The Monge-Ampère operator and geodesics in the space of Kähler potentials

By Duong H. Phong, Jacob Sturm. 2005

121 citations recorded by the source corpus.

It is shown that geodesics in the space of K\"ahler potentials can be uniformly approximated by geodesics in the spaces of Bergman metrics. Two important tools in the proof are the Tian-Yau-Zelditch approximation theorem for K\"ahler potentials and the pluripotential theory of Bedford-Taylor, suitably adapted to K\"ahler manifolds.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4f7b4d4077f12d4bfcf8d5ddbb4b11e585309b22

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

Lectures on Stability and Constant Scalar Curvature (2008, methodology): “74 12.3.4 Variations on the Ansatz and rates of convergence . . . . . . . . . . 74”
The Dirichlet problem for degenerate complex Monge-Ampere equations (2009, methodology): “Using the Monge–Ampère equation, geodesic segments in K have been constructed by Chen [8] using the method of a priori estimates, and by [24] using pluripotential theory.”
Bergman kernels and equilibrium measures for line bundles over projective manifolds (2007, result): “The result should also be compared to various C1,1-results for boundary value problems for complex Monge-Ampere equations on manifolds with boundary [13], [14], intimately related to the study of the geometry of the space of Kähler metrics on a Kähler manifold (see also [31], [9] for other relations”
Complex Monge-Ampere equations and totally real submanifolds (2009, background): “by Chen [16], Chen and Tian [17], Phong and Sturm [54], [55], [56], Blocki [10], and Berman and Demailly [7].”
C1,1 regularity for degenerate complex Monge–Ampère equations and geodesic rays (2017, background): “setting of algebraic test conﬁgurations. For background material and further references on geodesic rays, their regularity and relation to test conﬁguration and K-stability and we refer the reader to [3, 4, 5, 11, 12, 15, 20, 21, 23, 40, 32, 33, 34, 35, 36, 37, 38, 39, 41]. Next we prove a C1,1 regu”
Test configurations for K-stability and geodesic rays (2006, unknown): “Finally, we obtain, using (3.5), the following useful volume formula [27]: (n+ 1) ∫ X×Ua,b Ωn+1Φ = Ėω(b) − Ėω(a) (4.19) where Eω(t) = Eω(φt).”