Dirichlet principal eigenvalue comparison theorems in geometry with torsion

TL;DR

This paper develops min-max formulas for the principal eigenvalue of a V-drift Laplacian on geodesic balls in Riemannian manifolds, and establishes comparison theorems with model spaces under curvature and vector field conditions.

Contribution

It introduces new comparison theorems for the principal eigenvalue involving V-drift Laplacians, extending classical results to include vector fields and curvature conditions.

Findings

Derived min-max formulas for V-drift Laplacian eigenvalues

Established eigenvalue comparison theorems under curvature and vector field assumptions

Generalized known eigenvalue comparison results to include torsion and drift effects

Abstract

We describe min-max formulas for the principal eigenvalue of a $V$-drift Laplacian defined by a vector field $V$ on a geodesic ball of a Riemannian manifold $N$. Then we derive comparison results for the principal eigenvalue with the one of a spherically symmetric model space endowed with a radial vector field, under pointwise comparison of the corresponding radial sectional and Ricci curvatures, and of the radial component of the vector fields. These results generalize the known case $V=0$.