On the nonexistence of quasi-Einstein metrics

Details On the nonexistence of quasi-Einstein metrics On the nonexistence of quasi-Einstein metrics

2009-02-12

arxiv.org/abs/0902.2226v3

Pacific J. Math. 248-2 (2010), 277--284

Mathematics

Differential Geometry

Final version: Improved exposition of Section 2, corrected minor typos

Scientific paper

10.2140/pjm.2010.248.277

We study complete Riemannian manifolds satisfying the equation $Ric+\nabla^2 f-\frac{1}{m}df\otimes df=0$ by studying the associated PDE $\Delta_f f + m\mu e^{2f/m}=0$ for $\mu\leq 0$. By developing a gradient estimate for $f$, we show there are no nonconstant solutions. We then apply this to show that there are no nontrivial Ricci flat warped products with fibers which have nonpositive Einstein constant. We also show that for nontrivial steady gradient Ricci solitons, the quantity $R+|\nabla f|^2$ is a positive constant.

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