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ON THE INVARIANT SPECTRUM OF S 1 METRICS ON S 2
 

Summary: ON THE INVARIANT SPECTRUM OF S 1
-INVARIANT
METRICS ON S 2
MIGUEL ABREU AND PEDRO FREITAS
Abstract. A theorem of J. Hersch (1970) states that for any smooth metric
on S 2 , with total area equal to 4#, the first nonzero eigenvalue of the Laplace
operator acting on functions is less than or equal to 2 (this being the value
for the standard round metric). For metrics invariant under the standard
S 1 ­action on S 2 , one can restrict the Laplace operator to the subspace of
S 1 ­invariant functions and consider its spectrum there. The corresponding
eigenvalues will be called invariant eigenvalues, and the purpose of this paper
is to analyse their possible values.
We first show that there is no general analogue of Hersch's theorem, by
exhibiting explicit families of S 1 ­invariant metrics with total area 4# where
the first invariant eigenvalue ranges through any value between 0 and #. We
then restrict ourselves to S 1 ­invariant metrics that can be embedded in R 3
as surfaces of revolution. For this subclass we are able to provide optimal
upper bounds for all invariant eigenvalues. As a consequence, we obtain an
analogue of Hersch's theorem with an optimal upper bound (greater than 2
and geometrically interesting). This subclass of metrics on S 2 includes all

  

Source: Abreu, Miguel - Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa

 

Collections: Mathematics