Conjuntos de nível de aplicações harmônicas para o círculo e curvatura escalar em 3-variedades

Abstract

For closed surfaces in R^3, the Gauss-Bonnet theorem gives us a relation between the Gaussian curvature and the topology of this surface. Given this, there is a search for results in dimension 3, as well as in other dimensions, Schoen-Yau [22] brings a relationship between scalar curvature of 3-dimensional manifolds and minimal stable surfaces. We will present an integral relationship between scalar curvature of the environment manifold and the topology of level sets of harmonic maps u : M^3 → S^1. This one result was proved by D. Stern [26]. To do this, we will present some relationships between 1-harmonic forms, in the context of Hodge theory, and harmonic maps, being these critical point of the Dirichet energy functional. Given this result, we will present as application of the proof of Geroch’s conjecture, that the torus T^3 does not admit metric of positive scalar curvature.