A Lichnerowicz estimate for the spectral gap of the sub-Laplacian

TL;DR

This paper extends the classical Lichnerowicz estimate to a class of second order operators satisfying the strong Hörmander condition on compact manifolds, providing bounds for their spectral gap.

Contribution

It introduces a Lichnerowicz-type estimate for the spectral gap of sub-Laplacians under Hörmander conditions, broadening the scope beyond Riemannian Laplacians.

Findings

Derived a spectral gap bound for sub-Laplacians satisfying Hörmander's condition

Included horizontal lifts of Laplacians on Riemannian submersions with minimal leaves

Generalized classical estimates to a wider class of differential operators

Abstract

For a second order operator on a compact manifold satisfying the strong H\"ormander condition, we give a bound for the spectral gap analogous to the Lichnerowicz estimate for the Laplacian of a Riemannian manifold. We consider a wide class of such operators which includes horizontal lifts of the Laplacian on Riemannian submersions with minimal leaves.