A minimal hypersurface is a hypersurface whose mean curvature vector vanishes. It is a codimension-one
minimal submanifold, and is stationary for
volume under compactly
supported variations. The two-dimensional case in 
is a minimal surface
(do Carmo 1992).
An explicit periodic construction in 
has isolated singularities. Let 
, where 
is the lemniscate
function and 
is the lemniscate constant, so that 
has real period 2. Tkachev (2026) proved that the regular
part of
 |
is a minimal hypersurface. The zero set is invariant under the D4 lattice and its singular set is 
, the vertices
of the regular 24-cell honeycomb. Near each singular
point it is an analytic image of the cone over a Clifford
torus. Its quotient by the period lattice has exactly three singular points.
