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Jeff Cheeger: Lectures

April 10, 2018
TITLE: Curvature and injectivity radius estimates for Einstein 4-manifolds

ABSTRACT: We will review some old joint work with Gang Tian ([JAMS 2005]) on the structure of possibly collapsed Einstein 4-manifolds with a definite bound on the Euler characteristic. Among the main results is an \epsilon-regularity theorem in which the L^2-norm of the curvature is just assumed to be sufficiently small independent of the collapsing on a ball B_r(p). It leads to a bound on the curvature on the ball B_{r/2} (p), the topology of which need not be that of a Euclidean ball, though the possiblities for the topology are known; essentially it is a tube around a nilmanifold. Another theorem can be referred to as “collapse implies concentration of curvature”. There is also a theorem whose conclusion can be regarded as a kind of counterpart of the Margulis Lemma, for the case of negative Einstein constant. The hope remains that these results, and the corresponding techniques of proof, could lead to a more general theory of Einstein 4-manifolds.

Slides of lecture

September 9, 2016
TITLE: Ricci curvature: Some recent progress and open questions

ABSTRACT: We will survey some recent (and less recent) progress on Ricci curvature and mention a few questions
which remain open.

Slides of lecture