Infinite geodesic rays in the space of Kähler potentials

By C. Arezzo, IT GangTianParma-, Mit - Usa. 2002

54 citations recorded by the source corpus.

In this paper we prove the existence of solutions of a degenerate complex Monge-Ampere equation on a complex manifold. Applying our existence result to a special degeneration of complex structure, we show how to associate to a change of complex structure an infinite length geodetic ray in the space of potentials. We also prove an existence result for the initial value problem for geodesics. We end this paper with a discussion of a list of open problems indicating how to relate our reults to the existence problem for extremal metrics.

Subfields: Applied Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/4d9fa9f2bf84e7f4f5e59ea90921b5359201b6a1

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In Applied Mathematics

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

Lectures on Stability and Constant Scalar Curvature (2008, methodology): “In [4], Arezzo and Tian showed that, given a test configuration T for L → X with smooth central fiber X0, then one can use the Cauchy-Kowalevska theorem to find families of local analytic solutions to the geodesic equations near infinity.”
Space of Kähler metrics III – On the lower bound of the Calabi energy and geodesic distance (2006, methodology): “HCMA equation 2.5 on the rectangle domain �l = [−l, l] ×[0, 1 ] with boundary value”
The Dirichlet problem for degenerate complex Monge-Ampere equations (2009, methodology): “This problem has received considerable attention recently, and geodesic rays have been constructed in [1, 8–10, 25, 26] by several different methods.”
Test configurations for K-stability and geodesic rays (2006, unknown): “D G ]”
Smooth and singular Kahler-Einstein metrics (2014, result): “This generalized previous constructions of Arezzo–Tian, Phong– Sturm, Song–Zelditch, that constructed such rays out a degeneration in the sense of the previous paragraph [2, 201, 228].”