The Bernstein Conjecture, Minimal Cones, and Critical Dimensions

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.

19 pages, no figure; v2: comments added in footnote 1 and 2

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Cited by in corpus (6)

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• The universality of one half in commutative nonassociative algebras with identities
• Thick brane solutions and topology change transition on black hole backgrounds
• Stability of branes trapped by d-dimensional black holes
• Uniqueness of static spherically symmetric vacuum solutions in the IR limit of Hořava-Lifshitz gravity