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.