The Dirichlet problem for degenerate complex Monge-Ampere equations

By Duong H. Phong, Jacob Sturm. 2009

116 citations recorded by the source corpus.

The Dirichlet problem for a Monge-Ampere equation corresponding to a nonnegative, possible degenerate cohomology class on a Kaehler manifold with boundary is studied. C^{1,\alpha} estimates away from a divisor are obtained, by combining techniques of Blocki, Tsuji, Yau, and pluripotential theory. In particular, C^{1,\alpha} geodesic rays in the space of Kaehler potentials are constructed for each test configuration

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b8ea42e804a191176cf08a861491890dd0899e1e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Fully non-linear parabolic equations on compact Hermitian manifolds (2017, methodology): “It was pointed out in [50] that, under an extra concavity condition on f , C(2) estimates can be derived directly from C estimates in the elliptic case, using a test function introduced in [39].”
C1,1 regularity for degenerate complex Monge–Ampère equations and geodesic rays (2017, methodology): “To prove regularity of φ we begin with an approximation argument, following Phong– Sturm [34].”
Fu–Yau Hessian equations (2018, methodology): “This idea is also used to derive a C(1) estimate for the complex Monge-Ampère equation, [2, 14, 24, 25, 33].”
A numerical criterion for generalised Monge-Ampère equations on projective manifolds (2020, unknown): “For a function (akin to [30]) γ : [0,∞)→ R defined as γ(x) = Ax − 1 1 + x + 1,”
Convergence of the parabolic complex Monge-Amp\`ere equation on compact Hermitian manifolds (2010, background): “Related work can be found in [2], [5], [6], [9], [14], [15], [19], [20], and [21].”
The Chern–Ricci flow on complex surfaces (2012, methodology): “In the next lemma we make use of a trick of Phong-Sturm [40], which we employed in our previous paper [58].”
The Monge-Ampère equation for (n − 1)-plurisubharmonic functions on a compact Kähler manifold (2013, methodology): “By comparison, in the case of the usual complex Monge-Ampère equation on Hermitian manifolds the estimate trωω̃ 6 Ce A(u−infM u) was proved in [52] by adapting a trick of Phong-Sturm [41, 42].”
Yau-Tian-Donaldson correspondence for K-semistable Fano manifolds (2013, background): “For details, see [35].”