seminar: The Energy of a Kähler class
speaker: Santiago R. Simanca (University of New Mexico)abstract: On a compact complex manifold of Kähler type, the infimum of the squared L2-norm of the scalar curvarture of Kähler metrics representing a Kähler class Ω defines the energy of the class, E(Ω), and a Kähler metric that realizes this value is extremal. A strongly extremal metric is an extremal metric representing a critical point of E(Ω) when restricted to the set of Kähler classes of fixed positive top cup product. We study the existence of strongly extremal metrics and critical points ofE(Ω) on certain admissible manifolds, producing a number of non-trivial examples of manifolds that carry this type of metrics, and where in many of the cases, the class that they represent is one other than the first Chern class, and some examples of manifolds where these special metrics and classes do not exist. We also provide a detailed analysis of the gradient flow of E(Ω) on admissible ruled surfaces, show that this dynamical system can be extended to one beyond the Kähler cone, and analyze the convergence of solution paths at infinity in terms of conditions on the initial data, in particular proving that for any initial data in the Kähler cone, the corresponding path is defined for all t, and converges to a unique critical class of E(Ω) as time approaches infinity. Finally, we show how our class flow can be used to uniformize at least complex surfaces of Kähler type with positive first Chern class.
timetable:
Mon 10 Mar, 10:00 - 11:00, Sala Conferenze Centro De Giorgi
Thu 20 Mar, 15:30 - 16:30, Sala Conferenze Centro De Giorgi
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