Minimality in families of solutions of $\Delta u = Pu$ on Riemannian manifolds
Kwang-Nan Chow · Bulletin of the American Mathematical Society · 1971
The notion of minimality of solutions of Au = Pu was first introduced by C. Constantinescu and A. Cornea in 1958.On a Riemann surface, M. Nakai has given a complete characterization of the minimality in the monotone closure of the family of all Dirichlet-finite harmonic functions.His characterization is in terms of a positive bounded regular Borel (representing) measure on the Royden boundary of the Riemann surface (cf.[ó]).In this paper we announce that not only his work for the harmonic functions can be generalized for the solutions of the elliptic differential equation Au -Pu on a Riemannian manifold, but more significantly, the property of the existence of a minimal function is to a large extent an intrinsic part of the manifold, perhaps a quasi-conformal or quasi-isometric invariant.Consider a Riemannian manifold R and the elliptic differential equation Au-Pu on R, where P is nonnegative C 1 .For simplicity, solutions of Au~Pu will be called solutions.Let R* be the Royden compactification, T -R*\R the Royden boundary and A the harmonic boundary of R (cf.[ó]).The open subset A p = {qÇzAlq lias a neighborhood U in R* with f unit P< °° } of A introduced in [2] is crucial for solutions.Following the pattern that Nakai has established in [ó] for harmonic functions and using the results of Glasner-Katz [l, Theorems 1 2], we can construct a positive bounded regular Borel (representing) measure m p on T centered at z^CzR with support equal to the closure of A p characterized by u(z 0 ) -fru dm p for every solution u with finite energy integral (the so-called PE-iunction).Moreover, using Harnack's inequality we can also construct a nonnegative kernel K p (z, q) on RXT with the property u(z) = /r K p (z, q)u{q) dm p (q) for all z(ER and for all PE-i unctions u.Note that when P^O, m° and K°(z, q) are the corresponding measure and kernel for harmonic functions constructed by Nakai in [ó].Nakai's characterization for HD~-functions and HZ)~-minimal A MS 1970 subject classifications.Primary31B10, 31B25, 31B35; Secondary 31B05.