The Pluripotential Cauchy-Dirichlet problem for the Complex Monge-Ampère flow with a general measure on the right-hand side
Bowoo Kang
TL;DR
This work addresses the solvability of the pluripotential Cauchy-Dirichlet problem for the complex Monge-Ampère flow with a general right-hand side $dt \wedge d\mu$, allowing $d\mu$ to be dominated by a bounded PSH Monge-Ampère measure and dropping the strict positivity requirement. The authors develop convergence tools for time derivatives and measure data, enabling limit passages in the parabolic equation when approximating measures are used. They prove a bounded subsolution theorem in the parabolic pluripotential setting, obtaining existence of bounded solutions under a subsolution barrier $\varphi$ with $(dd^c\varphi)^n \ge d\mu$ and zero boundary trace, and extend solvability from compactly supported to general measures by approximation. They also analyze boundary conditions, showing the pointwise Cauchy limit can fail in general, but holds under an admissible pair $(\mu,h_0)$, with practical criteria ensuring admissibility. Overall, the results broaden the applicability of parabolic pluripotential theory to measures beyond strictly positive densities, with implications for parabolic Kähler-Ricci-type flows and complex Monge-Ampère evolutions.
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 $dt \wedge dμ$ where $dμ$ is dominated by a Monge-Ampère measure of a bounded plurisubharmonic function. In particular, we remove the strict positivity assumption on $dμ$. We use this result to prove the parabolic version of the bounded subsolution theorem due to Kolodziej in pluripotential theory.