Smooth Compactness of Self-shrinkers

We prove a smooth compactness theorem for the space of embedded selfshrinkers in R. Since self-shrinkers model singularities in mean curvature flow, this theorem can be thought of as a compactness result for the space of all singularities and it plays an important role in studying generic mean curvature flow. 0. Introduction A surface Σ ⊂ R is said to be a self-shrinker if it satisfies (0.1) H = 〈x,n〉 2 , where H = divn is the mean curvature, x is the position vector, and n is the unit normal. This is easily seen to be equivalent to that Σ is the t = −1 time-slice of a mean curvature flow (“MCF”) moving by rescalings, i.e., where the time t slice is given by √ −tΣ. Self-shrinkers play an important role in the study of mean curvature flow. Not only are they the simplest examples (those where later time slices are rescalings of earlier), but they also describe all possible blow ups at a given singularity of a mean curvature flow. The idea is that we can rescale a MCF in space and time to obtain a new MCF, thereby expanding a neighborhood of the point that we want to focus on. Huisken’s monotonicity, [H3], and Ilmanen’s compactness theorem, [I1], give a subsequence converging to a limiting solution of the MCF; cf. [I1], [W1]. This limit, which is called a tangent flow , achieves equality in Huisken’s monotonicity and, thus, its time t slice is √ −tΣ where Σ is a self-shrinker. The main result of this paper is the following smooth compactness theorem for selfshrinkers in R that is used in [CM1]. Theorem 0.2. Given an integer g ≥ 0 and a constant V > 0, the space of smooth complete embedded self-shrinkers Σ ⊂ R with • genus at most g, • ∂Σ = ∅, • Area (BR(x0) ∩ Σ) ≤ V R for all x0 ∈ R and all R > 0 is compact. Namely, any sequence of these has a subsequence that converges in the topology of C convergence on compact subsets for any m ≥ 2. The surfaces in this theorem are assumed to be homeomorphic to closed surfaces with finitely many disjoint disks removed. The genus of the surface is defined to be the genus of the corresponding closed surface. For example, an annulus is a sphere with two disks The authors were partially supported by NSF Grants DMS 0606629 and DMS 0405695. 1In [H3], self-shrinkers are time t = − 1 2 slices of self-shrinking MCFs; these satisfy H = 〈x,n〉. 1 2 TOBIAS H. COLDING AND WILLIAM P. MINICOZZI II removed and, thus, has genus zero. Below, we will use that the genus is monotone in the sense that if Σ1 ⊂ Σ2, then the genus of Σ1 is at most that of Σ2. As mentioned, the main motivation for this result is that self-shrinkers model singularities in mean curvature flow. Thus, the above theorem can be thought of as a compactness result for the space of all singularities. In practice, scale-invariant local area bound, smoothness, and the genus bound will automatically come from corresponding bounds on the initial surface in a MCF. Namely: • Area bounds are a direct consequence of Huisken’s monotonicity formula, [H3]. • Ilmanen proved that in R tangent flows at the first singular time must be smooth and have genus at most that of the initial surface; see theorem 2 of [I1] and page 21 of [I1], respectively. Conjecturally, the smoothness and genus bound hold at all singular times: • Ilmanen conjectured that tangent flows are smooth and have multiplicity one at all singularities. If this conjecture holds, then it would follow from Brakke’s regularity theorem that near a singularity the flow can be written as a graph of a function with small gradient over the tangent flow. Combining this with the above mentioned monotonicity of the genus of subsets and a result of White, [W3], asserting that the genus of the evolving surfaces are always bounded by that of the initial surface, we get conjecturally that the genus of the tangent flow is at most that of the initial surface. Our compactness theorem will play an important role in understanding generic mean curvature flow in [CM1]. Namely, in [CM1], we will see that it follows immediately from compactness together with the classification of (entropy) stable self-similar shrinkers proven in [CM1] that given an integer m and δ > 0, there exists an ǫ = ǫ(m, δ, V, g) > 0 such that: • For any unstable self-similar shrinker in R satisfying the assumptions of Theorem 0.2, there is a surface δ-close to it in the C topology and with entropy less than that of the self-similar shrinker −ǫ. This is, in particular, a key to showing that mean curvature flow that disappears in a compact point does so generically in a round point; see [CM1] for details and further applications. The simplest examples of self-shrinkers in R are the plane R, the sphere of radius 2, and the cylinder S ×R1 (where the S has radius √ 2). Combining [H3], [H4], and theorem 0.18 in [CM1] it follows that these are the only smooth embedded self-shrinkers with H ≥ 0 and polynomial volume growth. It follows from this that spheres and cylinders are isolated (among all self-shrinkers) in the C-topology. On the other hand, by Brakke’s theorem, [Br], any self-shrinker with entropy sufficiently close to one (which is the entropy of the plane) must be flat, so planes are also isolated and we see that all three of the simplest self-shrinkers are isolated. Moreover, one of the key results of [CM1] (see theorem 0.7 there) was to show that these are the only (entropy) stable self-shrinkers. In sum, self-shrinkers either with H ≥ 0 or that are stable are one of the three simplest types and all of those are isolated 2See, for instance, corollary 2.13 in [CM1]. 3Huisken, [H3], [H4], showed that these are the only smooth embedded self-shrinkers with H ≥ 0, |A| bounded, and polynomial volume growth. In [CM1], we prove that this is the case even without assuming a bound on |A|. COMPACTNESS OF SELF-SHRINKERS 3 among all self-shrinkers. 4 However, there are expected to be many examples of self-shrinkers in R where H changes sign or that are unstable. In particular, Angenent, [A], constructed a self-shrinking torus of revolution and there is numerical evidence for a number of other examples; cf. Chopp, [Ch], Angenent-Chopp-Ilmanen, [AChI], Ilmanen, [I2], and Nguyen, [N1], [N2]. These examples suggest that compactness fails to hold without a genus bound. There are three key ingredients in the proof of the compactness theorem. The first is a singular compactness theorem that gives a subsequence that converges to a smooth limit away from a locally finite set of points. Second, we show that if the convergence is not smooth, then the limiting self-shrinker is L-stable, where L-stable means that for any compactly supported function u we have