-Simon type gradient inequality for the Ricci flow and estimate the motion in the gauge directions. Similar dynamical stability results have been obtained by other authors using the Ricci-DeTurck flow.