等参超曲面导出的 Bernoulli 自由边界问题的奇点模型
Singularity models for the Bernoulli free boundary problem from isoparametric hypersurfaces
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中文总结 AI 辅助
本文从等参叶状结构构造 Bernoulli 自由边界问题的齐次解，产生无穷多新奇点模型，建立几何刚性定理，并猜想最低密度锥体的形式。
中文摘要 AI 辅助
我们发展了一种从球面的等参叶状结构构造 Bernoulli 自由边界问题的齐次解以及球面上一般极值域的通用方法。我们的构造产生了无穷多个新例子，这些例子具有与球面极小曲面通过光滑插值毛细曲面族相联系的复杂拓扑，其中包含低维情形下的新型奇点模型，并恢复了大多数已知的单相齐次解。我们还为每个这样的齐次映射引入了一种几何表示，并建立了一个几何刚性定理：对于每个给定的等参叶状结构，特别是 $O(n)$ 的余齐一子群，其径向几何唯一地确定了相应的齐次解。最后，我们研究了所构造解的密度，并猜想在给定维数 $n \geq 5$ 中，非平坦锥体的最低密度由 $O(k) \times O(n-k)$ 不变锥体达到。
英文摘要
We develop a general construction of homogeneous solutions to the Bernoulli free boundary problem, as well as general extremal domains on the sphere, from isoparametric foliations of the sphere. Our construction produces rich families of infinitely many new examples with sophisticated topologies connected to minimal surfaces of the sphere by smooth families of interpolating capillary surfaces, which include novel singularity models in low dimensions and recover most known homogeneous one-phase solutions. We also introduce a geometric representation for every such homogeneous map and establish a geometric rigidity theorem: for every prescribed isoparametric foliation, and in particular for cohomogeneity-one subgroups of $O(n)$, its radial geometry uniquely determines the corresponding homogeneous solution. Finally, we study the density of the constructed solutions and conjecture that the lowest density among non-flat cones in a given dimension $n \geq 5$ is attained by an $O(k) \times O(n-k)$-invariant cone.
发表机构
- MIT（麻省理工学院）
- Columbia University（哥伦比亚大学）
- Johns Hopkins University（约翰斯·霍普金斯大学）
机构由 AI 辅助整理，请以论文原文为准。
