拓扑和几何限制、自由边界问题和自引力流体
Topological and geometrical restrictions, free-boundary problems and self-gravitating fluids
摘要：
设（P1）是黎曼流形（M，g）上的一类椭圆自由边界问题。本文研究了（P1）的解f的纤维（水平集）在拓扑和几何上的限制。我们给出了一种技术的基础上的纤维（解析表示属性）的某些显着的属性，从初始PDE到一个全球性的分析表征的纤维（平衡分区条件）。我们研究这个分析的特点，并获得几个拓扑和几何性质的纤维的解决方案必须拥有，根据拓扑结构的M和度量张量g。我们将这些结果应用到经典的物理问题中的分类的平衡形状的牛顿和相对论静态自引力流体。我们还建议的关系，等距的黎曼流形。
英文摘要：
Let (P1) be certain elliptic free-boundary problem on a Riemannian manifold (M,g). In this paper we study the restrictions on the topology and geometry of the fibres (the level sets) of the solutions f to (P1). We give a technique based on certain remarkable property of the fibres (the analytic representation property) for going from the initial PDE to a global analytical characterization of the fibres (the equilibrium partition condition). We study this analytical characterization and obtain several topological and geometrical properties that the fibres of the solutions must possess, depending on the topology of M and the metric tensor g. We apply these results to the classical problem in physics of classifying the equilibrium shapes of both Newtonian and relativistic static self-gravitating fluids. We also suggest a relationship with the isometries of a Riemannian manifold.
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