TY  - THES
A1  - Kröncke, Klaus
T1  - Stability of Einstein Manifolds
T1  - Stabilität von Einstein-Mannigfaltigkeiten
N2  - 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.
N2  - Die vorliegende Arbeit beschäftigt sich mit Einsteinmetriken und Ricci-Fluss auf kompakten Mannigfaltigkeiten. Wir studieren die zweite Variation des Einstein- Hilbert Funktionals auf Einsteinmetriken. Im ersten Teil der Arbeit finden wir Krümmungsbedingungen, die die Stabilität von Einsteinmannigfaltigkeiten bezüglich des Einstein-Hilbert Funktionals sicherstellen, d.h. die zweite Varia- tion des Einstein-Hilbert Funktionals ist nichtpositiv in Richtung transversaler spurfreier Tensoren. Der zweite Teil der Arbeit widmet sich dem Studium des Ricci-Flusses und wie dessen Verhalten in der Nähe von Einsteinmetriken durch das Variationsver- halten des Einstein-Hilbert Funktionals beeinflusst wird. Wir finden Bedinun- gen, die dynamische Stabilität oder Instabilität von Einsteinmetriken bezüglich des Ricci-Flusses implizieren und wir drücken diese Bedingungen in Termen der Stabilität der Metrik bezüglich des Einstein-Hilbert Funktionals und Eigen- schaften des Spektrums des Laplaceoperators aus.
KW  - Einstein-Mannigfaltigkeiten
KW  - Ricci-Fluss
KW  - Variationsstabilität
KW  - Einstein-Hilbert-Wirkung
KW  - Einstein manifolds
KW  - Ricci flow
KW  - variational stability
KW  - Einstein-Hilbert action
Y1  - 2013
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-69639
ER  - 
TY  - JOUR
A1  - Kröncke, Klaus
T1  - Stability and instability of Ricci solitons
JF  - Calculus of variations and partial differential equations
N2  - 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).
Y1  - 2015
U6  - https://doi.org/10.1007/s00526-014-0748-3
SN  - 0944-2669
SN  - 1432-0835
VL  - 53
IS  - 1-2
SP  - 265
EP  - 287
PB  - Springer
CY  - Heidelberg
ER  - 
TY  - JOUR
A1  - Kröncke, Klaus
T1  - Rigidity and Infinitesimal Deformability of Ricci Solitons
JF  - The journal of geometric analysis
N2  - 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.
KW  - Ricci solitons
KW  - Moduli space
KW  - Linearized equation
KW  - Integrability
Y1  - 2016
U6  - https://doi.org/10.1007/s12220-015-9608-4
SN  - 1050-6926
SN  - 1559-002X
VL  - 26
SP  - 1795
EP  - 1807
PB  - Springer
CY  - New York
ER  - 
TY  - JOUR
A1  - Kröncke, Klaus
T1  - On infinitesimal Einstein deformations
JF  - Differential geometry and its applications
N2  - 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.
Y1  - 2015
U6  - https://doi.org/10.1016/j.difgeo.2014.11.007
SN  - 0926-2245
SN  - 1872-6984
VL  - 38
SP  - 41
EP  - 57
PB  - Elsevier
CY  - Amsterdam
ER  -