Partial differential equations and variational approaches to constant scalar curvature metrics in Kähler geometry
 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:
 2015
 Type:
 Dissertations
 Department(s):
 Mathematics
 Persistent URL:
 http://dx.doi.org/10.7916/D8HD7TMG
 Notes:
 Ph.D., Columbia University.
 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 MongeAmpère 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
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 Suggested Citation:
 Daniel Ilan Rubin, 2015, Partial differential equations and variational approaches to constant scalar curvature metrics in Kähler geometry, Columbia University Academic Commons, http://dx.doi.org/10.7916/D8HD7TMG.