Weak Solutions to the complex Monge-Ampère flows on compact Kähler manifolds : general measures on the right-hand side
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866918384351838208 |
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| author | Kang, Bowoo |
| author_facet | Kang, Bowoo |
| contents | We show the existence of a bounded solution to the Cauchy problem for the complex Monge-Ampère flow on a compact Kähler manifold, with the right-hand side of the form $dt \wedge dμ$ where $dμ$ is dominated by a Monge-Ampère measure of a Hölder continuous quasi-plurisubharmonic function. We also prove that for a given semi-positive big from $θ$, the $t$-slice of the solution is locally Hölder continuous on $\rm{Amp(θ)}$ for all $t \in (0, T)$. Next, we prove a comparison principle when $dμ$ is dominated by a Monge-Ampère measure of a bounded quasi-plurisubharmonic function, which implies the uniqueness of the solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_11882 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Weak Solutions to the complex Monge-Ampère flows on compact Kähler manifolds : general measures on the right-hand side Kang, Bowoo Complex Variables Analysis of PDEs Differential Geometry We show the existence of a bounded solution to the Cauchy problem for the complex Monge-Ampère flow on a compact Kähler manifold, with the right-hand side of the form $dt \wedge dμ$ where $dμ$ is dominated by a Monge-Ampère measure of a Hölder continuous quasi-plurisubharmonic function. We also prove that for a given semi-positive big from $θ$, the $t$-slice of the solution is locally Hölder continuous on $\rm{Amp(θ)}$ for all $t \in (0, T)$. Next, we prove a comparison principle when $dμ$ is dominated by a Monge-Ampère measure of a bounded quasi-plurisubharmonic function, which implies the uniqueness of the solution. |
| title | Weak Solutions to the complex Monge-Ampère flows on compact Kähler manifolds : general measures on the right-hand side |
| topic | Complex Variables Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2603.11882 |