Canonical identification between scales on Ricci-flat manifolds

Let $M$ be a complete Ricci-flat manifold with Euclidean volume growth. A theorem of Colding-Minicozzi states that if a tangent cone at infinity of $M$ is smooth, then it is the unique tangent cone. The key component in their proof is an infinite dimensional Lojasiewicz-Simon inequality, which implies rapid decay of the $L^2$-norm of the trace-free Hessian of the Green function. In this talk we discuss how this inequality can be exploited to identify two arbitrarily far apart scales in $M$ in a natural manner through a diffeomorphism. We also prove a pointwise Hessian estimate for the Green function when there is an additional condition on sectional curvature, which is an analogue of various matrix Harnack inequalities obtained by Hamilton and Li-Cao in different time-dependent settings.

— Organized by Prof. Shing-Tung Yau