Minimal $(n-2)$-umbilic submanifolds of the Euclidean space

TL;DR

This paper characterizes minimal $(n-2)$-umbilic submanifolds in Euclidean space, linking them to singular minimal surfaces, and provides a method for their local parametrization, expanding understanding of these geometric structures.

Contribution

It establishes a correspondence between minimal $(n-2)$-umbilic submanifolds and singular minimal surfaces, and develops a Weierstrass-type parametrization method for these surfaces.

Findings

Minimal $(n-2)$-umbilic submanifolds are $(n-2)$-rotational with singular minimal surface profiles.

A Weierstrass type method for local parametrization of singular minimal surfaces is developed.

All minimal $n$-dimensional hypersurfaces with a principal curvature of multiplicity $(n-2)$ are characterized.

Abstract

This paper investigates minimal $n$-dimensional submanifolds in the Euclidean space that are $(n-2)$-umbilic, meaning they carry an umbilical distribution of rank $n-2$. We establish a correspondence between the class of minimal ($n-2$)-umbilic submanifolds and the class of ($n-2$)-singular minimal surfaces. These surfaces are the critical points of its "energy potential" and have been previously studied in various contexts, including physics and architecture where, for instance, they model surfaces with minimal potential energy under gravitational forces. We show that minimal, generic, ($n-2$)-umbilic submanifolds, $n\geq4$, are ($n-2$)-rotational submanifolds whose profile is an ($n-2$)-singular minimal surface and vise versa. Furthermore, we develop a Weierstrass type method of local parametrization of all ($n-2$)-singular minimal surfaces, enabling a parametric description of all minimal $n$-dimensional, $n\geq4$, hypersurfaces of the Euclidean space with a nowhere vanishing principal curvature of multiplicity $n-2$.