3.4. Existence of Minima in Dimension > 3
Let (
M
,
g
) be a compact orientable Riemannian manifold. As well known the identity map
is a harmonic map.
is a harmonic map.
Definition 3.24 T. Nagano,
[221]
A Riemannian manifold (
M
,
g
) is said to be
stable
if the identity map is stable. Otherwise
M
is said to be
unstable
.
By a result of R.T. Smith,
[273]
, if (
M
,
g
) is a compact
n
-dimensional Enstein manifold then
M
is stable if and only if
where λ
1
is the first nonzero eigenvalue of the Laplace-Beltrami operator on functions and
ρ
is the scalar curvature. ...
where λ
1
is the first nonzero eigenvalue of the Laplace-Beltrami operator on functions and
ρ
is the scalar curvature. ...Get Harmonic Vector Fields now with the O’Reilly learning platform.
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