First eigenvalue estimates on complete balanced Hermitian manifolds

Abstract

We establish lower bounds for the first positive eigenvalue of the Laplace--de Rham operator on complete balanced Hermitian manifolds in terms of curvature of the Strominger--Bismut connection. Under a positive lower bound for its holomorphic Ricci curvature, we prove a Lichnerowicz--Obata type estimate and characterize the equality case in the K\"ahler setting. We also derive Li--Yau and Zhong--Yang type estimates from lower bounds on the same holomorphic Ricci curvature, including estimates under weaker assumptions only along a first eigendirection in the compact case. Finally, under a positive lower bound for the holomorphic sectional curvature of the Strominger--Bismut connection and a torsion-commutator condition along a first eigendirection, we obtain a lower bound for the first eigenvalue. In the compact case, the commutator condition follows from vanishing of the Strominger--Bismut torsion in that eigendirection. These results extend several classical K\"ahler and Riemannian spectral estimates to the balanced non-K\"ahler setting.