拟正则曲线的重标度原理及其在双曲性中的应用
A rescaling principle for quasiregular curves with applications to hyperbolicity
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中文总结 AI 辅助
研究拟正则曲线到校准流形的重标度原理，通过引入适应设定的布罗迪双曲性等方法，证明了相关等价性，量化了局部连续性模，回答了问题并构造新例子给出布罗迪双曲性障碍。
中文摘要 AI 辅助
我们证明了拟正则曲线到校准流形的Miniowitz-Zalcman重标度原理。有两个主要应用。一是引入适应此设定的布罗迪双曲性，证明其与从欧几里得单位球到目标的拟正则曲线族的正规性等价，当正规性成立时，利用目标的单射半径下界和截面曲率上界量化拟正则曲线的局部连续性模。二是在共形曲线到闭校准流形的特殊情况下，证明小林双曲性和布罗迪双曲性的等价性，回答了Broder-Iliashenko-Madnick提出的问题。还构造了新的非恒定整拟正则曲线的例子，给出了布罗迪双曲性的障碍。
英文摘要
We prove a Miniowitz--Zalcman rescaling principle for quasiregular curves into calibrated manifolds. We have two main applications. First, we introduce Brody hyperbolicity adapted to our setting and prove its equivalence to the normality of the family of quasiregular curves from the Euclidean unit ball into the target. When normality holds, we quantify the local modulus of continuity for quasiregular curves using an injectivity radius lower bound and a sectional curvature upper bound of the target. Second, in the special case of conformal curves into closed calibrated manifolds, we prove the equivalence of Kobayashi and Brody hyperbolicity. This answers a question posed by Broder--Iliashenko--Madnick. As an intermediate result, we prove an analogue of Marty's theorem from complex analysis in this setting. Additionally, we construct new examples of non-constant entire quasiregular curves factoring through a Special Lagrangian submanifold of a closed Calabi--Yau manifold, an associative submanifold of a closed $G_2$ manifold, and four-dimensional analogues thereof, providing obstructions to Brody hyperbolicity.
