The nonexistence of quasi-einstein metrics

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We study complete Riemannian manifolds satisfying the equation by studying the associated PDE Δff+ mμ exp(2f/m) = 0 for μ < 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 that have nonpositive Einstein constant. We also show that for nontrivial steady gradient Ricci solitons, the quantity R + |∇f|2 is a positive constant.

Original languageEnglish (US)
Pages (from-to)277-284
Number of pages8
JournalPacific Journal of Mathematics
Issue number2
StatePublished - 2010

All Science Journal Classification (ASJC) codes

  • General Mathematics


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