Colbois, Bruno

2003, Colbois, Bruno, Matei, Ana-Maria

,We consider a 1-parameter family of hyperbolic surfaces M(t) of genus v which degenerate as t --> 0 and we obtain a precise estimate of lambda(1,p)(t), the first eigenvalue of the p-Laplacian (p > 1) on M(t). In some cases we also give a precise estimate of the first eigenfunctions. As a direct application, we obtain that the quotient (lambda1,q)(1/q)/(lambda1,p)(1/p) (which is invariant under scaling of the metric) is unbounded even on the set of Riemannian manifolds with constant sectional curvature. This is to our knowledge, the first example of a family of manifolds with this property. To prove our results we use in an essential way the geometry of hyperbolic surfaces which is very well known. We show that an eigenfunction for lambda(1,p)(t) of L-p norm one is almost constant in the L-p sense (as t --> 0) on the parts of M(t) with large injectivity radius, and we estimate precisely its p-energy on the parts with small injectivity radius.