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Harmonic Maps of Riemann Surfaces and Applications in Geometry


Sagman, Nathaniel Levi (2022) Harmonic Maps of Riemann Surfaces and Applications in Geometry. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/14dw-zh58.


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.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Differential geometry
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Ni, Yi
Thesis Committee:
  • Katz, Nets H. (chair)
  • Ni, Yi
  • Gukov, Sergei
  • Chen, Lei
Defense Date:25 May 2022
Record Number:CaltechTHESIS:06012022-020454667
Persistent URL:
Related URLs:
URLURL TypeDescription adapted for Ch. III adapted for Ch. IV adapted for Ch. V adapted for Ch. VI adapted for Ch. VII
Sagman, Nathaniel Levi0000-0002-8485-7073
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:14659
Deposited By: Nathaniel Sagman
Deposited On:02 Jun 2022 19:55
Last Modified:04 Aug 2022 23:30

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