Examples of scalar-flat hypersurfaces in Rn+1\mathbb{R}^{n+1}

Given a hypersurface $M$ of null scalar curvature in the unit sphere $\mathbb{S}^n$, $n\ge 4$, such that its second fundamental form has rank greater than 2, we construct a singular scalar-flat hypersurface in \Rr^{n+1} as a normal graph over a truncated cone generated by $M$. Furthermore, this graph is 1-stable if the cone is strictly 1-stable.Comment: Paper accepted to publication in Manuscripta Mathematic

The Dirichlet Problem for Curvature Equations in Riemannian Manifolds

We prove the existence of classical solutions to the Dirichlet problem for a class of fully nonlinear elliptic equations of curvature type on Riemannian manifolds. We also derive new second derivative boundary estimates which allows us to extend some of the existence theorems of Caffarelli, Nirenberg and Spruck [4] and Ivochkina, Trundinger and Lin [19] to more general curvature functions and less convex domains.Comment: 32 pages, no figures. Final version. Paper accepted to publication in Indiana University Mathematics Journa