On gradient Ricci solitons

Details On gradient Ricci solitons On gradient Ricci solitons

2009-10-06

arxiv.org/abs/0910.1105v2

Mathematics

Differential Geometry

to appear in J. Geom. Anal

Scientific paper

In the first part of the paper we derive integral curvature estimates for complete gradient shrinking Ricci solitons. Our results and the recent work of Lopez-Rio imply rigidity of gradient shrinking Ricci solitons with harmonic Weyl tensor. In the second part of the paper we address the issue of existence of harmonic functions on gradient shrinking K\"{a}hler and gradient steady Ricci solitons and show that if the total energy of a harmonic function on such a manifold is finite then the function is constant. Consequences to the structure of the manifold at infinity are also discussed.

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