Premium
Examples of Non‐Homeomorphic Harmonic Maps Between Negatively Curved Manifolds
Author(s) -
Farrell F. T.,
Jones L. E.,
Ontaneda P.
Publication year - 1998
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609397004190
Subject(s) - mathematics , diffeomorphism , conjecture , counterexample , harmonic map , homotopy , pure mathematics , equivalence (formal languages) , harmonic , combinatorics , topology (electrical circuits) , physics , quantum mechanics
Let M and N be closed non‐positively curved manifolds, and let f : M → N be a smooth map. Results of Eells and Sampson [ 1 ] show that f is homotopic to a harmonic map φ, and Hartman [ 6 ] showed that this φ is unique when N is negatively curved and f ∗(π 1 M ) is not cyclic. Lawson and Yau conjectured that if f : M → N is a homotopy equivalence, where M and N are negatively curved, then the unique harmonic map φ homotopic to f would be a diffeomorphism. Counterexamples to this conjecture appeared in [ 2 ], and later in [ 7 ] and [ 5 ]. There remains the question of whether a ‘topological’ Lawson–Yau conjecture holds. 1991 Mathematics Subject Classification 53C20, 55P10, 57C25, 58E20.