The Weyl functional on 4-manifolds of positive Yamabe invariant

2021/08/09 by Sung, Chanyoung · 1 citation
#57R18 #58E99 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2108.03808

Abstract

It is shown that on every closed oriented Riemannian 4-manifold (M,g) with positive scalar curvature, ∫_M|W⁺_g|²dμ_g≥ 2π²(2χ(M)+3τ(M))-(8π²)/(|π₁(M)|), where W⁺_g, χ(M) and τ(M) respectively denote the self-dual Weyl tensor of g, the Euler characteristic and the signature of M. This generalizes Gursky's inequality \citegur for the case of b₁(M)>0 in a much simpler way. We also extend all such lower bounds of the Weyl functional to 4-orbifolds including Gursky's inequalities for the case of b₂⁺(M)>0 or δ_gW⁺_g=0, and obtain topological obstructions to the existence of self-dual orbifold metrics of positive scalar curvature.

Cited by

• On the Yamabe constants of product manifolds

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