Simons cone
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In geometry and geometric measure theory, the Simons cone refers to a specific minimal hypersurface in \mathbb R^8 that plays a crucial role in resolving Bernstein's problem in higher dimensions. It is named after American mathematician Jim Simons.
In geometry and geometric measure theory, the Simons cone refers to a specific minimal hypersurface in \mathbb R^8 that plays a crucial role in resolving Bernstein's problem in higher dimensions. It is named after American mathematician Jim Simons.
Definition
The Simons cone is defined as the hypersurface given by the equation S = \{x \in \mathbb R^8 | x_1^2 + x_2^2 + x_3^2 + x_4^2 = x_5^2 + x_6^2 + x_7^2 + x_8^2 \} \subset \mathbb R^8 . This 7-dimensional cone has the distinctive property that its mean curvature vanishes at every point except at the origin, where the cone has a singularity.
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