On the Area Functional of the Second Fundamental Form of Ovaloids

TL;DR

This paper investigates the variation of the area functional of the second fundamental form of hypersurfaces, revealing that spheres uniquely serve as critical points under various constraints due to their mean curvature properties.

Contribution

It introduces new characteristic properties of spheres related to the mean curvature of the second fundamental form and proves their uniqueness as critical points among ovaloids.

Findings

Spheres are the only ovaloids critical for the area functional of the second fundamental form.

The paper establishes new properties of the mean curvature of the second fundamental form.

Critical point conditions are characterized under different geometric constraints.

Abstract

The expression for the variation of the area functional of the second fundamental form of a hypersurface in a Euclidean space involves the so-called "mean curvature of the second fundamental form". Several new characteristic properties of (hyper)spheres, in which the mean curvature of the second fundamental form occurs, are given. In particular, it is shown that the spheres are the only ovaloids which are a critical point of the area functional of the second fundamental form under various constraints.