Angenent torus

In differential geometry, the Angenent torus is a smooth embedding of the torus into three-dimensional Euclidean space, with the property that it remains self-similar as it evolves under the mean curvature flow. Its existence shows that, unlike the one-dimensional curve-shortening flow (for which every embedded closed curve converges to a circle as it shrinks to a point), the two-dimensional mean-curvature flow has embedded surfaces that form more complex singularities as they collapse.

The Angenent torus is named after Sigurd Angenent, who published a proof that it exists in 1992. However, as early as 1990, Gerhard Huisken wrote that Matthew Grayson had told him of "numerical evidence" of its existence.

To prove the existence of the Angenent torus, Angenent first posits that it should be a surface of revolution. Any such surface can be described by its cross-section, a curve on a half-plane (where the boundary line of the half-plane is the axis of revolution of the surface). Following ideas of Huisken, Angenent defines a Riemannian metric on the half-plane, with the property that the geodesics for this metric are exactly the cross-sections of surfaces of revolution that remain self-similar and collapse to the origin after one unit of time. This metric is highly non-uniform, but it has a reflection symmetry, whose symmetry axis is the half-line that passes through the origin perpendicularly to the boundary of the half-plane.

By considering the behavior of geodesics that pass perpendicularly through this axis of reflectional symmetry, at different distances from the origin, and applying the intermediate value theorem, Angenent finds a geodesic that passes through the axis perpendicularly at a second point. This geodesic and its reflection join up to form a simple closed geodesic for the metric on the half-plane. When this closed geodesic is used to make a surface of revolution, it forms the Angenent torus.

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