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

TL;DR

This paper develops a Feynman-Kac formula for differential forms on manifolds with boundary, enabling new insights into harmonic forms, geometric obstructions, and boundary conditions in Riemannian geometry.

Contribution

It extends the Feynman-Kac formula to differential forms with boundary conditions, linking stochastic analysis with geometric properties of manifolds.

Findings

Constructed $L^2$ harmonic forms from bounded ones using stochastic methods.

Identified geometric obstructions to metrics with 2-convex boundary and positive $R_2$.

Extended Feynman-Kac formulas to spinors with boundary conditions.

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.