The aim of this note is to give a sharp upper bound on the ratio
formula here
where ϕ is a nonconstant eigenfunction for the Laplace–Beltrami
operator on a
connected compact Riemannian manifold without boundary. This ratio is always
positive, since maxϕ>0 and minϕ<0 for every nonconstant eigenfunction.
We
assume that maxϕ[ges ]−minϕ, in order to simplify the notation.
For the case of
a two-dimensional manifold with nonnegative Ricci curvature, our theorem
implies
that the above ratio is less than the ratio of the maximum divided by the
absolute
value of the minimum of the Bessel function of order zero.
The proof is based on a gradient estimate from a previous paper of the
author
(see [5]), which in turn was proved using the maximum
principle technique. In
contrast to the standard applications of gradient estimates, which are
based on
integration along geodesics, we arrive at a contradiction by integrating
the gradient
estimate over small spheres centred at a point where the absolute value
of the
eigenfunction attains its maximum.
The main motivation for our work is that the ratio of the maximum and
the minimum of an eigenfunction plays a role in estimates of the corresponding
eigenvalues (see [5] and [7]).
More precisely, our theorem implies that there are
minimizing sequences of compact manifolds such that the first eigenvalues
of the
manifolds approach the corresponding lower bound for the first eigenvalue
obtained
in [5, Theorem 2] for every possible ratio
of the maximum and the minimum of the corresponding eigenfunction.