We establish a Feynman–Kac-type formula for differential forms
satisfying absolute boundary conditions on Riemannian manifolds
with boundary and of bounded geometry. We use this to construct
-harmonic
forms out of bounded ones on the universal cover of a compact Riemannian
manifold whose geometry displays a positivity property expressed in
terms of a certain stochastic average of the Weitzenböck operator
acting
on
-forms
and the second fundamental form of the boundary. This extends previous work by
Elworthy, Li and Rosenberg on closed manifolds to this more general setting. As an
application we find a new obstruction to the existence of metrics with positive
(in
particular, positive isotropic curvature) and 2-convex boundary. We also discuss a
version of the Feynman–Kac formula for spinors under suitable boundary conditions
and use this to prove a semigroup domination result for the corresponding Dirac
Laplacian under a mean convexity assumption.
Keywords
Feynman–Kac formula, absolute boundary conditions, heat
flow on differential forms, Brownian motion on manifolds
