A Feynman-Kac formula for differential forms on manifolds with boundary and applications

Abstract

We prove a Feynman-Kac formula for differential forms satisfying absolute boundary conditions on Riemannian manifolds with boundary and of bounded geometry. We use this to construct $L^2$ 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 $R_p$ acting on $p$-forms and the second fundamental form of the boundary. This extends previous work by Elworthy-Li-Rosenberg on closed manifolds to this setting. As an application we find a geometric obstruction to the existence of metrics with 2-convex boundary and positive $R_2$ in this stochastic sense. We also discuss a version of the Feynman-Kac formula for spinors under suitable boundary conditions.

20 pages; no figures

References in corpus (1)

• Formulae for the derivatives of heat semigroups

Cited by in corpus (4)

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