Doctor of Philosophy, The Ohio State University, 2018, Mathematics

This work deals with various phenomena relating to complex geometry. We are particularly interested in non-Kahler Hermitian manifolds, and most of the work here was done to try to understand the geometry of these spaces by understanding the torsion. Chapter 1 introduces some background material as well as various equations and inequalities on Hermitian manifolds. We are focused primarily on the inequalities that are useful for the analysis that we do later in the thesis. In particular, we focus on k-Gauduchon complex structures, which were initially defined by Fu, Wang, and Wu. Chapter 2 discusses the spectral geometry of Hermitian manifolds. In particular, we estimate the real eigenvalues of the complex Laplacian from below. In doing so, we prove a theorem on non-self-adjoint drift Laplace operators with bounded drift. This result is of independent interest, apart from its application to complex geometry. The work in this section is largely based on the Li-Yau estimate as well as an ansatz due to Hamel, Nadirashvili and Russ. Chapter 3 considers orthogonal complex structures to a given Riemannian metric. Much of the work in this section is conjectural in nature, but we believe that this is a promising approach to studying Hermitian geometry. We do prove several concrete results as well. In particular, we show how the moduli space of k-Gauduchon orthogonal complex structures is pre-compact.

Committee: Fangyang Zheng (Advisor); Bo Guan (Committee Member); King-Yeung Lam (Committee Member); Jean-Francois Lafont (Committee Member); Mario Miranda (Committee Member) Subjects: Mathematics