On the Ranges of Eigenfunctions on Compact Manifolds

The aim of this note is to give a sharp upper bound on the ratio formula] where is a nonconstant eigenfunction for the Laplace–Beltramioperator on a connected compact Riemannian manifold withoutboundary. This ratio is always positive, since max>0 andmin<0 for every nonconstant eigenfunction. We assume thatmax–min, in order to simplify the notation. For the caseof a two-dimensional manifold with nonnegative Ricci curvature,our theorem implies that the above ratio is less than the ratioof the maximum divided by the absolute value of the minimumof the Bessel function of order zero. The proof is based on a gradient estimate from a previous paperof the author (see 5]), which in turn was proved using themaximum principle technique. In contrast to the standard applicationsof gradient estimates, which are based on integration alonggeodesics, we arrive at a contradiction by integrating the gradientestimate over small spheres centred at a point where the absolutevalue of the eigenfunction attains its maximum. The main motivation for our work is that the ratio of the maximumand the minimum of an eigenfunction plays a role in estimatesof the corresponding eigenvalues (see 5] and 7]). More precisely,our theorem implies that there are minimizing sequences of compactmanifolds such that the first eigenvalues of the manifolds approachthe corresponding lower bound for the first eigenvalue obtainedin 5, Theorem 2] for every possible ratio of the maximum andthe minimum of the corresponding eigenfunction. 1991 MathematicsSubject Classification 58G25.