标题： 子黎曼长度最小曲线的尖锐正则性 显示英文标题

标题： Sharp regularity of sub-Riemannian length-minimizing curves

摘要： 在子黎曼几何中，一个长期未解决的问题是长度最短曲线（弧长参数化）的光滑性。 在[6]中，这个问题被否定回答，给出了一个实解析（甚至多项式）子黎曼结构的$C^2$但不是$C^3$的长度最短曲线的例子。 在本文中，我们研究了一类推广了[6]中所展示的例子的子黎曼结构，并证明在这些例子中，长度最短曲线至少属于$C^2$类。 特别是，我们证明了[6]中的定理1.1是精确的。 显示英文摘要

摘要： A longstanding open question in sub-Riemannian geometry is the smoothness of (the arc-length parameterization of) length-minimizing curves. In [6], this question is negative answered, with an example of a $C^2$ but not $C^3$ length-minimizer of a real-analytic (even polynomial) sub-Riemannian structure. In this paper, we study a class of examples of sub-Riemannian structures that generalizes that presented in [6], and we prove that length-minimizing curves must be at least of class $C^2$ within these examples. In particular, we prove that Theorem 1.1 in [6] is sharp.