SOME ANALYTIC AND GEOMETRIC ASPECTS OF THE P-LAPLACIAN ON RIEMANNIAN MANIFOLDS

In this thesis, we deal with some problems related to the existence, uniqueness and triviality of the p-harmonic representative in the homotopy class of a map. In the first part, we prove that a map f with finite p-energy, p>2, from a complete Riemannian manifold M into a non-positively curved, compact manifold N is homotopic to a constant, provided the negative part of the Ricci curvature of the domain manifold is small in a suitable spectral sense. The result relies on a Liouville-type theorem for finite q-energy, p-harmonic maps under spectral assumptions. In the second part, we prove some comparison theorem for p-harmonic maps when the domain manifold is p-parabolic. First, we prove that, in general, given a p-harmonic map F from M to N and a convex real function H defined on N, the composition $H\circ F$ is not p-subharmonic, if p>2. This answers in the negative an open question arisen from a paper by Lin and Wei and suggests that the standard techniques used in the harmonic case can not be trivially adapted to $p\neq 2$. Then, we prove comparison results for the p-laplacian in the special case of both real-valued and vector-valued maps with finite p-energy. Finally, we obtain a general comparison result for homotopic finite p-energy continuous p-harmonic maps u and v, assuming that M is p-parabolic and N is complete and non-positively curved. In particular, we construct a homotopy through constant p-energy maps, which turn out to be p-harmonic when N is compact. Moreover, we obtain uniqueness in the case of negatively curved N. This generalizes a well known result in the harmonic setting due to R. Schoen and S.T. Yau

ON SOME ASPECTS OF OSCILLATION THEORY AND GEOMETRY

This thesis aims to discuss some of the relationships between oscillation theory for linear ordinary differential equations on the real line (shortly, ODE) and the geometry of complete Riemannian manifolds. In this respect, we prove new results in both directions. For instance, we improve on classical oscillation and nonoscillation criteria for ODE's, and we find sharp spectral estimates for a number of geometric differential operator on Riemannian manifolds. We apply these results to achieve topological and geometric properties. In the first part of the thesis, we collect some material which often appears in the literature in various forms and for which we give, in some instances, new proofs according to our specific point of view