Estimates for the Complex Monge-Ampère Equation on Hermitian and Balanced Manifolds

By Valentino Tosatti, B. Weinkove. 2009

113 citations recorded by the source corpus.

We generalize Yau's estimates for the complex Monge-Ampere equation on compact manifolds in the case when the background metric is no longer Kahler. We prove $C^{\infty}$ a priori estimates for a solution of the complex Monge-Ampere equation when the background metric is Hermitian (in complex dimension two) or balanced (in higher dimensions), giving an alternative proof of a theorem of Cherrier. We relate this to recent results of Guan-Li.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0fed59ae9228ac92a1ec812279b5e7ece7baa8da

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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): “2, we can follow closely the arguments of [45].”
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]).”
Convergence of the parabolic complex Monge-Amp\`ere equation on compact Hermitian manifolds (2010, methodology): “5) follows along the method of Tosatti and Weinkove in [25], but there are extra terms to control that arrive in the parabolic case.”
Hessian equations on closed Hermitian manifolds (2015, background): “For further background, we re fe the reader to [10], [11], [5], [17] and the references therein.”
$$C^{2,\alpha }$$C2,α estimates for nonlinear elliptic equations in complex and almost complex geometry (2014, methodology): “In [55], a direct Evans–Krylov argument was given, following a similar approach to that given in the notes of Siu [50].”
Complex Monge-Ampere equations and totally real submanifolds (2009, background): “Therefore one can consider extensions of Calabi–Yau theorem for Sij̄ ; see [67].”
Gauduchon metrics with prescribed volume form (2015, background): “(2) When n = 2 the conjecture was proved by Cherrier [4] in 1987 by solving a complex Monge-Ampère equation (see also [35, 16] for different proofs).”
A numerical criterion for generalised Monge-Ampère equations on projective manifolds (2020, background): “equations were ﬁrst studied in [13]. There is a big body of literature studying various generalisations of these equations to the Hermitian setting (ﬁrst initiated in the complex MongeAmp`ere case by [38, 39]). We recovermanyotherexisting resultswhen they arespecialisedto the projective case. If Gis”
Fully non-linear parabolic equations on compact Hermitian manifolds (2017, methodology): “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].”
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]).”
On the evolution of a Hermitian metric by its Chern-Ricci form (2011, methodology): “1 via a new second order estimate which we had previously established [53] using the maximum principle only in the cases n = 2 or (M,ω) balanced.”