Caltech/UCLA Joint Analysis Seminar

We prove that any quasi-conformal map of the (n−1)-dimensional sphere when n > 2 can be extended to a smooth quasi-isometry F of the n-dimensional hyperbolic space such that the heat flow starting with F converges to a quasi-isometric harmonic map. This implies the Schoen-Li-Wang conjecture that every quasi-conformal map of the (n−1)-sphere can be extended to a harmonic quasi-isometry when n > 2 (that result was recently proved for n=2,3 without proving convergence of the heat flow). The main tools to show the convergence of the heat flow are the Hamilton parabolic maximum principle for sub-solutions of the heat equation, the diffusion of heat in hyperbolic space and the Mostow rigidity. This is joint work with Vladimir Markovic.