Morse homology for the heat flow – Linear theory

Consider the linear parabolic partial differential equation Duξ = 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 apriori estimates. We show that for nondegenerate asymptotic boundary conditions the solutions decay exponentially in L in forward and backward time. In this case Du viewed as linear operator from parabolic Sobolev space W to L is Fredholm whenever p > 1. We close with an L 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.