A Priori Estimates for the Homogeneous Monge-Ampère Equation on Kähler Manifolds
Abstract
In connection with the question of geodesics in the space of Kähler metrics on a compact Kähler manifold (Mn , ω), in [11] Donaldson studied smooth solutions v ∈ C∞(U×M, R) for the following Dirichlet problem: Here U ⊂ C is the unit disc and ω denotes the pullback of ω by the projection U × M → M. Donaldson showed that the space of boundary functions F for which MA(F) admits a smooth solution is open, but that there exist boundary functions with no smooth solution. This thesis further investigates the existence of smooth solutions to MA(F), proving a priori estimates on the leaves of the foliation that corresponds to smooth solutions. We demonstrate that sequences of leaves of Monge-Amp`ere foliations converge to holomorphic disks.
Degree
Ph.D.
Advisors
Lempert, Purdue University.
Subject Area
Mathematics
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