Variedades compactas de dimensão 4 com curvatura positiva e parabolicidade de sólitons Ricci-harmônicos não-compactos

Abstract

The purpose of this work is to study four-dimensional compact Riemannian manifolds with positive biorthogonal (sectional) curvature and parabolicity of steady Ricciharmonic solitons. In the rst part, we obtain classi cation theorems for submanifolds with positive biorthogonal curvature. Moreover, we use the concept of biorthogonal curvature to obtain a pinching condition which ensures that a compact four-manifold is de nite. In the third part, we show that, under a pinching condition on the scalar curvature, a noncompact Ricci-harmonic soliton has at most one end. In addition, we obtain volume estimates for the geodesic balls of steady Ricci-harmonic solitons.