November 2011
Intermediate to advanced
528 pages
10h 30m
English
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3.3. Instability of Hopf Vector Fields
By
Corollary 3.16
the Hopf vector fields on
S
3
are absolute minima for
hence stable critical points of the energy functional. This is a rather surprising bias from the theory of harmonic maps where it is known that
hence stable critical points of the energy functional. This is a rather surprising bias from the theory of harmonic maps where it is known that
Theorem 3.17 Y.L. Xin,
[321]
Let N be Riemannian manifold. Then any nonconstant harmonic map
is unstable provided that n
≥ 3.
is unstable provided that n
≥ 3.
In particular the Hopf vector fields ξ on
S
3
are unstable as harmonic maps
. What about the stability of Hopf vector fields on a sphere
S
2
m
+1
(with
m
> 1)? To answer ...
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