Riemannian曲面和最小曲面的Calderón问题
偏微分方程分析
2024-06-26 v1 数学物理
微分几何
math.MP
摘要
本文证明了两个结果。第一个结果表明，Riemannian曲面上算子Δ_g+q的Dirichlet-Neumann映射可以确定其拓扑、微分和度结构。此前的工作假设曲面为平面域[36]或几何结构先验已知[29]。我们将将此结果应用于研究嵌入在3维黎曼流形中的最小曲面的几何逆问题。特别是，我们将表明，嵌入最小曲面的体积知识不仅能确定其拓扑和微分结构，还能确定其作为嵌入曲面的黎曼结构。此类几何逆问题部分灵感来自AdS/CFT对应关系提出的物理模型。消除平面域假设的关键要素是从Δ_g+q的Dirichlet-Neumann映射中确定全纯函数的边界迹。这需要一种新型论证，涉及Carleman估计和构造CGO，其相位函数不像[29]中的情况那样为Morse型。我们预计这些技术可能用于研究其他几何逆问题和PDE问题。
引用
@article{arxiv.2406.16944,
title = {The Calder\'on problem on Riemannian surfaces and of minimal surfaces},
author = {Cătălin I. Cârstea and Tony Liimatainen and Leo Tzou},
journal= {arXiv preprint arXiv:2406.16944},
year = {2024}
}
备注
The authors are grateful to Matti Lassas who was involved in an earlier iteration of this work, but graciously removed himself as an author from the current, much more general, version. Based on an advice from arXiv moderators we submit this new version of the paper with different authors to replace the earlier version arXiv:2310.14268. The earlier version will not ever be published anywhere