A dynamical approach to the Sard problem in Carnot groups

The Sard problem in step 2 and in filiform Carnot groups

We study the Sard problem for the endpoint map in some well-known classes of Carnot groups. Our first main result deals with step 2 Carnot groups, where we provide lower bounds (depending only on the algebra of the group) on the codimension of the abnormal set; it turns out that our bound is always at least 3, which improves the result proved in arXiv:1503.03610 and settles a question emerged in arXiv:1709.02854. In our second main result we characterize the abnormal set in filiform groups and show that it is either a horizontal line, or a 3-dimensional algebraic variety.

Sard Property for the endpoint map on some Carnot groups

In Carnot-Caratheodory or sub-Riemannian geometry, one of the major open problems is whether the conclusions of Sard's theorem holds for the endpoint map, a canonical map from an infinite-dimensional path space to the underlying finite-dimensional manifold. The set of critical values for the endpoint map is also known as abnormal set, being the set of endpoints of abnormal extremals leaving the base point. We prove that a strong version of Sard's property holds for all step-2 Carnot groups and several other classes of Lie groups endowed with left-invariant distributions. Namely, we prove that …

Sard-正则Carnot群上的曲率指数与测地维数

Curvature exponent and geodesic dimension on Sard-regular Carnot groups

本文在Sard-正则Carnot群上刻画了测地维数并给出曲率指数的新下界，通过构造二步Carnot群实例证明了$N_{CE} > N_{GEO}$，回答了Rizzi提出的问题。

无穷维多项式映射的Sard性质

Sard properties for polynomial maps in infinite dimension

本文研究了无穷维多项式映射的Sard性质，提出了Sard定理在该设定下的精确量化条件，并应用于亚黎曼几何中的Sard猜想证明。