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Jacobi fields along harmonic 2-spheres in and are not all integrable
In a previous paper, we showed that any Jacobi field along a harmonic map
from the 2-sphere to the complex projective plane is integrable (i.e., is
tangent to a smooth variation through harmonic maps). In this paper, in
contrast, we show that there are (non-full) harmonic maps from the 2-sphere to
the 3-sphere and 4-sphere which have non-integrable Jacobi fields. This is
particularly surprising in the case of the 3-sphere where the space of harmonic
maps of any degree is a smooth manifold, each map having image in a totally
geodesic 2-sphere.Comment: 43 pages. Some typos corrected; introduction expande
Topological restrictions for circle actions and harmonic morphisms
Let be a compact oriented smooth manifold which admits a smooth circle
action with isolated fixed points which are isolated as singularities as well.
Then all the Pontryagin numbers of are zero and its Euler number is
nonnegative and even. In particular, has signature zero. Since a
non-constant harmonic morphism with one-dimensional fibres gives rise to a
circle action we have the following applications: (i) many compact manifolds,
for example , surfaces, () where
is the closed surface of genus can never be the domain of a
non-constant harmonic morphism with one-dimensional fibres whatever metrics we
put on them; (ii) let be a compact orientable four-manifold and
a non-constant harmonic morphism. Suppose that one of
the following assertions holds: (1) is half-conformally flat and its
scalar curvature is zero, (2) is Einstein and half-conformally flat,
(3) is Hermitian-Einstein. Then, up to homotheties and Riemannian
coverings, is the canonical projection between flat tori.Comment: 18 pages; Minor corrections to Proposition 3.1 and small changes in
Theorem 2.8, proof of Theorem 3.3 and Remark 3.
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