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September 12, 2025Open Access

The Pluripotential Cauchy-Dirichlet Problem for the Complex Monge-Ampère Flow with A General Measure on the Right-Hand Side

Key Points

The solvability of the Cauchy-Dirichlet problem is established under a less restrictive measure condition.
Importantly, the strict positivity requirement for the measure is removed, broadening applicability.
The results enable a proof of the parabolic version of the bounded subsolution theorem in pluripotential theory.
This approach utilizes properties of both Monge-Ampère measures and bounded plurisubharmonic functions.

Abstract

Abstract We show that the pluripotential Cauchy-Dirichlet problem for the complex Monge-Ampère flow is solvable for the right-hand side of the form dt d d t ∧ d μ where d d μ is dominated by a Monge-Ampère measure of a bounded plurisubharmonic function. In particular, we remove the strict positivity assumption on d d μ. We use this result to prove the parabolic version of the bounded subsolution theorem due to Kołodziej in pluripotential theory.

The Pluripotential Cauchy-Dirichlet Problem for the Complex Monge-Ampère Flow with A General Measure on the Right-Hand Side

Key Points

Abstract

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