Differential Geometry and Topology

course: Heat diffusion and mean curvature

speaker: Alessandro Savo (Sapienza Università di Roma)

abstract: Given any smooth function (initial temperature distribution) on a Riemannian manifold with smooth boundary, we study its evolution by the heat equation under Dirichlet boundary conditions (i.e. temperature distribution at time t assuming absolute refrigeration on the boundary). Integrating over the manifold we obtain the total heat content at time t, whose asymptotic behavior for small times is described by a countable family of Riemannian invariants. Our main result is an explicit recursive formula which computes all these invariants. In particular, we show that the invariants under consideration can be expressed in terms of an algebra of operators with only two generators, the Laplacian and a first order boundary operator written in terms of the mean curvature.

Finally, we relate this algebra to the pointwise heat flow on the boundary and give a unified solution to other heat diffusion problems which appeared in the literature.

The talk is based on the paper Asymptotics of the heat flow on a manifold with smooth boundary, to appear in Commun. Analysis Geometry.

timetable:
Mon 25 Oct, 18:15 - 19:15, Sala Conferenze Centro De Giorgi
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