Morse homology for the heat flow – Linear theory |
| |
Authors: | Joa Weber |
| |
Affiliation: | Universidade de S?o Paulo, Departamento de Matemática, Instituto de Matemática e Estatística, Rua do Mat?o 1010, Cidade Universitária, S?o Paulo, SP 05508‐090, Brazil. Phone: +55?11?3091?6136, Fax: +55?11?3091?6131 |
| |
Abstract: | Consider the linear parabolic partial differential equation ${mathcal {D}}_uxi =0$ which arises by linearizing the heat flow on the loop space of a Riemannian manifold M. The solutions are vector fields along infinite cylinders u in M. For these solutions we establish regularity and a priori estimates. We show that for nondegenerate asymptotic boundary conditions the solutions decay exponentially in L2 in forward and backward time. In this case ${mathcal {D}}_u$ viewed as linear operator from the parabolic Sobolev space ${mathcal {W}}^{1,p}$ to Lp is Fredholm whenever p > 1. We close with an Lp estimate for products of first order terms which is a crucial ingredient in the sequel 13 to prove regularity and the implicit function theorem. The results of the present text are the base to construct in 13 an algebraic chain complex whose homology represents the homology of the loop space. |
| |
Keywords: | Heat flow exponential decay Fredholm theory Morse complex MSC (2010) 35K05 58J35 |
|
|