Topics in Low Dimensional Geometric Analysis

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Numerous advances in topology have arisen from applications of ideas from geometry and dynamics. This project aims to continue this line of investigation. More specifically, we aim to study 3-manifolds, harmonic maps, and Riemann surfaces. The study of 3-manifolds is in essence the study of the possible shapes of the universe and as such sits naturally at the intersection of mathematics and theoretical physics. Surfaces are two-dimensional spaces like the surface of a ball or a doughnut. Their topological and geometric properties have been extensively studied and this research lies at the intersection of all fields of mathematics. The parameter space of all geometric structures on a given surface is called the moduli space. We study moduli spaces because to understand one particular structure on a space we must understand the structure of the space of all structures. Theory of harmonic maps originates in the theory of minimal surfaces. It aims to establish important properties of harmonic maps which in turn provide useful information about minimal surfaces like the physical description of a soap film spanning a wire boundary.The principal investigator will study geometry of Teichmueller spaces, existence of equivariant harmonic maps for surface group representations, maps from surfaces into 3-manifolds, and the solutions of the 2D-Euler equation for incompressible fluids. The main goals are to: Classify Teichmueller discs where the Caratheodory and Teichmueller metrics agree; To build on the proof of the Schoen conjecture and establish results about the existence of equivariant harmonic maps corresponding to representations of the fundamental groups of compact surfaces with boundary and of surfaces of infinite rank; To investigate the simple loop conjecture for hyperbolic 3-manifolds; To consider whether the gradient of the vorticity in the 2D-Euler equation can achieve the double exponential growth.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.