论文标题
在管域上的积极弯曲的kähler指标及其在最佳运输中的应用
Positively curved Kähler metrics on tube domains and their applications to optimal transport
论文作者
论文摘要
在本文中，我们研究了$ \ mathbb {c}^n $中的管域上定义的Kähler歧管，尤其是那些具有$ O（n）\ times \ times \ mathbb {r}^n $对称性的kähler歧管。为此，我们证明了一个独特性结果表明，任何完整且具有非负正交双弯曲曲率（$ n \ geq 3 $）或非负性双义曲率（$ n \ geq 2 $）的歧管都是生物态均与$ \ mathbb insotic isstricricricricricricricricricricricricricricricricricricricric in $ \ \ Mathbbbb {C c}^n $。我们还考虑了另一种称为\ emph {正交反合曲率}的曲率张量。我们发现完全$ o（n）$ - 对称管域具有非负性正交抗异构曲率的必要条件，并提供了满足这种情况的完整指标的几个示例。最后，我们在最佳运输中讨论了这些空间的一些应用。特别是，我们研究了非平滑几何形状的“合成”曲率边界，以及如何将它们应用于Monge诱导的粗糙几何形状，费用为$ C（x，y）= \ | x-y \ | $。
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 $c(x,y)=\|x-y\|$.
