New Minimal Hypersurfaces in R(k+1)(2k+1) and S(2k+3)k

TL;DR

This paper introduces a new class of minimal hypersurfaces in specific high-dimensional spaces, constructed via Pfaffians and determinants of antisymmetric matrices, expanding the known examples in differential geometry.

Contribution

The paper constructs novel minimal hypersurfaces in R(k+1)(2k+1) and S(2k+3)k using Pfaffians and determinants, extending the catalog of known minimal hypersurfaces.

Findings

H(1) and H(2) are congruent to known hypersurfaces

H(k>2) are new special harmonic cones

These hypersurfaces are invariant under certain symmetry groups

Abstract

We find a class of minimal hypersurfaces H(k) as the zero level set of Pfaffians, resp. determinants of real 2k+2 dimensional antisymmetric matrices. While H(1) and H(2) are congruent to a 6-dimensional quadratic cone resp. Hsiang's cubic su(4) invariant in R15, H(k>2) (special harmonic so(2k+2)-invariant cones of degree>3) seem to be new.