Vol. 49, No. 1, 1973

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ISSN: 0030-8730
HD-minimal but no HD-minimal

Young K. Kwon

Vol. 49 (1973), No. 1, 171–176
Abstract

Let UHDk (resp. UHDk) 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

U 1HD ⊂ U1˜
HD

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

U kHD ⊂ UkH˜D

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
Authors
Young K. Kwon