Conformal capacities and conformally invariant functions on Riemannian manifolds
Let \(M\) be a Riemannian manifold of class \(C^1\) with dimension \(n \geq 2\), \(C(M)\) the linear space of continuous real-valued functions on \(M\), and \(H(M) = C(M) \cap L^1_n(M)\) the subspace of functions \(u\) admitting a generalized \(L^n\)-integrable differential, denoted \(\nabla u\) by identification with a gradient, satisfying \(I(u,M) = \int_M |\nabla u|^2 dr<\infty\), where \(dr\) is the volume element of \(M\) and \(|\nabla u|\) the norm of \(\nabla u\) defined by the Riemannian structure of \(M\). Let \((C_0,C_1)\) be a pair of closed sets in \(M\). The capacity of the condenser \(\Gamma(C_0,C_1,M)\) is \[ \text{Cap}_M(C_0,C_1) = \inf_u I(u,M), \] where the infimum is taken over the set \(A(C_0,C_1)\) of all functions \(u\in H(M)\) satisfying \(u = 0\) on \(C_0\), \(u = 1\) on \(C_1\) and \(0 \leq u(x) \leq 1\) for all \(x\), these functions being called admissible for \(\Gamma(C_0,C_1)\). The paper under review develops the theory of conformal invariants for a Riemannian manifold \(M\). The author constructs and studies four conformally invariant functions \(\rho_M\), \(\nu_M\), \(\mu_M\), \(\lambda_M\), respectively depending on 4, 3, 2 points on \(M\) defined as extremal capacities for condensers associated with those points. These functions have similarities with classical invariants on \(S^n\), \(\mathbb{R}^n\), or \(H^n\). Their properties, and especially their continuity, are efficient tools for solving some problems of conformal geometry in the large.
scientific article; zbMATH DE number 883879