Niels Martin Moller -- Mean Curvature Flow - Pictures of Mean Curvature Self-Shrinkers

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Embedded Mean Curvature Flow Self-Shrinkers in R^3

A self-shrinker under Mean Curvature Flow is a smooth hypersurface in R^3 satisfying this nonlinear elliptic partial differential "soliton" equation:

H = <X,v>/2,

where H is the mean curvature, X is the position vector and v is a unit normal vector field. Such a surface is also a minimal surface in R^3 (more generally R^{n+1}), but w.r.t. a Gaussian metric.

Here follows a list of all known embedded, (half-)complete self-shrinkers in Euclidean 3-space [last updated Nov 30th, 2011]:

1. Flat hyperplanes

2. Round cylinders R x S^1

3. Round 2-sphere S^2

4. Angenent's (non-round) torus self-shrinker (~1988)

	5. Ilmanen's "Planosphere" self-shrinkers (~1994 & 2010) Complete, embedded of genus g. One non-compact end, asymptotic to a regular cone w/ 4g+g symmetries. Converges to the union of a sphere and a plane, as g → ∞. Ref.: [1] T. Ilmanen. Lectures on Mean Curvature Flow and Related Equations, Conference on Partial Differential Equations & Applications to Geometry, 1995, ICTP, Trieste. [2] N. Kapouleas, S. Kleene, N.M. Møller. Mean curvature self-shrinkers of high genus: Non-compact examples. [Rigorous proof of existence/construction via minimal surface gluing. Link: arxiv:1106.5454]
	6. Self-shrinker "Trumpets": Asymptotically conical ends w/ rotational symmetry (2009) Embedded, non-compact ends (not complete). Asymptotic to symmetric cones, interpolating between plane and cylinder (unique shrinker for each such cone). Ref.: [3] S. Kleene, N.M. Møller. Self-shrinkers with a rotational symmetry. [Link: arxiv:1008.1609]
	7. Self-shrinker "Toruspheres" (2011) Embedded, closed of even genus g and 2g symmetries. Ref.: [4] N.M. Møller. Closed self-shrinking surfaces in R^3 via the torus. [Link: arxiv:1111.7318]