Prof. Dr. Bernd Ammann, Universität Regensburg, Vortrag: The Yamabe constant of products and applications

Thursday, 24.01.2013 16:30 im Raum M5

Mathematik und Informatik

Collaboration with M. Dahl, Stockholm, and E. Humbert, Tours The conformal Yamabe constant of a compact connected riemannian mani- fold (M; g0) is de ned as Y (M; [g0]) := inf Z M scalg dvg where the in mum runs over all metrics g of volume 1 in the conformal class [g0]. This in mum is attained, and such a minimizer is called a Yamabe metric. Yamabe metrics are of constant scalar curvature. In the talk we raise the question: Assume gi is a Yamabe metric of positive scalar curvature on Mi, i = 1; 2. Is then the product metric on M1 x M2 again a Yamabe metric? We will see that up to scaling and a "small" error, this is indeed true. We will then explain how this product formula can be used to give lower bounds for the (smooth) Yamabe invariant of compact manifolds. The smooth Yamabe invariant of M is de ned as sup Y (M; [g0]) where the supremum runs over all conformal classes on M. This invariant is positive i M carries a metric of positive scalar curvature, however for most manifolds it is extremely diffcult to calculate it. Our methods yield explicit lower bounds in the positive case, e.g. the (smooth) Yamabe invariant of a simply connected 6-dimensional manifold is at least 49.9.

Angelegt am Wednesday, 10.10.2012 11:53 von shupp_01
Geändert am Monday, 07.01.2013 12:09 von shupp_01
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