Heat Kernel and Hard Estimates for Locally Euclidean Manifolds with Fractal Boundaries.

E.B. Davies; M. Lianantonakis

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In this talk we shall present some joint work with A. Grigory’an. Upper and lower estimates on the rate of decay of the heat kernel on a complete non-compact riemannian manifold have recently been obtained in terms of the geometry at infinity of the manifold, more precisely in terms of a kind of $L^{2}$ isoperimetric profile. The main point is to connect the decay of the $L^{1} - L^{\infty}$ norm of the heat semigroup with some adapted Nash or Faber-Krahn inequalities, which is done by functional analytic methods....

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