@phdthesis{Kroencke2013,
  author    = {Kr{\"o}ncke, Klaus},
  title     = {Stability of Einstein Manifolds},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-69639},
  school      = {Universit{\"a}t Potsdam},
  year      = {2013},
  abstract  = {This thesis deals with Einstein metrics and the Ricci flow on compact mani- folds. We study the second variation of the Einstein-Hilbert functional on Ein- stein 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 be- haviour 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.},
  language  = {en}
}
@article{Kroencke2015,
  author    = {Kr{\"o}ncke, Klaus},
  title     = {Stability and instability of Ricci solitons},
  series = {Calculus of variations and partial differential equations},
  volume    = {53},
  journal   = {Calculus of variations and partial differential equations},
  number    = {1-2},
  publisher = {Springer},
  address   = {Heidelberg},
  issn      = {0944-2669},
  doi       = {10.1007/s00526-014-0748-3},
  pages     = {265 -- 287},
  year      = {2015},
  abstract  = {We consider the volume- normalized Ricci flow close to compact shrinking Ricci solitons. We show that if a compact Ricci soliton (M, g) is a local maximum of Perelman's shrinker entropy, any normalized Ricci flowstarting close to it exists for all time and converges towards a Ricci soliton. If g is not a local maximum of the shrinker entropy, we showthat there exists a nontrivial normalized Ricci flow emerging from it. These theorems are analogues of results in the Ricci- flat and in the Einstein case (Haslhofer and Muller, arXiv:1301.3219, 2013; Kroncke, arXiv: 1312.2224, 2013).},
  language  = {en}
}
@article{Kroencke2016,
  author    = {Kr{\"o}ncke, Klaus},
  title     = {Rigidity and Infinitesimal Deformability of Ricci Solitons},
  series = {The journal of geometric analysis},
  volume    = {26},
  journal   = {The journal of geometric analysis},
  publisher = {Springer},
  address   = {New York},
  issn      = {1050-6926},
  doi       = {10.1007/s12220-015-9608-4},
  pages     = {1795 -- 1807},
  year      = {2016},
  abstract  = {In this paper, an obstruction against the integrability of certain infinitesimal solitonic deformations is given. Using this obstruction, we show that the complex projective spaces of even complex dimension are rigid as Ricci solitons although they have infinitesimal solitonic deformations.},
  language  = {en}
}
@article{Kroencke2015,
  author    = {Kr{\"o}ncke, Klaus},
  title     = {On infinitesimal Einstein deformations},
  series = {Differential geometry and its applications},
  volume    = {38},
  journal   = {Differential geometry and its applications},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0926-2245},
  doi       = {10.1016/j.difgeo.2014.11.007},
  pages     = {41 -- 57},
  year      = {2015},
  abstract  = {We study infinitesimal Einstein deformations on compact flat manifolds and on product manifolds. Moreover, we prove refinements of results by Koiso and Bourguignon which yield obstructions on the existence of infinitesimal Einstein deformations under certain curvature conditions. (C) 2014 Elsevier B.V. All rights reserved.},
  language  = {en}
}