Реферат: In problems of classical analysis and its generalizations, an important role is played by graph mapping. For example, the classes of minimal and maximal surfaces (for details on these surfaces, their properties, and applications, see [1]–[3] and the references given there) are locally representable as graphs. In addition, at the beginning of the 21st century, a link was found between neuroscience problems on construction of visualization models and properties of certain minimal surfaces in the sub-Riemannian geometry [4]. Metric properties of graphs in sub-Riemannian geometry are actively studied, both the classical method of construction (of the form \((x,f(x))\)), and the method matched with the group structure and multiplication operation (see, for example, [5]–[7], etc.) are considered. A distinctive feature here is that if the method of construction is based on the multiplication operation, and the mapping is Lipschitz continuous in the sub-Riemannian sense, then the graph itself is not Lipschitz continuous in the general case. Hence analogs of metric properties like an explicit area formula for a surface and a characteristic of local distortion of measure are derived in terms of new objects, which generally have quite involved structure. In other words, an “explicit form” result is often a formality, despite its rigor and accuracy. An alternative approach was proposed by the author of the present paper (see, in particular, [8] and [9]); however, some criteria involve constraints on the classes of mappings from which the graphs are constructed (such mappings are required to feature additional smoothness with respect to some variables). Under such constraints, the graph itself becomes, in fact, a smooth noncontact mapping. In addition, on the image of the graph one defines not a sub-Riemannian analog of the Hausdorff measure, which is constructed via an intrinsic quasimetric, but rather its modification, which depends on approximation of this noncontact mapping. This therefore suggests the natural problem of derivation (and existence) of basic properties of graph mappings, whose method of construction agrees with the sub-Riemannian structure and which are Lipschitz continuous with respect to sub-Riemannian (quasi)metrics. In particular, this problem calls for the description of classes of mappings whose corresponding graphs are Lipschitz continuous. In the present paper, we consider a model case of Heisenberg groups and solve the problem of existence of Lipschitz graphs on these groups. The methods involved in our analysis allow us to deal with classes of mappings of two-step Carnot groups and construct examples of Lipschitz and nonLipschitz mappings on modified Heisenberg groups whose graphs are Lipschitz continuous in the sub-Riemannian sense. We first recall the necessary definitions and facts and describe our assumptions on the image and preimage of the mappings we consider. We start with the definition of a Carnot group, because most of our results will be formulated also for these structures.

Библиографическая ссылка: Karmanova M.B.
Lipschitz Graphs on Heisenberg groups and Related Problems
Mathematical Notes. 2025. V.118. N1. P.205-211. DOI: 10.1134/s0001434625603545 WOS Scopus OpenAlex

Оригинальная: Карманова М.Б.
Липшицевы графики на группах Гейзенберга и связанные задачи
Математические заметки. 2025. Т.118. №1. С.154-158. DOI: 10.4213/mzm14704 OpenAlex

Даты:

Поступила в редакцию:	10 апр. 2025 г.
Принята к публикации:	12 апр. 2025 г.
Опубликована в печати:	16 дек. 2025 г.
Опубликована online:	16 дек. 2025 г.

Идентификаторы БД:

Web of science:	WOS:001642030600019
Scopus:	2-s2.0-105025205251
OpenAlex:	W4417405557

Цитирование в БД: Пока нет цитирований

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