HD-minimal but no HD-minimal

Young K. Kwon

Vol. 49 (1973), No. 1, 171–176

DOI: 10.2140/pjm.1973.49.171

Abstract

Let U_HD^k (resp. U_HD^k) be the class of Riemannian n-manifolds (n ≧ 2) on which there exist k non-proportional HD-minimal (resp. HD-minimal) functions. The purpose of the present paper is to construct a Riemannian n-manifold n ≧ 3 which carries a unique (up to constant factors) HD-minimal function but no HD-minimal functions. Thus the inclusion relation

is strict for n ≧ 3. By welding k copies of this Riemannian n-manifold, it is then established that the inclusion relation

is strict for all k ≧ 1 and n ≧ 3. The problem still remains open for n = 2.

Mathematical Subject Classification 2000

Primary: 53C20

Secondary: 31B05

Milestones

Received: 21 August 1972

Revised: 17 January 1973

Published: 1 November 1973