Estimates for the first eigenvalue of the p-Laplacian on Riemannian manifolds

Abstract

The following thesis aims to study estimates of the first eigenvalue of the p-Laplacian operator on compact Riemannian manifolds and complete non-compact manifolds. We established a linearized operator for the divergence-type p-Laplacian, which resulted in a Bochner-type formula. From this, we initially obtained lower bounds for the first eigen- value of the p-Laplacian through the norm of the second fundamental form for p ≥ 2, with

characterization of equality. Next, we demonstrated a similar result for submanifolds with prescribed scalar curvature and for submanifolds with constant mean curvature. In each

case reported above, we presented a generalization for manifolds with non-empty bound- ary through a Reilly-type formula for the linearized operator. Additionally, we presented

an analytical version of the previous results for the singular case with 3

2 < p < 2. Finally,

we developed a Liouville-type theorem for complete non-compact manifolds, with appli- cations in warped products.