Dirichlet principal eigenvalue comparison theorems in geometry with torsion

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 $N$. Using these formulas, under pointwise upper or lower bounds of the the radial sectional and Ricci curvatures of $N$ and of the radial component of $V$, we derive comparison results for the principal eigenvalue with the one of a spherically symmetric model space endowed with a radial vector field. These results generalize the well known case $V=0$.