Differential Geometry Seminar Schedule for
Fall 2009
Fridays 3:00 - 3:50pm, SH 6617
10/9 Rugang Ye, UCSB
``G2 geometry and the Laplacian flow I"
10/16 Rugang Ye, UCSB
``G2 geometry and the Laplacian flow II"
10/30 Guihua Gong, University of Puerto Rico
``Positive scalar curvature and
non-commutative geometry"
11/6, Jeffrey Case, UCSB
``On the nonexistence of quasi-Einstein metrics"
Abstract: We study Riemannian manifolds satisfying the equation $Ric+Hess(f)-\frac{1}{m}df^2=0$ by studying the associated PDE $\Delta_f f + ke^{2f/m}=0$ for $k\leq 0$. We develop a gradient estimate for such functions, and show that the only solutions on a complete manifold are constant functions. As an application, we show that there are no nontrivial Ricci flat warped products whose fibers are Einstein with nonpositive scalar curvature. We also show that one can take the limit $m\to\infty$ and get that for nontrivial steady gradient Ricci solitons $R+|df|^2$ is a positive constant.
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