Noncompact self-shrinkers for mean curvature flow with arbitrary genus

Abstract

In his lecture notes on mean curvature flow, Ilmanen conjectured the existence of noncompact self-shrinkers with arbitrary genus. Here, we employ min-max techniques to give a rigorous existence proof for these surfaces. Conjecturally, the self-shrinkers that we obtain have precisely one (asymptotically conical) end. We confirm this for large genus via a precise analysis of the limiting object of sequences of such self-shrinkers for which the genus tends to infinity. Finally, we provide numerical evidence for a further family of noncompact self-shrinkers with odd genus and two asymptotically conical ends.

Infinitely many pairs of free boundary minimal surfaces with the same topology and symmetry group

“…Our argument here relates to the analysis of the catenoidal annuli in Section 3.1 of the present article. Following the ideas in [39] we formulate a full proof of the convergence result Proposition D.3 with a similar approach as in Section 3 of [4]. Especially [4, Lemma 2.9], which we restate here for the convenience of the reader, is crucial for several of the arguments we are about to present.…”

“…= {(0, 0, t) : t ∈ R} and contains the horizontal disc B 2 (cf. Claim 1 in section 3 of[4]). With slight abuse of notation, let A g (in lieu of A Γg ) denote the second fundamental form of Γ g and B r (x) the open ball of radius r > 0 around some x ∈ B 3 .…”

“…4 } and likewise if we simultaneously replace B by K and B 2 by K b m,ξ . Using item (iii.ii) of Proposition 3.18, definition (4.22), and the estimates (4.29), we then get…”