#### Vol. 292, No. 1, 2018

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

### Levi Lopes de Lima

Vol. 292 (2018), No. 1, 177–201
##### Abstract

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 ${L}^{2}$-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 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 ${R}_{2}$ (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
##### Mathematical Subject Classification 2010
Primary: 53C21, 53C27
Secondary: 58J35, 58J65