Vol. 292, No. 1, 2018

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ISSN: 0030-8730
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

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 L2-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 Rp 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 R2 (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.

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
Received: 19 June 2016
Revised: 4 May 2017
Accepted: 2 June 2017
Published: 22 September 2017
Levi Lopes de Lima
Departamento de Matemática
Universidade Federal do Ceará (UFC)