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PhD Thesis colloquium

Title: Harmonic map heat flow and framed surface-group representations
Speaker: Gobinda Sau (IISc Mathematics)
Date: 11 August 2023
Time: 11:15 am
Venue: LH-1, Mathematics Department

This thesis concerns the construction of harmonic maps from certain non-compact surfaces into hyperbolic 3-space $\mathbb{H}^3$ with prescribed asymptotic behavior and has two parts.

The focus of the first part is when the domain is the complex plane. In this case, given a finite cyclic configuration of points $P \subset \partial\mathbb{H}^3=\mathbb{CP}^1$, we construct a harmonic map from $\mathbb{C}$ to $\mathbb{H}^3$ that is asymptotic to a twisted ideal polygon with ideal vertices contained in $P$. Moreover, we prove that given any ideal twisted polygon in $\mathbb{H}^3$ with rotational symmetry, there exists a harmonic map from $\mathbb{C}$ to $\mathbb{H}^3$ asymptotic to that polygon. This generalizes the work of Han, Tam, Treibergs, and Wan which concerned harmonic maps from $\mathbb{C}$ to the hyperbolic plane $\mathbb{H}^2$.

In the second part, we consider the case of equivariant harmonic maps. For a closed Riemann surface $X$, and an irreducible representation $\rho$ of its fundamental group into $PSL_2(\mathbb{C})$, a seminal theorem of Donaldson asserts the existence of a $\rho$-equivariant harmonic map from the universal cover $\tilde{X}$ into $\mathbb{H}^3$. In this thesis, we consider domain surfaces that are non-compact, namely marked and bordered surfaces (introduced in the work of Fock-Goncharov). Such a marked and bordered surface is denoted by a pair $(S, M)$ where $M$ is a set of marked points that are either punctures or marked points on boundary components. Our main result in this part is: given an element $X$ in the enhanced Teichmuller space $\mathcal{T}^{\pm}(S, M)$, and a non-degenerate type-preserving framed representation $(\rho,\beta):(\pi_1(X), F_{\infty})\rightarrow (PSL_2(\mathbb{C}),\mathbb{CP}^1)$, where $F_\infty$ is the set of lifts of the marked points in the ideal boundary, there exists a $\rho$-equivariant harmonic map from $\mathbb{H}^2$ to $\mathbb{H}^3$ asymptotic to $\beta$.

In both cases, we utilize the harmonic map heat flow applied to a suitably constructed initial map. The main analytical work is to show that the distance between the initial map and the final harmonic map is uniformly bounded, proving the desired asymptoticity.

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 11 Apr 2024