Bernstein猜想、极小锥和临界维数
The Bernstein Conjecture, Minimal Cones, and Critical Dimensions
摘要：
极小曲面和畴壁在时空物理和材料科学的各种背景下起着重要的作用。本文首先回顾了Bernstein猜想及其在高维空间中的失败，Bernstein猜想认为平面是平坦空间中超平面上的单值图的极小曲面方程的唯一全局良定解。然后，我们回顾如何在四维和高维时空，这是弯曲的，甚至奇异的顶点，最小锥，可用于提供反例的猜想。这些反例在弯曲时空中的物理意义从不同的角度进行了讨论，从经典的广义相对论，膜物理，和全息模型的基本相互作用。
英文摘要：
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.
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