Global Convergence of the Yamabe Flow in Dimension 6 and Higher.
November 01, 2005 11:00 AM to 12:00 PM
Speakers:
Brendle, Simon
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Abstract: |
Let M be a compact manifold of dimension n \geq 3. Along the Yamabe flow, a Riemannian metric on M is deformed according to the equation \frac{\partial g}{\partial t} = -(R_g - r_g) \, g, where R_g is the scalar curvature associated with the metric g and r_g denotes the mean value of R_g. It is known that the Yamabe flow exists for all time. Moreover, if 3 \leq n \leq 5 or M is locally conformally flat, then the solution approaches a metric of constant scalar curvature as t \to \infty. I will describe how this result can be generalized to dimensions 6 and higher under a technical condition on the Weyl tensor. The proof is based on the construction of a family of test functions. Each of these test functions has Yamabe energy less than the Yamabe constant of the standard sphere S^n. |
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