Realization of Lipschitz Geometry at Infinity on Complex Analytic Sets

• Дата: 2025
• Автор: Стахів, Ростислав
• Формат: Стаття
• Мова: Українська
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2025
• Онлайн доступ: http://mcm-math.kpnu.edu.ua/article/view/341604
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

The article provides an in-depth analysis of the development and application of Lipschitz geometry at infinity in the study of complex analytic sets, aimed at establishing the relationship between their algebraic nature and the global metric structure. A generalized definition of Lipschitz and Bialpischitz homeomorphisms at infinity in terms of metric spaces is considered, which provides the possibility of classifying analytic sets according to their asymptotic behavior and introducing the concept of Lipschitz equivalence outside compact domains. A class of pure d-dimensional entire complex analytic subsets is investigated, for which a criterion of algebraicity is formulated and proven due to the existence of a unique tangent cone at infinity, which turns out to be a d-dimensional complex algebraic set. The equivalence of three fundamental properties is proved: algebraicity of an analytic set; uniqueness of its tangent cone at infinity; of the Bialpischitz homeomorphism at infinity to a complex algebraic set. Special attention is paid to the role of the Meeks’ Conjecture III, which concerns the uniqueness of tangent cones of minimal surfaces in R3 with quadratic growth of area, and its connection with the concept of Lipschitz regularity at infinity. It is shown that metric invariants, in particular Bialpischitz homeomorphisms, allow us to describe the asymptotic rigidity and stability of analytic structures. The results obtained generalize and deepen the theorems of Chow, Stoll-Bishop, and Lojasevich, introducing new criteria for identifying algebraic sets by their global metric characteristics. The proposed approach forms a conceptual basis for further studies of the asymptotic rigidity, stability, and deformations of complex analytic objects within the framework of modern complex and differential geometry