Minimal submanifolds confined in space

TL;DR

This paper demonstrates that minimal submanifolds confined in space are highly restricted, establishing structural results and growth conditions that extend classical Bernstein theorems across all dimensions.

Contribution

It proves that confined minimal submanifolds with sublinear height growth must have Euclidean volume growth, generalizing Bernstein-type results to higher dimensions.

Findings

Minimal submanifolds confined in space are structurally restricted.

Proper minimal immersions with sublinear height growth have Euclidean volume growth.

An optimal Bernstein theorem is established for stable hypersurfaces with sublinear height growth.

Abstract

Already in $\bf{R}^4$, there are many known examples of minimal hypersurfaces, yet few structural results. We show that minimal submanifolds, of any dimension, that are confined in space are very restricted. It is well-known that the half-space theorem fails already for hypersurfaces in $\bf{R}^4$, where there are many examples contained in a slab. In $\bf{R}^3$ the height of the catenoid grows at a logarithmic rate, whereas in higher dimension the height of the catenoid remains bounded. We will see that even in high dimensions, minimal submanifolds that are confined in space must satisfy strong structural restrictions. We show that any proper minimal immersion whose height grows sublinearly must have Euclidean volume growth. A consequence is an optimal Bernstein theorem in any dimension for stable hypersurfaces with sublinearly growing height that generalizes results of Moser, Bombieri-De Giorgi-Miranda, Trudinger, Caffarelli-Nirenberg-Spruck and Ecker-Huisken.