论文标题
Kähler-Ricci流保持负反见曲率
Kähler-Ricci Flow preserves negative anti-bisectional curvature
论文作者
论文摘要
在最近的工作（纯应用肛门2（2020），397-426）中，第一名命名作者和J. Zhang发现了最佳运输的规则性理论与Kähler歧管的曲率之间的联系。特别是，我们证明了成本函数的MTW张量$ c（x，y）=ψ（x-y）$可以理解为在管域上定义的相关Kähler度量的抗两种曲率。在这里，抗两种曲率定义为$ r（\ Mathcal {x}，\ edline {\ MathCal {\ MathCal {y}}，\ Mathcal {X}，\ edrowline {\ Mathcal {y Mathcal {y}}}}}）向量和$ r $是曲率张量。抗两种曲率和MTW张量之间的对应关系提供了一种有意义的意义，其中抗碰撞曲率可以具有符号（即，为正或负）。 在本文中，我们研究了Kähler-Icci流动下的抗两种曲率的行为。我们发现，在流动下保留了非阳性抗分离曲率。在复杂的维度二中，我们还表明，非负性正交抗脉冲曲率（即MTW（0）条件）保留在流动下。我们提供了这些结果的几种应用 - 复杂的几何形状，最佳传输和仿射几何形状。
In recent work (Pure Appl. Anal. 2 (2020), 397-426), the first named author and J. Zhang found a connection between the regularity theory of optimal transport and the curvature of Kähler manifolds. In particular, we showed that the MTW tensor for a cost function $c(x,y)=Ψ(x-y)$ can be understood as the anti-bisectional curvature of an associated Kähler metric defined on a tube domain. Here, the anti-bisectional curvature is defined as $R(\mathcal{X}, \overline{ \mathcal{Y}},\mathcal{X}, \overline{ \mathcal{Y}}) $ where $\mathcal{X}$ and $\mathcal{Y}$ are polarized $(1,0)$ vectors and $R$ is the curvature tensor. The correspondence between the anti-bisectional curvature and the MTW tensor provides a meaningful sense in which the anti-bisectional curvature can have a sign (i.e., be positive or negative). In this paper, we study the behavior of the anti-bisectional curvature under Kähler-Ricci flow. We find that non-positive anti-bisectional curvature is preserved under the flow. In complex dimension two, we also show that non-negative orthogonal anti-bisectional curvature (i.e., the MTW(0) condition) is preserved under the flow. We provide several applications of these results -- in complex geometry, optimal transport, and affine geometry.
