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Heat conservation for generalized Dirac Laplacians on manifolds with boundary. (arXiv:1712.06372v1 [math.DG])
来源于:arXiv
We consider a notion of conservation for the heat semigroup associated to a
generalized Dirac Laplacian acting on sections of a vector bundle over a
noncompact manifold with a (possibly noncompact) boundary under mixed boundary
conditions. Assuming that the geometry of the underlying manifold is controlled
in a suitable way and imposing uniform lower bounds on the zero order
(Weitzenb\"ock) piece of the Dirac Laplacian and on the endomorphism defining
the mixed boundary condition we show that the corresponding conservation
principle holds. A key ingredient in the proof is a domination property for the
heat semigroup which follows from an extension to this setting of a Feynman-Kac
formula recently proved in \cite{dL1} in the context of differential forms.
When applied to the Hodge Laplacian acting on differential forms satisfying
absolute boundary conditions, this extends previous results by Vesentini
\cite{Ve} and Masamune \cite{M} in the boundaryless case. Along the way we also
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