Complex Monge-Ampere equations and totally real submanifolds

By Bo Guan, Qun Li. 2009

143 citations recorded by the source corpus.

We study the Dirichlet problem for complex Monge-Ampere equations in Hermitian manifolds with general (non-pseudoconvex) boundary. Our main result extends the classical theorem of Caffarelli, Kohn, Nirenberg and Spruck in the flat case. We also consider the equation on compact manifolds without boundary, attempting to generalize Yau's theorems in the Kaehler case. As applications of the main result we study some connections between the homogeneous complex Monge-Ampere ({\em HCMA}) equation and totally real submanifolds, and a special Dirichlet problem for the HCMA equation related to Donaldson's conjecture on geodesics in the space of Kaehler metrics.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1b27ca15ab8864e0186b61c6af445ab4d576b1ca

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

On Bismut flat manifolds (2016, unknown): “We refer the readers to [37], [9], [10], [11], [12], [13], [24], [41], [19], [16], [17], [18], [33], [34], [35], [36], [25], [26], [27], [39], [40], [38] and the references therein for some recent progress in this area.”
The Chern–Ricci flow on complex surfaces (2012, methodology): “Gill’s result makes use of the C0 estimate of the authors [57] for the Hermitian complex Monge-Ampère equation (see also [13, 25, 14, 3]).”
Fully non-linear parabolic equations on compact Hermitian manifolds (2017, result): “Generalizations of Yau’s theorem had attracted a lot of attention, and many partial results had been obtained before, including those of Cherrier [3], Guan-Li [20], and others.”
The Monge-Ampère equation for (n − 1)-plurisubharmonic functions on a compact Kähler manifold (2013, methodology): “Rather than following the method of proof of the complex Monge-Ampère equation (ω + √ −1∂∂u) = eω, which was solved by Yau [58] when ω is Kähler and by the authors [51] for ω Hermitian (see also Cherrier [7] and Guan-Li [24]), we instead take an approach used for the complex Hessian equations.”
The Monge–Ampère equation for non-integrable almost complex structures (2016, background): “[5, 24] for instances where a real Hessian bound is obtained for the complex Monge-Ampère equation, with J integrable, after having first obtained a complex Hessian bound).”
Convergence of the parabolic complex Monge-Amp\`ere equation on compact Hermitian manifolds (2010, unknown): “To start the proof, we need to perform a change of coordinates made possible by the following lemma from [12].”
On the evolution of a Hermitian metric by its Chern-Ricci form (2011, methodology): “We carry out the computation here using tensorial quantities, following [10], rather than using a particular choice of complex coordinates as in [19, 21, 43, 53], for example.”
$$C^{2,\alpha }$$C2,α estimates for nonlinear elliptic equations in complex and almost complex geometry (2014, methodology): “Guan–Li [30] proved a C2,α estimate by first establishing a bound on the real Hessian of u using a maximum principle argument and then applying the usual Evans–Krylov theory.”
On a class of fully nonlinear elliptic equations on Hermitian manifolds (2013, methodology): “In this paper we prove the following existence result which is new even in the case when M is a bounded domain in Cn and χ = 0; we assume 2 ≤ α ≤ n − 2 as the cases α = 1 and α = n − 1 were considered in [17,18], while for the complex Monge-Ampère equation (α = n) it was proved in [16].”
Gauduchon metrics with prescribed volume form (2015, background): “Some other works on this topic include [6, 7, 13, 14, 15, 16, 21, 22, 23, 25, 26, 30, 31, 39, 40, 42, 43, 44].”
Non-Kähler Calabi-Yau manifolds (2014, methodology): “When ω is Kähler, (2) is Yau’s solution of the Calabi Conjecture [84], and in the non-Kähler case (2) follows from work of Cherrier [16] when n = 2 and of Weinkove and the author [71] in general (see also [41, 72]).”
Non-K\"ahler Calabi-Yau manifolds (2014, methodology): “When ω is Kähler, (2) is Yau’s solution of the Calabi Conjecture [84], and in the non-Kähler case (2) follows from work of Cherrier [16] when n = 2 and of Weinkove and the author [71] in general (see also [41, 72]).”