A Feynman-Kac formula for differential forms on manifolds with boundary and geometric applications (Q2408271)

This paper contains novel comparison results for Riemannian manifolds with boundary, achieved by stochastic methods. More precisely, let \((N, h)\) be a Riemannian manifold with boundary of dimension strictly greater than three which is complete and bounded in an appropriate sense if \(N\) is non-compact. Consider the Weitzenböck decomposition \(\Delta_q = \Delta_q^B + R_q\) of the Hodge Laplacian on \(q\)-forms into the sum of the Bochner Laplacian and a curvature expression \(R_q\). Write \(r_{(q)}(x)\) for the smallest eigenvalue of \(R_q(X)\) at the point \(x\in M\) giving rise to a function \(r_{(q)}:N\rightarrow\mathbb{R}\) and likewise, write \(\rho_{(q)}(y)\) for the sum of the first \(q\) smallest principal curvatures at \(y\in \partial N\) yielding a function \(\rho_{(q)}:\partial N\rightarrow\mathbb{R}\). Regarding the stochastic ingredient, consider a reflecting Brownian motion \(\{x^t\}\) on \((N,h)\) (for a brief introduction to this diffusion process see Section 5). Then one can talk about so-called strong stochastic positivity (s.s.p) of various pairs \((r_{(q')}, \rho_{q''})\) along \(N\) with respect to \(\{x^t\}\); roughly speaking s.s.p captures in what extent these pairs of functions lack being positive (for a precise definition see the equation on the top of page 179). In the main results of the paper, the author gives conditions under which \((N,h)\) does not admit metrics with certain pairs \((r_{(q')}, \rho_{q''})\) meeting the s.s.p property (see Theorem 1.3 and Corollary 1.1). A crucial ingredient in the proof of these results is the application of a Donnelly-Xavier-type eigenvalue estimate (Section 3) as well as a Feynman-Kac-type Formula for differential forms satisfying absolute boundary conditions; a variant of the Feynman-Kac formula on spin\({}^c\)-manifolds for spinors with suitably boundary conditions is also introduced (concerning both, see Section 5).