Harmonic Map Heat Flow and Framed Surface-group Representations

PhD;2023;Mathematics

This thesis concerns the construction of harmonic maps from certain non-compact surfaces into hyperbolic 3-space H3 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 twisted ideal polygon, there exists a harmonic map heat flow ut such that the image of ut is asymptotic to that polygon for all t ∈ [0, ∞). Moreover, we prove that given any twisted ideal polygon in H3 with rotational symmetry, there exists a harmonic map from C to H3 asymptotic to that polygon. This generalizes the work of Han, Tam, Treibergs, and Wan which concerned harmonic maps from C to the hyperbolic plane H2. In the second part, we consider the case of equivariant harmonic maps. For a closed Riemann surface X, and an irreducible representation ρ of its fundamental group into PSL2(C), a seminal theorem of Donaldson asserts the existence of a ρ-equivariant har- monic map from the universal cover ˜X into H3. 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 T ±(S, M ), and a non-degenerate type-preserving framed representation (ρ, β) : (π1(X), F∞) → (PSL2(C), CP1), where F∞ is the set of lifts of the marked points in the ideal boundary, there exists a ρ-equivariant harmonic map from H2 to H3 asymptotic to β. 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.