Laplacian Processes on Submanifolds

The Laplacian controls a variety of very natural
phenomena ranging from heat diffusion to volume
growth. It is the purpose of this project to show
precise bounds for specific values of measures
such as the mean exit time for Brownian motion
and isoperimetric inequalities for well chosen
subsets of the submanifolds under consideration.
The geometric structures under consideration are
minimal (or close to minimal) submanifolds and
minimal (or close to minimal) metric graphs. The
bounds alluded to are obtained via comparison
with corresponding values in constant curvature
spaces and in other tailor-made warped product
spaces.