The complex Monge-Ampère equation on compact Hermitian manifolds

By Valentino Tosatti, B. Weinkove. 2009

233 citations recorded by the source corpus.

We show that, up to scaling, the complex Monge-Ampere equation on compact Hermitian manifolds always admits a smooth solution.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ade86090db20aa46da47e640f222dcd239a22755

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

The Monge–Ampère equation for non-integrable almost complex structures (2016, methodology): “This was carried out in full generality in [46] where it was shown that (1.”
Fully non-linear parabolic equations on compact Hermitian manifolds (2017, result): “Thus we shall require the following version of the Evans-Krylov theorem for uniformly parabolic and concave equations, with the precise dependence of constants spelled out, and which can be proved using the arguments of Trudinger [60], and more particularly Tosatti-Weinkove [53] and Gill [17].”
Convergence of the parabolic complex Monge-Amp\`ere equation on compact Hermitian manifolds (2010, methodology): “1 completes the proof of the main theorem, and also provides a parabolic proof of the main theorem in [26].”
A numerical criterion for generalised Monge-Ampère equations on projective manifolds (2020, background): “There is a big body of literature studying various generalisations of these equations to the Hermitian setting (first initiated in the complex MongeAmpère case by [38, 39]).”
An extension of a theorem of Wu-Yau (2015, methodology): “1 still true in this more general situation? While the analogous complex Monge-Ampère equation in the Hermitian case is solved in [2] (see also [11, 12]), the argument with the Schwarz Lemma uses the Kähler condition for Royden’s lemma (which requires the Kähler symmetries of the curvature tensor), ”
On Chern-Yamabe problem (2015, background): “One motivation for considering exactly such “complex curvature scalar”, between other natural ones [12, 13], comes from the importance of the Chern Ricci curvature in non-Kähler Calabi-Yau problems (compare, for example, [21]).”
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]).”
$$C^{2,\alpha }$$C2,α estimates for nonlinear elliptic equations in complex and almost complex geometry (2014, background): “3) was established by Cherrier [13] for n = 2 (and in higher dimensions with additional hypotheses) and by the first and third-named authors in general [56].”