In this paper we present a new family of non-compact properly embedded, self-shrinking, asymptotically conical, positive mean curvature ends Σ n ⊆ R n+1 that are hypersurfaces of revolution with circular boundaries. These hypersurface families interpolate between the plane and half-cylinder in R n+1 , and any rotationally symmetric self-shrinking non-compact end belongs to our family. The proofs involve the global analysis of a cubic-derivative quasi-linear ODE.We also prove the following classification result: a given complete, embedded, self-shrinking hypersurface of revolution Σ n is either a hyperplane R n , the round cylinder R × S n−1 of radius 2(n − 1), the round sphere S n of radius √ 2n, or is diffeomorphic to an S 1 × S n−1 (i.e. a "doughnut" as in the paper by Sigurd B. Angenent, 1992, which when n = 2 is a torus). In particular, for self-shrinkers there is no direct analogue of the Delaunay unduloid family. The proof of the classification uses translation and rotation of pieces, replacing the method of moving planes in the absence of isometries.

Abstract. We give the first rigorous construction of complete, embedded self-shrinking hypersurfaces under mean curvature flow, since Angenent's torus in 1989. The surfaces exist for any sufficiently large prescribed genus g, and are non-compact with one end. Each has 4g + 4 symmetries and comes from desingularizing the intersection of the plane and sphere through a great circle, a configuration with very high symmetry.Each is at infinity asymptotic to the cone in R 3 over a 2π/(g + 1)-periodic graph on an equator of the unit sphere S 2 ⊆ R 3 , with the shape of a periodically "wobbling sheet". This is a dramatic instability phenomenon, with changes of asymptotics that break much more symmetry than seen in minimal surface constructions.The core of the proof is a detailed understanding of the linearized problem in a setting with severely unbounded geometry, leading to special PDEs of Ornstein-Uhlenbeck type with fast growth on coefficients of the gradient terms. This involves identifying new, adequate weighted Hölder spaces of asymptotically conical functions in which the operators invert, via a Liouville-type result with precise asymptotics.

Abstract. We construct infinitely many complete, immersed self-shrinkers with rotational symmetry for each of the following topological types: the sphere, the plane, the cylinder, and the torus.

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Abstract. Given a compact closed subset M of a line segment in R 3 , we construct a sequence of minimal surfaces Σ k embedded in a neighborhood C of the line segment that converge smoothly to a limit lamination of C away from M . Moreover, the curvature of this sequence blows up precisely on M , and the limit lamination has non-removable singularities precisely on the boundary of M .