Capacity boundary elements in Riemannian manifolds and generalized boundaries

D. A. Sboev

Novosibirsk State University, Novosibirsk, Russia

Abstract: Using the capacity metric on Riemannian manifolds, we introduce capacity boundary elements and study the boundary behavior of closed mappings with bounded distortion. We establish properties of the spaces of capacity boundary elements and examine the relations between boundary elements for different exponents. The paper also examines geometric properties of a (natural) generalized boundary in domains with locally finitely connected boundary in metric spaces. It describes conditions on the metric under which the generalized boundary is unique up to homeomorphism and provides examples of such metrics in various domains. In domains with locally finitely connected boundary on Riemannian manifolds, it is shown that every element of the generalized boundary is contained in the support of some capacity boundary element. As a consequence, results on the relationship between prime ends and capacity boundary elements are obtained.

Keywords: Riemannian manifold, capacity metric, capacity boundary element, generalized boundary, Mazurkiewicz metric, prime end, boundary behavior, closed mapping, mapping with bounded distortion.

UDC: 517.54+517.518

MSC: 35R30

Received: 19.11.2024
Revised: 19.11.2024
Accepted: 25.04.2025

DOI: 10.33048/smzh.2025.66.314

 English version: Siberian Mathematical Journal, 2025, 66:3, 763–787