The Bernstein Conjecture, Minimal Cones, and Critical Dimensions

TL;DR

This paper reviews the Bernstein conjecture on minimal surfaces, explores counterexamples in higher dimensions using minimal cones, and discusses their implications in physics such as general relativity and brane models.

Contribution

It provides a comprehensive review of the Bernstein conjecture, highlights counterexamples in higher dimensions, and discusses their physical significance across multiple theoretical frameworks.

Findings

Bernstein conjecture holds in low dimensions but fails in higher dimensions.

Minimal cones serve as counterexamples to the conjecture in curved spacetimes.

Counterexamples have implications for theories in general relativity and brane physics.

Abstract

Minimal surfaces and domain walls play important roles in various contexts of spacetime physics as well as material science. In this paper, we first review the Bernstein conjecture, which asserts that a plane is the only globally well defined solution of the minimal surface equation which is a single valued graph over a hyperplane in flat spaces, and its failure in higher dimensions. Then, we review how minimal cones in four- and higher-dimensional spacetimes, which are curved and even singular at the apex, may be used to provide counterexamples to the conjecture. The physical implications of these counterexamples in curved spacetimes are discussed from various points of view, ranging from classical general relativity, brane physics, and holographic models of fundamental interactions.