论文标题
$ \ bar {\ partial} $ - neumann问题的紧凑性在有界面的固有几何形状上
Compactness of the $\bar{\partial}$-Neumann problem on domains with bounded intrinsic geometry
论文作者
论文摘要
通过考虑固有的几何条件，我们在复杂的欧几里得空间中引入了一类新的域。该类是在生物形态下不变的，包括强烈的pseudoconvex域，二维域中有限类型域，凸域，$ \ mathbb {c} $ - 凸域和同质域。对于这类域，我们表明$ \ bar {\ partial} $ - $（0，q）$的neumann运算符的紧凑性与不包含任何$ q $维分析品种的边界等效（假设边界仅是拓扑submanifold）。我们还证明，对于这类域而言，伯格曼度量标准等同于Kobayashi指标，并且Pluricomplex Green功能在伯格曼度量方面满足了某些局部估计。
By considering intrinsic geometric conditions, we introduce a new class of domains in complex Euclidean space. This class is invariant under biholomorphism and includes strongly pseudoconvex domains, finite type domains in dimension two, convex domains, $\mathbb{C}$-convex domains, and homogeneous domains. For this class of domains, we show that compactness of the $\bar{\partial}$-Neumann operator on $(0,q)$-forms is equivalent to the boundary not containing any $q$-dimensional analytic varieties (assuming only that the boundary is a topological submanifold). We also prove, for this class of domains, that the Bergman metric is equivalent to the Kobayashi metric and that the pluricomplex Green function satisfies certain local estimates in terms of the Bergman metric.
