Harmonic Maps of Riemann Surfaces and Applications in Geometry

Author: Sagman, Nathaniel Levi

Year: 2022

Degree: Dissertation (Ph.D.)

Advisor: Ni, Yi

Committee Members: Katz, Nets H.; Ni, Yi; Gukov, Sergei; Chen, Lei

Option: Mathematics

DOI: 10.7907/14dw-zh58

Abstract

Harmonic maps are fundamental objects in differential geometry. They play an important role in studying deformations of geometric structures and in various rigidity problems. In this thesis, we present three projects, all of which involve harmonic mappings of Riemann surfaces.

In the first project, we study infinite energy harmonic maps and spacelike maximal surfaces in pseudo-Riemannian manifolds, and give applications to domination for surface group representations and anti-de Sitter geometry. The culminating result is the existence of a new class of anti-de Sitter 3-manifolds and a parametrization of their deformation space.

The second project concerns moduli spaces of harmonic surfaces inside higher dimensional Riemannian manifolds. First, we prove a factorization theorem for harmonic maps. We then use infinite-dimensional transversality theory to prove results about the distribution of certain families of harmonic surfaces inside our moduli spaces.

The final project is motivated by the Labourie conjecture from Higher Teichmueller theory. We prove non-uniqueness results for minimal surfaces in products of hyperbolic surfaces and products of ℝ-trees, and we make a connection to classical minimal surfaces.

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