Geometric stability for complex Monge-Ampere equations

@misc{guo2026geometric,
  title = {{Geometric stability for complex Monge-Ampere equations}},
  author = {Bin Guo and Jian Song and Jacob Sturm},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28978},
  eprint = {2609.28978},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28978},
}

TY  - UNPB
TI  - Geometric stability for complex Monge-Ampere equations
AU  - Bin Guo
AU  - Jian Song
AU  - Jacob Sturm
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.28978
UR  - https://arxiv.org/abs/2609.28978
AB  - Let X be a compact Kahler manifold. The analytic stability theorem of Kolodziej for complex Monge-Ampere equation states that for any Kahler metrics ω and ω' in the same cohomology class, if their volume measures are bounded in L^p(X) (for some p>1) and close in L^1(X), then their Kahler potentials are close in L^∞(X). In this paper, we establish the geometric stability for complex Monge-Ampère equations that L^1-closeness of volume measures implies L^∞-closeness for the induced distance functions by ω and ω'. Consequently, we prove that any non-smooth Kahler current with volume measure bounded in L^p (for some p>1) and Ricci current bounded below induces a unique metric space, which turns out to be a compact RCD space homeomorphic to X itself.
ER  -