New York Journal of Mathematics
Volume 13 (2007) 423-435


Neil Course

f-harmonic maps which map the boundary of the domain to one point in the target

Published: November 11, 2007
Keywords: harmonic maps, f-harmonic, boundary, Riemannian surface, constant boundary data
Subject: 58E20 35J25 53C43

One considers the class of maps u:D → S2, which map the boundary of D to one point in S2. If u were also harmonic, then it is known that u must be constant. However, if u is instead f-harmonic -- a critical point of the energy functional 1/2 ∫D f(x) |∇ u(x)|2 -- then this need not be true. We shall see that there exist functions f:D → (0,∞) and nonconstant f-harmonic maps u:D → S2 which map the boundary to one point. We will also see that there exist nonconstant f for which, there is no nonconstant f-harmonic map in this class. Finally, we see that there exists a nonconstant f-harmonic map from the torus to the 2-sphere.


Research supported by Swiss National Science Foundation grant number 200020-107652/1 and EPSRC award number 00801877.

Author information

Département de Mathématiques, Université de Fribourg, Pérolles, CH-1700 Fribourg, Switzerland