Sharp regularity of sub-Riemannian length-minimizing curves

2025/02/01 by Socionovo, Alessandro
#28A75 #49K21 #53C17 #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2502.00403

Abstract

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 C² but not C³ 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 C² within these examples. In particular, we prove that Theorem 1.1 in [6] is sharp.

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