A Feynman-Kac formula for differential forms on manifolds with boundary and applications
A Feynman-Kac formula for differential forms on manifolds with boundary and applications
We prove a Feynman-Kac-type formula for the heat flow acting onÌýdifferential forms satisfying absolute boundary conditions on Riemannian manifoldsÌýwith boundary and of bounded geometry. We use this to construct L^2 harmonic formsÌýout of bounded ones on the universal cover of a compact Riemannian manifold whoseÌýgeometry displays a positivity property expressed in terms of a certain stochasticÌýaverage of the Weitzenbock operator acting on forms and the second fundamental formÌýof the boundary. As a geometric application we find a new obstruction to the existence of metricsÌýwith positive isotropic curvature and 2-convex boundary.We also present a version of the Feynman-Kac formula for spinors under suitableÌýboundary conditions and discuss potential applications.Ìý