管状域上正曲率凯勒度量及其在最优传输中的应用
Positively curved Kähler metrics on tube domains and their applications to optimal transport
AI总结：
本文研究管状域上具有 $O(n) \times \mathbb{R}^n$ 对称性的凯勒流形，证明非负双截曲率条件下的唯一性结果，刻画非负正交反双截曲率的充要条件，并探讨其在最优传输中合成曲率界的应用。
摘要：
本文研究定义在 $\mathbb{C}^n$ 中管状域上的一类凯勒流形，特别是具有 $O(n) \times \mathbb{R}^n$ 对称性的那些流形。对于这些流形，我们证明了一个唯一性结果：任何完备且具有非负正交双截曲率（$n \geq 3$）或非负双截曲率（$n \geq 2$）的此类流形都双全纯等距于 $\mathbb{C}^n$。我们还考虑了另一种曲率张量，称为正交反双截曲率。我们找到了完备 $O(n)$-对称管状域具有非负正交反双截曲率的充分必要条件，并给出了若干满足该条件的完备度量的例子。最后，我们讨论了这些空间在最优传输中的一些应用。特别地，我们研究了非光滑几何的“合成”曲率界，以及如何将它们应用于由 Monge 代价 $c(x,y)=\\|x-y\\|$ 诱导的粗糙几何。
英文摘要：
In this article, we study a class of Kähler manifolds defined on tube domains in $\mathbb{C}^n$, and in particular those which have $O(n) \times \mathbb{R}^n$ symmetry. For these, we prove a uniqueness result showing that any such manifold which is complete and has non-negative orthogonal bisectional curvature ($n \geq 3$) or non-negative bisectional curvature ($n \geq 2$) is biholomorphically isometric to $\mathbb{C}^n$. We also consider another curvature tensor called the \emph{orthogonal anti-bisectional curvature}. We find necessary and sufficient conditions for a complete $O(n)$-symmetric tube domain to have non-negative orthogonal anti-bisectional curvature and provide several examples of complete metrics which satisfy this condition. Finally, we discuss some applications of these spaces within optimal transport. In particular, we study "synthetic" curvature bounds for non-smooth geometries and how they can be applied to the rough geometry induced by the Monge cost $c(x,y)=\|x-y\|$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
