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PreprintarXiv licencearXiv

Deforming area-preserving maps between surfaces by mean curvature flow coupled with Ricci flow

  • Ping-Hung Lee
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Abstract

We study a natural way to deform area-preserving maps between compact Riemann surfaces. Specifically, we evolve the metrics on the two Riemann surfaces by the normalized Ricci flow and the graph of the area-preserving map by the mean curvature flow. We prove that the flow exists for all time, remains the graph of an area-preserving map, and converges smoothly and exponentially to a minimal Lagrangian graph with respect to the product of the limiting metrics. This generalizes earlier results of Wang and Smoczyk, in which the Riemann surfaces have constant scalar curvature.

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Cite

BibTeXRIS
BibTeX
@misc{lee2026deforming,
  title = {{Deforming area-preserving maps between surfaces by mean curvature flow coupled with Ricci flow}},
  author = {Ping-Hung Lee},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.19724},
  eprint = {2609.19724},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.19724},
}
RIS
TY  - UNPB
TI  - Deforming area-preserving maps between surfaces by mean curvature flow coupled with Ricci flow
AU  - Ping-Hung Lee
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.19724
UR  - https://arxiv.org/abs/2609.19724
AB  - We study a natural way to deform area-preserving maps between compact Riemann surfaces. Specifically, we evolve the metrics on the two Riemann surfaces by the normalized Ricci flow and the graph of the area-preserving map by the mean curvature flow. We prove that the flow exists for all time, remains the graph of an area-preserving map, and converges smoothly and exponentially to a minimal Lagrangian graph with respect to the product of the limiting metrics. This generalizes earlier results of Wang and Smoczyk, in which the Riemann surfaces have constant scalar curvature.
ER  -