Use this URL to cite or link to this record in EThOS: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.543052
Title: Generalized Lagrangian mean curvature flow in almost Calabi-Yau manifolds
Author: Behrndt, Tapio
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2011
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Abstract:
In this work we study two problems about parabolic partial differential equations on Riemannian manifolds with conical singularities. The first problem we are concerned with is the existence and regularity of solutions to the Cauchy problem for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. By introducing so called weighted Hölder and Sobolev spaces with discrete asymptotics, we provide a complete existence and regularity theory for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. The second problem we study is the short time existence problem for the generalized Lagrangian mean curvature flow in almost Calabi-Yau manifolds, when the initial Lagrangian submanifold has isolated conical singularities that are modelled on stable special Lagrangian cones. First we use Lagrangian neighbourhood theorems for Lagrangian submanifolds with conical singularities to integrate the generalized Lagrangian mean curvature flow to a nonlinear parabolic equation of functions, and then, using the existence and regularity theory for the heat equation, we prove short time existence of the generalized Lagrangian mean curvature flow with isolated conical singularities by letting the conical singularities move around in the ambient space and the model cones to rotate by unitary transformations.
Supervisor: Joyce, Dominic Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.543052  DOI: Not available
Keywords: Differential geometry ; Partial differential equations ; Mathematics ; conical singularities ; mean curvature flow
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