Legendrian cycles and curvatures (Q253773)

In the present note, the authors study general Legendrian cycles in Euclidean space, first providing a simpler proof of a uniqueness result (Theorem 4.2) due to [\textit{J. H. G. Fu}, Indiana Univ. Math. J. 38, No. 3, 745--771 (1989; Zbl 0668.49010)], which states that a compactly supported Legendrian cycle is uniquely determined by its restriction to the so-called Gauss curvature form (introduced in Section 4). Then they establish a related uniqueness result (Theorem 5.3) under a suitable \textit{full-dimensionality assumption} by studying projections of such cycles onto the first component. In the process of establishing the latter theorem, the authors prove a constancy theorem for Lipschitz submanifolds (Theorem 3.1), which is itself of independent interest. The authors conclude the paper by dropping the full-dimensionality assumption to establish a weaker form (Proposition 6.1) of Theorem 5.3 and establish a result on the support of a strongly full-dimensional Legendrian cycle.