Lectures on Pluripotential Theory on Compact Hermitian Manifolds

Dinew, Sławomir

doi:10.1007/978-3-030-25883-2_1

Cited by 4 publications

(8 citation statements)

References 40 publications

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“…Then a De Giorgi type iteration argument of Kolodziej [24] (see also [11] and [8]) implies that φ u (s) = 0 for s ≥ S ∞ for some uniform constant…”

Section: Proof Of Theoremmentioning

confidence: 99%

Uniform entropy and energy bounds for fully non-linear equations

Guo¹,

Phong²

2022

Preprint

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Energy bounds which are uniform in the background metric are obtained from upper bounds for entropy-like quantities. The argument is based on auxiliary Monge-Ampère equations involving sublevel sets, and bypasses the Alexandrov-Bakelman-Pucci maximum principle. In particular, it implies uniform L ∞ bounds for systems coupling a fully non-linear equation to its linearization, generalizing the cscK equation.

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“…Then a De Giorgi type iteration argument of Kolodziej [24] (see also [11] and [8]) implies that φ u (s) = 0 for s ≥ S ∞ for some uniform constant…”

Section: Proof Of Theoremmentioning

confidence: 99%

Uniform entropy and energy bounds for fully non-linear equations

Guo¹,

Phong²

2022

Preprint

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“…In the context of parabolic equations the pluripotential estimates are also useful. For example, To [41] (independently, Nie [31] in particular cases) used results in [11,32] to prove a conjecture by Tosatti and Weinkove [38]. The geometric applications of pluripotential theory on Hermitian manifolds are discussed at length in surveys by Dinew [9,10].…”

Section: Theorem 12mentioning

confidence: 99%

“…Let (X , ω) be a compact Hermitian manifold of dimension n. The study of the complex Monge-Ampère equation in this setting was initiated by Cherrier [5], and the counterpart of the Calabi-Yau theorem [44] on compact Hermitian manifolds was proven by Tosatti and Weinkove [36]. Later Dinew and the authors, in a series of papers [11,24,26,27], obtained weak continuous solutions for more general densities on the right hand side of the equation, by extending the pluripotential methods employed before on Kähler manifolds. In this paper we deal with yet more general measures on the right hand side.…”

Section: Introductionmentioning

confidence: 99%

Continuous solutions to Monge–Ampère equations on Hermitian manifolds for measures dominated by capacity

Kołodziej

Nguyen

2021

Calc. Var.

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We prove the existence of a continuous quasi-plurisubharmonic solution to the Monge–Ampère equation on a compact Hermitian manifold for a very general measure on the right hand side. We admit measures dominated by capacity in a certain manner, in particular, moderate measures studied by Dinh–Nguyen–Sibony. As a consequence, we give a characterization of measures admitting Hölder continuous quasi-plurisubharmonic potential, inspired by the work of Dinh–Nguyen.

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“…In this case, for the Monge-Ampère equation, C 0 estimates in the non-Kähler case were first obtained by Tosatti-Weinkove [35], building on earlier works of Cherrier [7], and a pointwise upper bound for e nF was needed. The stronger version requiring only weaker entropy bounds as described in Theorem 1 was first obtained by Dinew and Kolodziej [11,12,13,27], using a non-trivial extension of pluripotential theory to the Hermitian setting. More recently a new approach using the theory of envelopes has been developed by Guedj and Lu [19].…”

Section: Introductionmentioning

confidence: 99%

“…unless extra conditions [11] are put on ω. So it is interesting to estimate the lower/upper bound of the volume of (X, ω ϕ ), i.e.…”

Section: Introductionmentioning

confidence: 99%

On $L^\infty$ estimates for fully nonlinear partial differential equations

Guo¹,

Phong²

2022

Preprint

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Sharp L ∞ estimates are obtained for general classes of fully non-linear PDE's on Hermitian manifolds. The method builds on the method of comparison with auxiliary Monge-Ampère equations introduced earlier in the Kähler case by the authors in joint work with F. Tong, but uses this time auxiliary equations on open balls with a Dirichlet condition. The role of Yau's theorem in the compact Kähler case is now played by the theorem of Caffarelli-Kohn-Nirenberg-Spruck for the Dirichlet problem. Global estimates are then deduced by combining arguments of Blocki with exponential estimates due to Kolodziej generalizing the classic inequality of Brezis-Merle. The method applies not just to compact Hermitian manifolds, but also to the Dirichlet problem, to open manifolds with a lower bound on their injectivity radii, and to (n − 1)-form Monge-Ampère equations.

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Lectures on Pluripotential Theory on Compact Hermitian Manifolds

Cited by 4 publications

References 40 publications

Uniform entropy and energy bounds for fully non-linear equations

Uniform entropy and energy bounds for fully non-linear equations

Continuous solutions to Monge–Ampère equations on Hermitian manifolds for measures dominated by capacity

On $L^\infty$ estimates for fully nonlinear partial differential equations

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