论文标题
地图和多尺寸的几何学程度
Degrees of maps and multiscale geometry
论文作者
论文摘要
我们研究了Riemannian流形之间的$ L $ -Lipschitz地图，证明了新的上限并构建了新的示例。例如，如果$ x_k $是$ K $的连接总和$ \ mathbb cp^2 $ for $ k \ ge 4 $，那么我们证明，$ l $ -lipschitz $ x_k $的最大程度是$ x_k $为$ c_1 l^4（\ log l^4（\ log l）更一般而言，我们仅将歧管与三种不同行为相连。每种类型都是由纯粹的拓扑标准定义的。对于可扩展的简单连接的$ n $ - manifolds，最大学位为$ \ sim l^n $。对于正式但不可估计的简单连接的$ n $ - manifolds，最大程度大致像$ l^n（\ log l）^{θ（1）} $。对于非正式的简单连接的$ n $ - manifolds，最大学位的限制为$ l^α$，对于某些$α<n $。
We study the degree of an $L$-Lipschitz map between Riemannian manifolds, proving new upper bounds and constructing new examples. For instance, if $X_k$ is the connected sum of $k$ copies of $\mathbb CP^2$ for $k \ge 4$, then we prove that the maximum degree of an $L$-Lipschitz self-map of $X_k$ is between $C_1 L^4 (\log L)^{-4}$ and $C_2 L^4 (\log L)^{-1/2}$. More generally, we divide simply connected manifolds into three topological types with three different behaviors. Each type is defined by purely topological criteria. For scalable simply connected $n$-manifolds, the maximal degree is $\sim L^n$. For formal but non-scalable simply connected $n$-manifolds, the maximal degree grows roughly like $L^n (\log L)^{θ(1)}$. And for non-formal simply connected $n$-manifolds, the maximal degree is bounded by $L^α$ for some $α< n$.
