通过平均曲率流与Ricci流耦合变形紧致黎曼曲面间的保面积映射
Deforming area-preserving maps between surfaces by mean curvature flow coupled with Ricci flow
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中文总结 AI 辅助
本文提出用归一化Ricci流演化度量、平均曲率流演化图的耦合方法变形紧致黎曼曲面间的保面积映射，证明流长期存在、保持图结构并指数收敛至极小球拉格朗日图，推广了常数量曲率情形。
中文摘要 AI 辅助
我们研究了一种自然的方式来变形紧致黎曼曲面之间的保面积映射。具体而言，我们在两个黎曼曲面上通过归一化Ricci流演化度量，并通过平均曲率流演化保面积映射的图。我们证明了该流对所有时间存在，保持为保面积映射的图，并关于极限度量的乘积光滑且指数地收敛到一个极小球拉格朗日图。这推广了Wang和Smoczyk的早期结果，其中黎曼曲面具有常数量曲率。
英文摘要
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.
发表机构
- Columbia University（哥伦比亚大学）
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