New York Journal of Mathematics
Volume 23 (2017) 1395-1415


Brian Weber

Harnack inequalities for critical 4-manifolds with a Ricci curvature bound

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Published: October 12, 2017
Keywords: Critical Riemannian metrics, ε-regularity, collapsing, Ricci curvature, Harnack inequalities
Subject: 53C21, 58J05 (primary); 53C25 (secondary)

We study critical Riemannian 4-manifolds with bounded Ricci curvature, but with no apriori analytic constraints such as on Sobolev constants. We derive elliptic-type estimates for the local curvature radius, which itself controls sectional curvature. The method is use degenerating, collapsing metrics to create a noncollapsed blow-up limit, and then use a geometric triviality result for complete Ricci-flat manifolds with a Killing field to rule out such a blow-up. The Cheeger-Tian ε-regularity theorem on Einstein manifolds is reproved as a byproduct.

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David Rittenhouse Lab., 209 South 33rd Street, Philadelphia, PA 19104-6395