|
|||||
|
|
|
|
|
|
|
|
|
The nonexistence of
quasi-Einstein metrics
Jeffrey S. Case |
|
Vol. 248 (2010), No. 2, 277–284
|
Abstract |
|
We study complete Riemannian manifolds satisfying the equation  by studying the associated PDE Δff + mμexp(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. |
Keywords
Einstein, warped product, quasi-Einstein
|
Mathematical Subject Classification 2000
Primary: 53C21
Secondary: 58J60
|
Milestones
Received: 9 September 2009
Revised: 23 January 2010
Accepted: 4 February 2010
Published: 1 December 2010
|
Authors |
| Jeffrey S. Case | |
| Department of Mathematics University of California Santa Barbara, CA 93106 United States |
|
| http://math.ucsb.edu/~casej | |
|
