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OP-11436Status unassessedGeometry and topology

Gradient Estimate

PDE methods in complex geometry · AIMPL workshop record

MSC 32 · Several complex variables and analytic spaces

Editorial mapping of AIMPL list topic: PDE methods in complex geometry. http://aimpl.org/pdecomplexgeom/ Not an author-supplied MSC code.

Workshop record · status unassessedAs of 2026-09-10 source snapshot; not a current-status certificationSource & evidence ↓
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Problem statement

Consider the equation

F[λ[u]]=ψ, F[\lambda[u]]=\psi,

where ω0,ωX\omega_{0},\omega_{X} are nonnegative Hermitian (1,1)(1,1)-forms and λ[u]\lambda[u] are the eigenvalues of ω0+−1∂∂‾u\omega_{0}+\sqrt{-1}\partial\overline{\partial}u with respect to ωX\omega_{X}. Assume the CC-subsolution condition holds, and that FF is concave. Is it possible to prove effective estimates (not through a contradiction argument) of the form

sup⁡X∣∇u∣≤C(ω0,ωX,ψ,F)? \sup_{X}|\nabla u|\leq C(\omega_{0},\omega_{X},\psi,F)?

For example, can one prove this using maximum principle techniques?

Blocki did this for the complex Monge-Ampere equation using maximum principle techniques, but so far there is no known direct proof in the case of the σk\sigma_{k} equations. It is also known in the case of deformed Hermitian-Yang-Mills.

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References
  1. [1]PDE methods in complex geometry · Original question link
  2. [2]AIMPL source bibliography link
  3. [3]AIMPL text license · CC BY-SA 3.0 linkAIM Problem Lists (AIMPL), original contributors; CC BY-SA 3.0. Adaptation: structural TeX converted to Markdown, documented source macros expanded, and source bibliography links added. Mathematical wording retained. These adapted source texts remain under CC BY-SA 3.0.

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