Gao, Zhiyong2009-06-042009-06-041991Anderson, John Patrick. "Harmonic diffeomorphisms between manifolds with bounded curvature." (1991) Diss., Rice University. <a href="https://hdl.handle.net/1911/16413">https://hdl.handle.net/1911/16413</a>.https://hdl.handle.net/1911/16413Let 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.67 p.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.MathematicsThesisThesis Math. 1991 Anderson