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dc.contributor.advisorYau, Shing-Tung
dc.contributor.authorLi, Yi
dc.date.accessioned2013-02-14T15:09:54Z
dc.date.issued2013-02-14
dc.date.submitted2012
dc.identifier.citationLi, Yi. 2012. Analysis of Some PDEs over Manifolds. Doctoral dissertation, Harvard University.en_US
dc.identifier.otherhttp://dissertations.umi.com/gsas.harvard:10092en
dc.identifier.urihttp://nrs.harvard.edu/urn-3:HUL.InstRepos:10288948
dc.description.abstractIn this dissertation I discuss and investigate the analytic aspect of several elliptic and parabolic partial differential equations arising from Rimannian and complex geometry, including the generalized Ricci flow, Gaussian curvature flow of negative power, mean curvature flow of positive power, harmonic-Ricci flow, vector field flow, and also Mabuchi-Yau functionals and Donaldson equation over complex manifolds. The new ingredient of this thesis is the first introduction to the vector field flow, a joint work with Professor Kefeng Liu at UCLA, which may lead to a new way to prove Hopf conjectures, which state that any positively curved manifold of even dimension has positive Euler characteristic and \(\mathbb{S}^2 x \mathbb{S}\) does not admit any Riemannian metric of positive sectional curvature, and give affirmative answer to Yau's question on the existence of an effective \(\mathbb{S}^1\)-action on positively curved manifolds. Furthermore, our flow is naturally related to Navier-Stokes equations. The vector field flow is defined as \(\partial_t(X_t)^i=\Delta_{LB}(X_t)^i+\nabla^idiv(X_t)+R^i_j(X_t)^j, X_0=X,\) where X is a fixed vector field on \(\mathcal{M}, \partial_t \doteqdot \frac {\partial} {\partial t}\) is the time derivative, and \(\Delta_{LB}\)denotes the Laplace-Beltrami operator on \(\mathcal{M}\). If we define \(Ric^\#\), the (1, 1)-tensor field associated to Ric, by \(g(Ric^\#(X),Y) \doteqdot Ric(X,Y)\), where X,Y are two vector fields, then \(Ric^\#\) is an operator on the space of vector fields, denoted by \(C^\infty (\mathcal{M},T\mathcal{M})\), and the above flow can be rewritten as \(\partial_tX_t=\Delta_{LB}X_t+\nabla div(X_t)+Ric^\#(X_t)\). We establish some basic properties, including long time existence and convergence, and formulate a conjecture on the vector field flow on positively curved manifold. This conjecture immediately implies Hopf conjectures and Yau's question.en_US
dc.description.sponsorshipMathematicsen_US
dc.language.isoen_USen_US
dash.licenseMETA_ONLY
dc.subjectmathematicsen_US
dc.titleAnalysis of Some PDEs over Manifoldsen_US
dc.typeThesis or Dissertationen_US
dash.embargo.until10000-01-01
thesis.degree.date2012en_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.grantorHarvard Universityen_US
thesis.degree.leveldoctoralen_US
thesis.degree.namePh.D.en_US
dc.contributor.committeeMemberYau, Shing-Tungen_US
dc.contributor.committeeMemberSternberg, Shlomoen_US
dc.contributor.committeeMemberTaubes, Cliffen_US


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