**Title: Harmonic maps and heat flows on hyperbolic spaces**

**Abstract:**

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. We

also prove the corresponding conjecture when n=2 (which was the original Schoen Conjecture), but this proof does not involve heat flows.

Wednesday, Nov. 11, 4:30 – 5:30 p.m.

Mathematics 520

Tea will be served at 4:00 p.m.