 

Neil Course
fharmonic maps which map the boundary of the domain to one point in the target


Published: 
November 11, 2007

Keywords: 
harmonic maps, fharmonic, boundary, Riemannian surface, constant boundary data 
Subject: 
58E20 35J25 53C43 


Abstract
One considers the class of maps u:D → S^{2}, which map the boundary of D to one point in S^{2}. If u
were also harmonic, then it is known that u must be constant. However, if u is instead
fharmonic  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 fharmonic maps u:D → S^{2} which map the boundary to one point. We will also
see that there exist nonconstant f for which, there is no nonconstant fharmonic map in this
class. Finally, we see that there exists a nonconstant fharmonic map from the torus to the 2sphere.


Acknowledgements
Research supported by Swiss National Science Foundation grant number 200020107652/1 and EPSRC award number 00801877.


Author information
Département de Mathématiques, Université de Fribourg, Pérolles, CH1700 Fribourg, Switzerland
neil.course@unifr.ch
http://www.neilcourse.co.uk

