Immersed self-shrinkers

Abstract

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.

Shrinking Doughnuts via Variational Methods

“…Because γ ∞ is simple and closed, Theorem 1 gives an embedded self-shrinking Σ that is a topological torus for all n ≥ 2. The existence of toroidal self-shrinkers was first proved by Angenent [3] using a shooting method for geodesics (see also Drugan [4] and Drugan-Kleene [5] for immersed tori). Our proof here uses variational methods and we do not know if our tori coincide with Angenent's shrinking doughnuts.…”

A Survey of Closed Self-Shrinkers with Symmetry

“…1. Embedded T 3 self-shrinker in R 4 : A detailed analysis of (6.3), adapted from the crossing arguments in [16] and [20], confirms there is a t 2 > 0 so that the geodesic T [t] has the initial shape illustrated on the left of Figure 23 for t ≥ t 2 . Numerics show that there is a t 1 > √ 6 so that T [t 1 ] has the following initial shape: Figure 24.…”

“…A rich source of examples of immersed self-shrinkers comes from hypersurfaces with rotational symmetry. Using the shooting method for geodesics (see Section 5), an infinite number of complete, self-shrinkers for each of the rotational topological types: S n , S 1 ×S n−1 , R n , and S 1 × S n−1 were constructed in [20]. In the following, we introduce the geodesic equation for the profile curve of a self-shrinker with rotational symmetry and highlight a few modern examples of closed self-shrinkers.…”

“…• Immersed self-shrinkers: Building on the work in [9,16,43], infinitely many immersed and non-embedded self-shrinkers for each of the rotational topological types: S n , S 1 × S n−1 , R n , and S 1 × R n−1 were constructed in [20]. The main idea for the construction is to study the behavior of solutions to the geodesic equation near two known self-shrinkers and use continuity arguments to find complete self-shrinkers between them.…”

“…In the first part of this section, we outline the analysis used in [16] to construct an immersed sphere self-shrinker with rotational symmetry. Then, we illustrate how the behavior of geodesics for three shooting problems can be used to generate more examples of closed self-shrinkers with rotational symmetry [20]. We end this section with a discussion on the role of continuity in the shooting method.…”