Paper · Oct 5, 2026

Complex Monge-Ampère equations on Hermitian Manifolds: From bounded to smooth solutions

Papa Badiane, Chinh H. Lu, Ahmed Zeriahi

math.CVarXiv:2610.06473
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Abstract

We study regularity of bounded weak solutions to complex Monge-Ampère equations on compact Hermitian manifolds with right-hand side possibly decreasing in the unknown. Our proof relies on the domination principle and a priori estimates, partially motivated by the Monge-Ampère eigenvalue problem. Our main result roughly says that any bounded solution is smooth in the regular locus of the data. When the right-hand side is strictly positive and smooth, we recover known results by Nie, Ko lodziej-Nguyen for the Hermitian case and Székelyhidi-Tosatti for the Kähler case, which rely on the regularizing property of the Kähler-Ricci flow.

Ricci flowWeak & singular flowsEigenvalues & spectrum

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  1. On a general class of free boundary Monge-Ampère equations

    Tristan C. Collins and Benjy Firester PDF

    Topics Eigenvalues & spectrum · Shared terms “Monge Ampère”, “Ampère equation”, “Ampère eigenvalue”

  2. Regularity of solutions to Monge–Ampère equations on compact Hermitian manifolds

    Quang-Tuan Dang PDF

    Topics Ricci flow · Shared terms “Hermitian manifolds”, “Ampère equation”, “Monge”

  3. Complex Monge-Ampère equation for positive -forms on compact Kähler manifolds

    Mathew George PDF

    Topics Ricci flow · Shared terms “complex Monge”, “Monge Ampère”, “Ampère equation”

  4. The estimate for parabolic complex Monge-Ampère equations

    Qizhi Zhao PDF

    Topics Ricci flow · Shared terms “complex Monge”, “Monge Ampère”, “Ampère equation”

  5. Singularities of the Chern-Ricci flow

    Quang-Tuan Dang PDF

    Topics Ricci flow · Shared terms “complex Monge”, “Monge Ampère”, “Ampère equation”

  6. Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces

    Ye-Won Luke Cho and Young-Jun Choi PDF

    Topics Ricci flow · Shared terms “complex Monge”, “Monge Ampère”, “Ampère equation”

  7. Kähler-Ricci flows coming out of metric spaces

    Alix Deruelle, Vincent Guedj, Henri Guenancia et al. PDF

    Topics Weak & singular flows · Same author Ahmed Zeriahi · Shared terms “Kähler Ricci”

  8. Eigenvalue lower bounds and splitting for modified Ricci flow

    Tobias Holck Colding and William P. Minicozzi PDF

    Topics Eigenvalues & spectrum

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) Eigenvalues & spectrum Perelman's λ-functional is the lowest eigenvalue of −4Δ+ R, and it never decreases along Ricci flow. Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.