Bismut hypoelliptic Laplacians for manifolds with boundaries

TL;DR

This paper studies boundary conditions for Bismut's hypoelliptic Laplacian on manifolds with boundaries, linking them to classical boundary conditions and analyzing their spectral properties.

Contribution

It introduces boundary conditions for the hypoelliptic Laplacian that correspond to classical Dirichlet and Neumann conditions, and examines their spectral implications.

Findings

Boundary conditions for hypoelliptic Laplacian are compatible with classical boundary conditions.

The commutation of differential with the resolvent is established.

Spectral properties like PT-symmetry are analyzed for these operators.

Abstract

Boundary conditions for Bismut's hypoelliptic Laplacian which naturally correspond to Dirichlet and Neumann boundary conditions for Hodge Laplacians are considered. Those are related with specific boundary conditions for the differential and its various adjoints. Once the closed realizations of those operators are well understood, the commutation of the differential with the resolvent of the hypoelliptic Laplacian is checked with other properties like the PT-symmetry, which are important for the spectral analysis.