In differential geometry, the Angenent torus is a smooth

embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is giv ...

of the

torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not ...

into three-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...

, with the property that it remains self-similar as it evolves under the

mean curvature flow In the field of differential geometry in mathematics, mean curvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold (for example, smooth surfaces in 3-dimensional Euclidean space). Intuitively, a family of s ...

. Its existence shows that, unlike the one-dimensional

curve-shortening flow In mathematics, the curve-shortening flow is a process that modifies a smooth curve in the Euclidean plane by moving its points perpendicularly to the curve at a speed proportional to the curvature. The curve-shortening flow is an example of a g ...

(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.

History

The Angenent torus is named after

Sigurd Angenent Sigurd Bernardus Angenent (born 1960) is a Dutch-born mathematician and professor at the University of Wisconsin–Madison. Angenent works on partial differential equations and dynamical systems, with his recent research focusing on heat equation a ...

, who published a proof that it exists in 1992.. However, as early as 1990,

Gerhard Huisken Gerhard Huisken (born 20 May 1958) is a German mathematician whose research concerns differential geometry and partial differential equations. He is known for foundational contributions to the theory of the mean curvature flow, including Huisk ...

wrote that Matthew Grayson had told him of "numerical evidence" of its existence..

Existence

To prove the existence of the Angenent torus, Angenent first posits that it should be a

surface of revolution A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation. Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending o ...

. 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 In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent spac ...

on the half-plane, with the property that the

geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...

s 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 In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two im ...

, 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 Simple or SIMPLE may refer to: * Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by John ...

closed geodesic In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that returns to its starting point with the same tangent direction. It may be formalized as the projection of a closed orbit of the geodesic fl ...

for the metric on the half-plane. When this closed geodesic is used to make a surface of revolution, it forms the Angenent torus. Other geodesics lead to other surfaces of revolution that remain self-similar under the mean-curvature flow, including spheres, cylinders, planes, and (according to numerical evidence but not rigorous proof) immersed topological spheres with multiple self-crossings. prove that the only complete smooth embedded surfaces of rotation that stay self-similar under the mean curvature flow are planes, cylinders, spheres, and topological tori. They conjecture more strongly that the Angenent torus is the only torus with this property.

Applications

The Angenent torus can be used to prove the existence of certain other kinds of singularities of the mean curvature flow. For instance, if a

dumbbell The dumbbell, a type of free weight, is a piece of equipment used in weight training. It can be used individually or in pairs, with one in each hand. History The forerunner of the dumbbell, halteres, were used in ancient Greece as lifting ...

shaped surface, consisting of a thin cylindrical "neck" connecting two large volumes, can have its neck surrounded by a disjoint Angenent torus, then the two surfaces of revolution will remain disjoint under the mean curvature flow until one of them reaches a singularity; if the ends of the dumbbell are large enough, this implies that the neck must pinch off, separating the two spheres from each other, before the torus surrounding the neck collapses.

Related shapes

Any shape that stays self-similar but shrinks under the mean curvature flow forms an

ancient solution In mathematics, an ancient solution to a differential equation is a solution that can be extrapolated backwards to all past times, without singularities. That is, it is a solution "that is defined on a time interval of the form .". The term was int ...

to the flow, one that can be extrapolated backwards for all time. However, the reverse is not true. In the same paper in which he published the Angenent torus, Angenent also described the

Angenent oval Angenent is a surname. Notable people with the surname include: *Henk Angenent (born 1967), Dutch speed skater *Sigurd Angenent (born 1960), Dutch-born American mathematician and professor **Angenent torus In differential geometry, the Angenent tor ...

s; these are not self-similar, but they are the only simple closed curves in the plane, other than a circle, that give ancient solutions to the

References

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External links

Angenent's torus
visualization by Dongsun Lee of UNIST Mathematical Sciences Differential geometry of surfaces