Evolution of relative Yamabe constant under Ricci Flow

2019/01/31 by Botvinnik, Boris, Lu, Peng
#53C27 #57R65 #58J05 #58J50 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1901.11169

Abstract

Let W be a manifold with boundary M given together with a conformal class C which restricts to a conformal class C on M. Then the relative Yamabe constant Y _C(W,M;C) is well-defined. We study the short-time behavior of the relative Yamabe constant Y_{[ gt]}(W,M;C) under the Ricci flow g_t on W with boundary conditions that mean curvature H _gt≡ 0 and g_t|_M∈ C = [g₀]. In particular, we show that if the initial metric g₀ is a Yamabe metric, then, under some natural assumptions, .(d)/(dt)|_t=0Y_{[ gt]}(W,M;C)≥ 0 and is equal to zero if and only the metric g₀ is Einstein.

Related

• Ricci curvature and Yamabe constants
• Evolution of Yamabe constant under Ricci flow
• Relative Yamabe Invariant
• On metrics of positive Ricci curvature conformal to MxR^m
• A Note on Conformal Ricci Flow