各向异性极小曲面的存在性与正则性讲义
Lectures on Existence and Regularity of Anisotropic Minimal Surfaces
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中文总结 AI 辅助
本迷你课程概述各向异性极小曲面的存在性与正则性理论进展，研究各向异性普拉托问题解、临界点正则性，并将结果应用于闭黎曼流形中最优正则各向异性极小超曲面的构造。
中文摘要 AI 辅助
针对多种自然现象，表面积泛函是一种初步近似。为捕捉微结构，应用科学中的众多模型采用了与方向相关的泛函，即各向异性能量。由于各向异性能量在刚体运动下不具有不变性，其临界点不具备与各向同性极小曲面相同的守恒律。例如，密度比的单调性公式尚不明确是否适用于一般各向异性能量的极小化子。因此，各向异性极小曲面的研究比各向同性对应物更具挑战性。在本迷你课程中，我们概述各向异性极小曲面的存在性与正则性理论的最新研究进展。特别地，我们首先聚焦于各向异性普拉托问题的解，随后转向各向异性能量临界点的可求积性与正则性理论研究。最后，我们将相关结果应用于极小极大理论，以在闭黎曼流形中构造闭的最优正则各向异性极小超曲面。
英文摘要
For several natural phenomena, the use of the surface area functional is a first approximation. In order to capture microstructures, numerous models in applied sciences employ directionally dependent functionals, known as anisotropic energies. Since anisotropic energies are not invariant under rigid motions, their critical points do not enjoy the same conservation laws as isotropic minimal surfaces. For instance, the monotonicity formula for the density ratio is not known to hold for minimizers of general anisotropic energies. Consequently, the study of anisotropic minimal surfaces is more challenging than the study of their isotropic counterparts. In this mini-course, we give an overview of the state of the art in the existence and regularity theory of anisotropic minimal surfaces. In particular, we first focus on solutions of the anisotropic Plateau problem and subsequently move to the investigation of the rectifiability and regularity theory for critical points of anisotropic energies. To conclude, we provide applications to the min-max theory for the construction of closed optimally regular anisotropic minimal hypersurfaces in closed Riemannian manifolds.
