# Harmonic diffeomorphisms between manifolds with bounded curvature

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 Title: Harmonic diffeomorphisms between manifolds with bounded curvature Anderson, John Patrick Gao, Zhiyong Doctor of Philosophy thesis Let compact n-dimensional Riemannian manifolds $(M,g),\ (\widehat M,\ g)$ a diffeomorphism $u\sb0: M\to \widehat M,$ and a constant $p > n$ be given. Then sufficiently small $L\sp{p}$ bounds on the curvature of $\widehat M$ and on the difference of $g$ and $u\sbsp{0}{\*}\ g$ guarantee that $u\sb0$ can be continuously deformed to a harmonic diffeomorphism. A vector field $v$ is constructed on the space of mappings $u$ which are $L\sp{2,p}$ close to $u\sb0$ by solving the nonlinear elliptic equation $\Delta v + \widehat{Rc}\ v = -\Delta u.$ It is shown that under sufficient conditions on $u\sb0$ and on the curvature $\widehat{Rm}$ of the target, the integral curve $u\sb t$ of this vector field converges to a harmonic diffeomorphism. Since the objects we work with, such as $v$ and its derivatives, live in bundles over $M$, to prove regularity results we must first adapt standard techniques and results of elliptic theory to the bundle case. Among the generalizations we prove are Moser iteration, a Sobolev embedding theorem, and a Calderon-Zygmund inequality. Anderson, John Patrick. (1991) "Harmonic diffeomorphisms between manifolds with bounded curvature." Doctoral Thesis, Rice University. http://hdl.handle.net/1911/16413. http://hdl.handle.net/1911/16413 1991