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 welldefined 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 higherdimensional 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.
 Publication:

Classical and Quantum Gravity
 Pub Date:
 September 2009
 DOI:
 10.1088/02649381/26/18/185008
 arXiv:
 arXiv:0906.0264
 Bibcode:
 2009CQGra..26r5008G
 Keywords:

 High Energy Physics  Theory;
 General Relativity and Quantum Cosmology
 EPrint:
 19 pages, no figure