科研之友
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摘要
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 Weitzenbock 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 semi-group domination result for the corresponding Dirac Laplacian under a mean convexity assumption.
- 出版日期2018-1
