Harmonic diffeomorphisms between manifolds with bounded curvature

[摘要] Let compact n-dimensional Riemannian manifolds $(M,g), (widehat M, g)$ a diffeomorphism $usb0: Mo widehat M,$ and a constant $p > n$ be given. Then sufficiently small $Lsp{p}$ bounds on the curvature of $widehat M$ and on the difference of $g$ and $usbsp{0}{*} g$ guarantee that $usb0$ can be continuously deformed to a harmonic diffeomorphism. A vector field $v$ is constructed on the space of mappings $u$ which are $Lsp{2,p}$ close to $usb0$ by solving the nonlinear elliptic equation $Delta v + widehat{Rc} v = -Delta u.$ It is shown that under sufficient conditions on $usb0$ and on the curvature $widehat{Rm}$ of the target, the integral curve $usb t$ 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.

[发布机构] Rice University