Parabolic Equations and the Geometry of Complete Kaehler Manifolds

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AbstractAward: DMS-0504792Principal Investigator: Lei NiThe principle investigator proposes to study the interplaybetween the geometry and the analysis on complete Kaehlermanifolds. The focus will be the linear and nonlinear parabolicequations on complete manifolds with various curvatureassumptions. These equations includes the linear heat equationand Laplacian operator, harmonic mapping heat equation, meancurvature flow, Hermitian-Einstein flow, Ricci/Kaehler-Ricciflow, etc. These equations have various physical and geometricorigins. But they share many common features such as geometricsymmetries and identities, monotonicity formulae, entropy likeconsiderations, differential Harnack (also calledLi-Yau-Hamilton) inequalities. They also connect to each otherin various ways. By studying them together, more light is shedon all of them.Differential Geometry is the study of the relationship betweenthe geometry of a space, a manifold in the mathematical notion,and the analytic properties of the functions and the differentialequations, on the underlying space. Geometric analysis is thestudy of the overall geometric and topological properties of aspace by piecing together the local information. Since the spacesare usually curved ones, the ``curvature" was introduced tomeasure the deviation from the Euclidean space and the techniquesare often `nonlinear' even dealing with a linear problem. Thestudy of this area of mathematics has close connection with thegeneral relativity and string theory in physics. The applicationscan be found in the study of thestructure of complicatedmolecules, liquid-gas boundaries, and even the large scalenetworks. This project on studying the Kaehler manifolds vialinear and nonlinear parabolic equations will enhance theunderstanding of geometric analysis, linear and nonlinear partialdifferential equations, algebraic geometry and mathematicalphysics.