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Geometric Flows of Diffeomorphisms

Carapetis, Anthony

Description

The idea of this thesis is to apply the methodology of geometric heat flows to the study of spaces of diffeomorphisms. We start by describing the general form that a geometrically natural flow must take and the implications this has for the evolution equations of associated geometric quantities. We discuss the difficulties involved in finding appropriate flows for the general case, and quickly restrict ourselves to the case of surfaces. In particular the main...[Show more]

dc.contributor.authorCarapetis, Anthony
dc.date.accessioned2018-04-09T05:09:51Z
dc.date.available2018-04-09T05:09:51Z
dc.identifier.urihttp://hdl.handle.net/1885/142453
dc.description.abstractThe idea of this thesis is to apply the methodology of geometric heat flows to the study of spaces of diffeomorphisms. We start by describing the general form that a geometrically natural flow must take and the implications this has for the evolution equations of associated geometric quantities. We discuss the difficulties involved in finding appropriate flows for the general case, and quickly restrict ourselves to the case of surfaces. In particular the main result is a global existence, regularity and convergence result for a geometrically defined quasilinear flow of maps u between flat surfaces, producing a strong deformation retract of the space of diffeomorphisms onto a finite-dimensional submanifold. Partial extensions of this result are then presented in several directions. For general Riemannian surfaces we obtain a full local regularity estimate under the hypothesis of bounds above and below on the singular values of the first derivative. We achieve these gradient bounds in the flat case using a tensor maximum principle, but in general the terms contributed by curvature are not easy to control. We also study an initial-boundary-value problem for which we can attain the necessary gradient bounds using barriers, but the delicate nature of the higher regularity estimate is not well-adapted for obtaining uniform estimates up to the boundary. To conclude, we show how appropriate use of the maximum principle can provide a proof of well-posedness in the smooth category under the assumption of estimates for all derivatives.
dc.language.isoen
dc.subjectdifferential geometry
dc.subjectgeometric flow
dc.subjectgeometric analysis
dc.subjectdiffeomorphism
dc.subjectparabolic equations
dc.subjectheat flow
dc.titleGeometric Flows of Diffeomorphisms
dc.typeThesis (PhD)
local.contributor.supervisorAndrews, Ben
local.contributor.supervisorcontactBen.Andrews@anu.edu.au
dcterms.valid2018
local.description.notesthe author deposited 9/04/2018
local.type.degreeDoctor of Philosophy (PhD)
dc.date.issued2017
local.contributor.affiliationMathematical Sciences Institute, The Australian National University
local.identifier.proquestYes
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