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Partial differential equations and variational approaches to constant scalar curvature metrics in Kähler geometry

Daniel Ilan Rubin

Title:
Partial differential equations and variational approaches to constant scalar curvature metrics in Kähler geometry
Author(s):
Rubin, Daniel Ilan
Thesis Advisor(s):
Phong, Duong Hong
Date:
Type:
Theses
Degree:
Ph.D., Columbia University
Department(s):
Mathematics
Persistent URL:
Abstract:
In this thesis we investigate two approaches to the problem of existence of metrics of constant scalar curvature in a fixed Kähler class. In the first part, we examine the equation for constant scalar curvature under the assumption of toric symmetry, thus reducing the problem to a fourth order nonlinear degenerate elliptic equation for a convex function defined in a polytope in ℝ^n. We obtain partial results on this equation using an associated Monge-Ampère equation to determine the boundary behavior of the solution. In the second part, we consider the asymptotics of certain energy functionals and their relation to stability and the existence of minimizers. We derive explicit formulas for their asymptotic slopes, which allows one to determine whether or not (X,L) is stable, and in some cases rule out the existence of a canonical metric.
Subject(s):
Mathematics
Item views
341
Metadata:
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Suggested Citation:
Daniel Ilan Rubin, , Partial differential equations and variational approaches to constant scalar curvature metrics in Kähler geometry, Columbia University Academic Commons, .

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