On gradient Ricci solitons with symmetry

We study gradient Ricci solitons with maximal symmetry. First we show that there are no nontrivial homogeneous gradient Ricci solitons. Thus, the most symmetry one can expect is an isometric cohomogeneity one group action. Many examples of cohomogeneity one gradient solitons have been constructed. H...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2009-06, Vol.137 (6), p.2085-2092
Main Authors: Petersen, Peter, Wylie, William
Format: Article
Language:English
Subjects:
Coordinate systems
Curvature
Distance functions
Exact sciences and technology
General mathematics
General, history and biography
Global analysis, analysis on manifolds
Infinity
Mathematical analysis
Mathematical manifolds
Mathematical theorems
Mathematics
Partial differential equations
Riemann manifold
Scalars
Sciences and techniques of general use
Solitons
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
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Summary:We study gradient Ricci solitons with maximal symmetry. First we show that there are no nontrivial homogeneous gradient Ricci solitons. Thus, the most symmetry one can expect is an isometric cohomogeneity one group action. Many examples of cohomogeneity one gradient solitons have been constructed. However, we apply the main result in our paper ``Rigidity of gradient Ricci solitons'' to show that there are no noncompact cohomogeneity one shrinking gradient solitons with nonnegative curvature.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-09-09723-8