The space of asymptotically conical self-expanders of mean curvature flow

Abstract

We show that the space of asymptotically conical self-expanders of the mean curvature flow is a smooth Banach manifold. An immediate consequence is that nondegenerate self-expanders -that is, those self-expanders that admit no non-trivial normal Jacobi fields that fix the asymptotic cone -are generic in a certain sense.

Rotational symmetry of solutions of mean curvature flow coming out of a double cone II

“…To conclude the section, we give a sufficient condition when there exists a flow coming out of C that has a singularity. The proof makes heavy use of the structure theory of self-expanders developed by Bernstein and Wang in a series of papers starting from [BW21]. It would be interesting to see if a simpler proof exists.…”

“…Let σ = L(C), and let W be the connected component of S n lying between the two connected components of σ. By Corollary 1.2 of [BW21], the set of generic cones (in the sense that there is no C 2 -asymptotically self-expander with nontrivial Jacobi fields that fix the infinity) whose link lie in W , is dense near C. These facts allow us to take a sequence of C 2,α -hypersurfaces σ i in S 2 such that…”

Topological Uniqueness for Self-expanders of Small Entropy

“…For the reader's convenience, we recall, in Sections 2.1-2.7, some of the notation and background introduced in our previous works [7,9]. In Section 2.8 we define an a.c.isotopy between two asymptotically conical hypersurfaces and discuss some basic properties of a.c.-isotopies.…”

“…In Section 6 we use a perturbation by the first eigenfunction of the stability operator for self-expanders together with results of the preceding section to deform any low entropy asymptotically conical unstable self-expander, in the a.c.-isotopy class and preserving the asymptotic cone, to a stable self-expander. In Section 7 we apply the analysis carried out in our previous work [7] and results from Section 5 to show that one may connect, via an a.c.-isotopy that does not move the asymptotic cones much along the path, any weakly stable self-expander to a self-expander asymptotic to a cone which is a generic perturbation of the asymptotic cone of the initial self-expander. In Section 8 we complete the proof of Theorem 1.1 and Theorem 1.2.…”

Rotational symmetry of solutions of mean curvature flow coming out of a double cone

“…We suspect that counterexamples exist. We refer to [BW17], [BW18], [BW19a], [BW20], and [Din19] for more information on self-expanders.…”