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This work deals with Einstein metrics and the Ricci flow on compact manifolds. We study the second variation of the Einstein-Hilbert functional on Einstein metrics. In the first part of the work, we find curvature conditions which ensure the stability of Einstein manifolds with respect to the Einstein-Hilbert functional, i.e. that the second variation of the Einstein-Hilbert functional at the metric is nonpositive in the direction of transverse-traceless tensors. The second part of the work is devoted to the study of the Ricci flow and how its behaviour close to Einstein metrics is influenced by the variational behaviour of the Einstein-Hilbert functional. We find conditions which imply that Einstein metrics are dynamically stable or unstable with respect to the Ricci flow and we express these conditions in terms of stability properties of the metric with respect to the Einstein-Hilbert functional and properties of the Laplacian spectrum.

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The author was born in 1986 in Vienna. He studied Mathematics, Physics and Geography at the University of Vienna from 2004-2010. From 2010-2013, he did his doctoral studies in Mathematics at the University of Potsdam. Currently, he is a research assistant at the Potsdam University.

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