Canonical identification between scales on Ricci-flat manifolds

DIFFERENTIAL GEOMETRY

Speaker:

Jiewon Park - MIT

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