Potenciais estáticos em variedades assintoticamente planas

Abstract

In this work, we study how the existence of static potentials in asymptotically at manifolds can in uence the geometry of this manifold. Firstly, we study a paper of Pengzi Miao and Luen-Fai Tam, \Static Potencial and Asymptoticaly Flat Manifolds", where is discussed questions about rigidity for asymptotically at 3 manifolds that admit a static potential. It is analyzed the dimension of the static potential space and the asymptotic behavior of the nonempty zero set, it is given com ditions to a asymptotically at 3-manifold have such set extending to in nty. Moreover, in this scope, are demonstrated results of rigidity for 3-manifolds without boundary. In a second moment, we study the papers of Lan-Hsuan Huang, Daniel Martin and Pengzi Miao, \Static Potentials and Area Minimizing Hypersurfaces" and Gregory J. Gal loway and Pengzi Miao, \Variational and Rigidity Properties of Static Potentials", where has been proven that if a asymptotically at manifold with horizon boundary has a global static potential, then this static potential must be zero on the boundary. Moreover, it is shown that if a asymptotically at manifold with horizon boundary has a unbounded static potential in a end, then the manifold must contain a complete non-compact area minimizing hypersurface.