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

TL;DR

This paper constructs singular scalar-flat hypersurfaces in Euclidean space as normal graphs over cones generated by hypersurfaces with null scalar curvature, extending understanding of scalar-flat geometry.

Contribution

It introduces a method to generate scalar-flat hypersurfaces in Euclidean space from hypersurfaces with null scalar curvature in the sphere, including stability conditions.

Findings

Constructed singular scalar-flat hypersurfaces in $ ^{n+1}$

Established stability conditions based on the cone

Extended the class of known scalar-flat hypersurfaces

Abstract

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.