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dc.contributor.advisorGupta, Subhojoy
dc.contributor.authorSau, Gobinda
dc.date.accessioned2024-01-12T07:17:07Z
dc.date.available2024-01-12T07:17:07Z
dc.date.submitted2023
dc.identifier.urihttps://etd.iisc.ac.in/handle/2005/6371
dc.description.abstractThis 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.en_US
dc.language.isoen_USen_US
dc.relation.ispartofseries;ET00383
dc.rightsI grant Indian Institute of Science the right to archive and to make available my thesis or dissertation in whole or in part in all forms of media, now hereafter known. I retain all proprietary rights, such as patent rights. I also retain the right to use in future works (such as articles or books) all or part of this thesis or dissertationen_US
dc.subjectharmonic mapsen_US
dc.subjectHeat flowen_US
dc.subjectframed representationsen_US
dc.subject.classificationResearch Subject Categories::MATHEMATICS::Algebra, geometry and mathematical analysis::Algebra and geometryen_US
dc.titleHarmonic Map Heat Flow and Framed Surface-group Representationsen_US
dc.typeThesisen_US
dc.degree.namePhDen_US
dc.degree.levelDoctoralen_US
dc.degree.grantorIndian Institute of Scienceen_US
dc.degree.disciplineFaculty of Scienceen_US


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