Immersed hypersurfaces with positive first Newton transformation

Abstract

We prove that every closed cooriented hypersurface immersion in $\R^{n+1}$, $n\ge 2$, with positive definite first Newton transformation bounds a compact immersed manifold with the prescribed boundary map and outward coorientation. Combined with the rigidity results of Ros and Pinkall, this filling theorem implies that every closed connected hypersurface immersed in $\R^{n+1}$ with constant scalar curvature is a round sphere.