The Bernstein conjecture, minimal cones and critical dimensions

Gary W. Gibbons, Kei Ichi Maeda, Umpei Miyamoto

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)


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.

Original languageEnglish
Article number185008
JournalClassical and Quantum Gravity
Issue number18
Publication statusPublished - 2009
Externally publishedYes

ASJC Scopus subject areas

  • Physics and Astronomy (miscellaneous)


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