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arXiv · 1512.01153

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

Levi Lopes de Lima
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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 Weitzenb\"ock 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.

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Levi Lopes de Lima. 2015-12-03. A Feynman-Kac formula for differential forms on manifolds with boundary and applications. https://doi.org/10.2140/pjm.2018.292.177

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