260130 VO Nonlinear elliptic equations in geometry (2015S)
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Advanced Graduate Lecture Course March 11 - 27, and April 14 - May 8, 2015: Wednesday and Friday 14:00 - 15:30.Venue: ESI, Schrödinger Lecture Hall (1090, Boltzmanngasse 9, top floor on the right)
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Sprache: Englisch
Lehrende
Termine
Registrierung über Univis von Mo 09.02.15 08:00 Uhr bis Di 30.06.15 23:00 Uhr
Information
Ziele, Inhalte und Methode der Lehrveranstaltung
Art der Leistungskontrolle und erlaubte Hilfsmittel
Mindestanforderungen und Beurteilungsmaßstab
The goal of this course is to highlight the role of weak solutions of the complex Monge-Ampère equation in finding singular canonical metrics in Kahler geometry or, equivalently, the limits of the Kähler-Ricci flow. The methods include: PDE a priori estimates and the continuity method, pluripotential theoretic approach based on the notion of the positive current and properties of plurisubharmonic functions, and the parabolic maximum principle for the Ricci flow.
Prüfungsstoff
Literatur
Zuordnung im Vorlesungsverzeichnis
MF 5
Letzte Änderung: Mo 07.09.2020 15:41
Basic notions of theory of distributions.
Elliptic and parabolic differential operators. Maximum principles. Schauder estimates.
Introduction to Kähler geometry. Ricci curvature. Canonical bundle. Kähler-Einstain metrics.Part II. The complex Monge-Ampère equation:Calabi-Yau theorem. Continuity method and a priori estimates.
Introduction to pluripotential theory: positive currents, capacities.
Weak solutions to the complex Monge-Ampère equation. Stability of those solutions.Part III. The Kähler-Ricci flow on compact Kähler manifolds:The evolution of geometric quantities along the flow.
Long time existence.
Convergence of the flow for definite Chern classes.
Singular Kähler-Einstain metrics.
The current research activity in the area. Similar equations and their geometrical context.