$p$-Laplace operator and diameter of manifolds.

Jean-Francois Grosjean (1)

1 IECN - Institut Élie Cartan de Nancy

Let $\var$ be a compact Riemannian manifold without boundary. In this paper, we consider the first nonzero eigenvalue of the $p$-Laplacian $\lap$ and we prove that the limit of $\sqrt[p]{\lap}$ when $p\rightarrow\infty$ is $2/d(M)$ where $d(M)$ is the diameter of $M$. Moreover if $\var$ is an oriented compact hypersurface of the Euclidean space $\R$ or $\Snpi$, we prove an upper bound of $\lap$ in term of the largest principal curvature $\kappa$ over $M$. As applications of these results we obtain optimal lower bounds of $d(M)$ in term of the curvature. In particular we prove that if $M$ is a hypersurface of $\R$ then : $d(M)\geq\pi/\kappa$.

Mots clés: p-laplacien; hypersurfaces; courbure

Domaines: Géométrie différentielle [math.DG]

Jean-Francois Grosjean. $p$-Laplace operator and diameter of manifolds.. Ann. Global Anal. Geom., 2005, 28, pp.257-270. ⟨10.1007/s10455-005-6637-4⟩. ⟨hal-00137924⟩