James Turetsky
Abstract: Let M be a compact connected Riemannian manifold and 
be the (Levi-Civita) Laplacian. Let 
be the fundamental solution
to the Cauchy initial value problem for the heat equation 
. We derive the upper bounds which say that the
n-th covariant logarithmic derivative of 
at x behaves
like a constant times
for 
. We then study the asymptotic behavior of logarithmic
derivatives. In particular, we show that this bound can be improved for
compact sets in the complement of the cut locus of y, and that a dramatic
change in behavior occurs for x at the cut locus of y.