Polynomial growth harmonic functions on complete Riemannian manifolds
In [Commun. Pure Appl. Math. 28, 201--228 (1975; Zbl 0291.31002)] \textit{S. T. Yau} generalized the classical Liouville theorem to complete Riemannian manifolds of nonnegative Ricci curvature, and then conjectured that on such a manifold, the space of harmonic functions of polynomial growth at most \(d\) is finite dimensional. Since then, much research has been done on this problem. The conjecture was settled by \textit{T. Colding} and \textit{W. P. Minicozzi} II [Ann. Math. (2) 146, No. 3, 725--747 (1997; Zbl 0928.53030)]. In this paper the author establishes specific upper bounds on \(\dim{\mathcal H}^d(M)\), for manifolds \(M\) satisfying various conditions. Here \({\mathcal H}^d(M)\) denotes the space of harmonic functions of polynomial growth of order at most \(d\). The author establishes a bound on \(\dim{\mathcal H}^d(M)\) for manifolds with finitely many ends, where each end satisfies a volume doubling condtion, a mean value inequality for nonnegative subharmonic functions and a finite covering condition. It is known that these conditions are satisfied for Riemannian manifolds with nonnegative Ricci curvature outside a compact set and finite first Betti number. For manifolds satisfying stronger conditions, e.g.~for a connected sum of manifolds satisfying a stronger doubling condition, the author obtains a sharp bound on the growth of \(\dim{\mathcal H}^d(M)\) in terms of \(d\). Finally, since the Poincaré inequality implies the mean value inequality and is invariant under rough isometries, the author is able to establish similar bounds for manifolds roughly isometric to the cases previously considered, under the extra assumption that these satisfy a Poincaré inequality.
scientific article; zbMATH DE number 2110189
scientific article