AN UNKNOTTEDNESS RESULT FOR NONCOMPACT SELF SHRINKERS

ALEXANDER MRAMOR

Abstract. In this article we extend an unknottedness theorem for compact self shrinkers to the mean curvature flow to shrinkers with finite topology and one asymptotically conical end, which conjecturally comprises the entire set of self shrinkers with finite topology and one end. The mean curvature flow itself is used in the argument presented.

1. Introduction Self shrinkers are the most basic singularity models to the mean curvature flow and hence are an important topic of study. In this article we extend (and reprove) the results of [61], where the author with S. Wang showed compact self shrinkers in R3 are topologically standard, to include some noncompact self shrinkers: Theorem 1.1. Let M 2 ⊂ R3 be a two-sided, possibly noncompact, self shrinker with no more than one end. Then if M has an asymptotically conical end or is compact, it is topologically standard. In [44] Ilmanen conjectured that a self shrinker with a cylindrical end must its self be the round shrinking cylinder, so in light of L. Wang’s theorem on the ends of noncompact self shrinkers with finite topology shown in [66], that the ends of such self shrinkers must be either cylindrical or conical, it seems reasonable that our result covers all self shrinkers with one end and finite topology. Note her result [67], for when a shrinker is very quickly asymptotic to a cylinder, gives some concrete evidence supporting Ilmanen’s cylinder conjecture. From the desingularization of the sphere and plane by Kapouleas, Kleene, and Møller [50] we see that the set of asymptotically conical shrinkers with one end and finite topology is nonempty and contains elements with nontrivial topology. arXiv:2005.01688v8 [math.DG] 31 Dec 2020 From theorem 2 of Brendle’s “genus 0 uniqueness” paper [6] noncompact shrinkers where any two curves have vanishing mod 2 intersection number must be the cylinder or plane and hence unknotted; otherwise to the author’s knowledge no other unknot- tedness results for noncompact self shrinkers are known besides L. Wang’s cylinder theorem (aside from symmetry or curvature convexity assumptions on the shrinker like mean convexity). The definition of standard embeddedness is given in section 4 1 UNKNOTTEDNESS 2 below but in layman’s terms it essentially means that a surface is “unknotted:” for example tubular neighborhoods of knotted S1 ⊂ R3 are not topologically standard. For technical reasons an argument appealing directly the Frankel-type theorems for self shrinkers, Frankel theorems being perhaps the most natural geometric point of contact for these types of statements, encounters some difficulties (at least until the end of our proof); these issues and related literature and techniques (including the Morse-theoretic ones as well) in the classical minimal surface case (of course, the proofs in the classical case don’t carry over immediately) are discussed in depth in the concluding remarks below. To work around these issues we will use the (renormal- ized) mean curvature flow. In a nutshell, its use will be that if a shrinker satisfying the conditions of theorem 1.1 is in fact topologically nonstandard, we may perturb it and use the flow described in the statement above to find another shrinker which must have impossible properties (stable but polynomial volume growth, alternately disjoint from the original shrinker). Note that what one might call elliptic techniques, in particular minimizing in an isotopy class as in Meeks, Simon, and Yau [57] aren’t immediately applicable because for instance (1) the Gaussian metric (for which self shrinkers are minimal) is incomplete and, less importantly, (2) they consider only isotopy classes with compact representatives. Besdes being of independent interest the flow provides a way to sidestep these issues faced by elliptic methods, which are discussed in more detail (along with how they could possibly be dealt with to give a different proof than given in this paper) towards the end of the concluding remarks as well. More precisely, the main result we use to carry this scheme out is the following shown in section 3:

3 X⊥ Theorem 1.2. Let M ⊂ R be an asymptotically conical surface such that H − 2 ≥ c(1 + |X|2)−α for some c, α > 0, and so that as R → ∞ |A(p)|2 → 0 for any c p ∈ M ∩ B(p, R) . Then there is a level set flow Lt out of M with respect to the renormalized mean curvature flow which is

(1) inward in that Lt1 ⊂ Lt2 for any t1 > t2. k (2) the Hausdorff limit of surgery flows St with initial data M.

The definitions of weak flows needed are given in section 3. Lt will also satisfy a number of other more technical properties (ex. support of a Brakke flow) which we discuss in the section as well. The level set flow for compact sets under the renormalized flow (and with more general forcing terms) has been well studied, see for example the work of Hershkovits and White [36] (see also [35] by the same authors for a use of the renormalized flow in studying the homotopy groups of self shrinkers). The renormalized mean curvature flow on noncompact hypersurfaces seems to be less well studied in situations where singularity formation is not ruled out, here we do so in a rather particular context. The point of theorem 1.2 is that we give an UNKNOTTEDNESS 3 explicit construction of the level set flow via a limit of (renormalized) mean curvature flows with surgery to ensure it has the properties one would probably expect to hold, drawing ideas from the author’s previous paper [59] and his joint work with S. Wang [62]. Many of these properties also seem to follow from section 7 of the very recent preprint [12] (which appeared roughly at the same time this work was being completed) but we point out that the construction of the level set flow as a limit of surgery flows, besides potentially being of independent interest, is also useful to show the flow is nonempty (see lemma 4.4 below) in its ultimate application to the proof of theorem 1.1: this isn’t very hard but it doesn’t seem to follow directly from preexisting literature on topological change under the flow (e.g. [71]). Acknowledgements: The author is grateful to Bill Minicozzi for asking him about unknottedness for noncompact shrinkers at the 2018 Barrett lectures held at the University of Tennessee–Knoxville, which inspired this work. He also thanks Jacob Bernstein, Letian Chen, Martin Lesourd, Peter McGrath, Joel Spruck, Ao Sun, Ryan Unger, Shengwen Wang, and Jonathan Zhu for stimulating discussions and feedback during various stages of this project.

2. A brief introduction the mean curvature flow In this section we discuss facts, some deep, concerning the mean curvature flow and self shrinkers which we will use below – note though that we elect to postpone some “preliminaries,” particularly weak notions of the flow, to other sections where they seem to fit more naturally into the discussion. Let X : M → N n+1 be an embedding of M realizing it as a smooth closed hypersurface of N, which by abuse of notation we also refer to as M. Then the mean curvature flow of Mt is given by the image of (overloading notation) X : M × [0,T ) → N n+1 satisfying where ν is the outward normal: dX = H~ = −Hν, X(M, 0) = X(M) (2.1) dt This is an interesting flow to consider for a variety of reasons, for example as a tool in topology – for a survey see [17]. There is a comparison principle for the mean cuvature flow, and for example by enclosing any compact hypersurface in Rn+1 with a sufficiently large sphere it is easy to see that singularities are common for mean curvature flows in Euclidean space. Generically the only noncompact singularities encountered will be modeled on round cylinders: generic mean curvature flow in R3 has been already rather well developed (see [13] and [12]) although there still might be situations where one is forced to consider “exotic” singularities, for example in potential applications of the flow to a family of surfaces considered simultaneously. To study these singularities, one may perform a tangent flow blowup which, as described by Ilmanen in his preprint [42] for flows of surfaces, will be modeled on UNKNOTTEDNESS 4 smooth self shrinkers: these are surfaces satisfying the following equivalent defini- tions: n n+1 X⊥ (1) M ⊂ R which satisfy H − 2 = 0, where X is the position vector −|x|2 (2) minimal surfaces in the Gaussian metric Gij = e 2n δij (3) surfaces M which√ give rise to ancient flows Mt that move by dilations by setting Mt = −tM Of course, as the degenerate neckpinch of Angenent and Velasquez [3] illustrate that tangent flows do not capture quite all the information about a developing sin- gularity but they are a natural starting point. The Gaussian metric is a poorly behaved metric in many regards; it is incomplete and by the calculations in [14] its scalar curvature at a point x is given by:

2   |x| n − 1 2 R = e 2n n + 1 − |x| (2.2) 4n We see that as |x| → ∞ the scalar curvature diverges, so there is no way to complete the metric. Also since R is positive for |x| small and negative for large |x|, there is no sign on sectional or Ricci curvatures. On the other hand it is f-Ricci positive, in the 1 2 sense of Bakry and Emery with f = − 2n |x| , suggesting it should satisfy many of the same properties of true Ricci positive metrics (see [68]). Indeed, this provides some idea as to why one might expect an unknottedness result for self shrinkers, because analogous unknottedness results hold in Ricci positive metrics on S3 (as discussed in more detail below). Crucial for our arguments, it turns out there are no stable minimal surfaces of polynomial volume growth in Rn endowed with the Gaussian metric as discussed in [14]. To see why this is so, the Jacobi operator for the Gaussian metric is given by: 1 1 L = ∆ + |A|2 − hX, ∇(·)i + (2.3) 2 2 1 The extra 2 term is essentially the reason such self shrinkers unstable in the Gaussian metric: for example owing to the constant term its clear in the compact case from this that one could simply plug in the function “1” to get a variation with Lu > 0 which doesn’t change sign implying the first eigenvalue is negative. To deal with this instability, in [13] Colding and Minicozzi introduced their entropy functional which is essentially an area that mods out by translations and dilations. They define the entropy λ(M) of M n ⊂ Rn+1 to be:

λ(M) = sup Fx0,r(M) (2.4) x0,r where the functionals Fx0,r are Gaussian areas shifted by x0 and rescaled by r – although it doesn’t concern us there are indeed entropy stable shrinkers namely UNKNOTTEDNESS 5 round spheres and cylinders. What does concern us is that the entropy by Huisken monotonicity [38] is nonincreasing under the flow and as shown lemma 2.9 in [13] a surface with finite entropy has polynomial volume growth. And in fact, every properly embedded shrinker has polynomial volume growth by Q. Ding and Y.L. Xin: Theorem 2.1 (Theorem 1.1 of [19]). Any complete non-compact properly immersed self-shrinker M n in Rn+m has Euclidean volume growth at most. We will combine these facts below to conclude the self shrinker we find via the renormalized flow is unstable in the Gaussian metric. Now we discuss some termi- nology describing possible behavior of the ends: 3 A regular cone in R is a surface of the form Cγ = {rγ}r∈(0,∞) where γ is smooth simple closed curve in S2. An end of a surface M 2 ,→ R3 is asymptotically conical 2 with asymptotic cross section γ if ρM → Cγ in the Cloc sense of graphs as ρ & 0 restricted to that end. Similarly we define asymptotically cylindrical ends to be ends which are asymptot- ically graphs over cylinders (with some precsribed axis and diameter) which converge 2 to that cylinder in Cloc on that end. The reason we focus on such ends is the following important result of L. Wang, which says that these are the only possible types of ends which may arise in the case of finite topology: Theorem 2.2 (theorem 1.1 of [66]). If M is an end of a noncompact self-shrinker in R3 of finite topology, then either of the following holds: −1 ∞ 3 3 (1) limτ→∞ τ M = C(M) in Cloc(R \ 0) for C(M) a regular cone in R −1 1 ∞ 3 3 (2) limτ→∞ τ (M − τv(M)) = Rv(M) × S in Cloc(R ) for a v(M) ∈ R \{0} In particular, theorem 2.2 applies to self shrinkers which arises as the tangent flow to compact mean curvature flows, although it is true one should expect shrinkers with more than one end to appear in a general blowup (for a trivial example consider a neckpinch). We end this discussion with a pseudolocality theorem. Pseudolocality roughly says that far away points are less consequential under the flow than nearby ones no matter their curvature and is a concrete artifact of the nonlinearity of the flow. In our case it is a consequence of the Ecker-Huisken estimates [20] but we give the formulation of B.L. Chen and L. Yin (see theorem 1.4 in [47] for a proof in Rn by controlling Gaussian densities). It will be heavily used in the extension of the flow with surgery used below: Theorem 2.3 (Theorem 7.5 of [10]). Let M be an n-dimensional manifold satisfying 3 P i 2 |∇ Rm| ≤ c0 and inj(M) ≥ i0 > 0. Then there is  > 0 with the following i=0 UNKNOTTEDNESS 6 property. Suppose we have a smooth solution Mt ⊂ M to the MCF properly embedded 2 2 in BM (x0, r0) for t ∈ [0,T ] where r0 < i0/2, 0 < T ≤  r0. We assume that at −1 time zero, x0 ∈ M0, and the second fundamental form satisfies |A|(x) ≤ r0 on M0 ∩ BM (x0, r0) and assume M0 is graphic in the ball BM (x0, r0). Then we have −1 |A|(x, t) ≤ (r0) (2.5) for any x ∈ BM (x0, r0) ∩ Mt, t ∈ [0,T ].

3. The renormalized mean curvature flow In this section we discuss the renormalized mean curvature flow (which we’ll abre- viate RMCF) ultimately to construct, via an adapted surgery flow, an inward level set flow for the RMCF; using the same notation as in the section above for surface M ⊂ R3 the RMCF is given by: dX X = H~ + (3.1) dt 2 Modding out by tangential directions of the flow makes the speed of the flow more transparent and is geometrically equivalent to 3.1: dX X⊥ = −(H − )ν (3.2) dt 2 Where here as before X is the position vector on M. It is related to the regular mean curvature flow by the following reparameterization; this will allow us to transfer many deep analytical properties of the MCF to the RMCF. Supposing that Mt is a mean curvature flow on [−1,T ), −1 < T ≤ 0 (T = 0 is the case for a self shrinker). ˆ Then the renormalized flow Mτ of Mt defined on [0, − log(−T )) is given by ˆ τ/2 Xτ = e X−e−τ , τ = − log (−t) (3.3) This is a natural flow for us to consider because it is up to a multiplicative term the gradient flow of the Gaussian area and fixed points with respect to it are precisely self shrinkers. More precisely, writing HG for the mean curvature of a surface with respect to the Gaussian metric: ⊥ |x|2 X H = e 4 (H − ) (3.4) G 2

Its clear from this that the RMCF should be better behaved then the MCF in the Guasisan metric then because of the missing exponential factor in the speed of the flow; in fact the surfaces we consider in the sequel will be well behaved with respect UNKNOTTEDNESS 7 to the RMCF but will have unbounded mean curvature in the Gaussian metric. Also since t = −e−τ , H = eτ/2Hˆ , and X⊥ = e−τ/2Xˆ ⊥ we have: X⊥ Xˆ ⊥ − tH + → e−τ/2(Hˆ − ) (3.5) 2 2 Under the reparameterization 3.3 above; this will be important in the sequel as well. Note that throughout when we refer only to the RMCF we will use the notation typical to the MCF (i.e. t instead of τ, etc.). Our main object of study in this section will be the following set, which we bold for emphasis; the asymptotics assumed are inspired by Bernstein and Wang [4] for use with Ecker and Huisken’s noncompact maximum principle in [20] as we’ll see shortly:

Definition 3.1. Denote by Σ the set of asymptotically conical hypersurfaces in R3 X⊥ 2 −α for which H − 2 ≥ c(1 + |X |) for some c, α > 0.

X⊥ Throughout, say that M is shrinker mean convex if H − 2 ≥ 0 at all points on M. First we note that short time existence of the RMCF of these surfaces:

Lemma 3.1. If M ∈ Σ then there exists some  > 0 for which the RMCF Mt of M exists for t ∈ [0, ). Proof: We can flow an element in Σ by the regular MCF for a short time by Ecker- Huisken [20]; then apply the reparameterization 3.3 to get a solution for short times for the RMCF. 

Our next lemma is that shrinker mean convexity is preserved under the RMCF and closely follows lemma 3.2 of [4]; in our future application to the flow with surgery, note that this must be reapplied (starting with t = 0) after every surgery time since high curvature regions will be removed:

Lemma 3.2. Let Mt be a smooth flow under RMCF on [0,T ]. Then if it is initially in Σ it remains so under the MCF and in fact: X⊥ (H − ) > ce−t/2(1 + |X|2 + 2n(t + 1))−α (3.6) 2 Proof: This could be deduced from the relation 3.5 above along with lemma 3.2 of [4], but we check it directly. Following the computation for the mean curvature flow, the evolution equation for the mean curvature is: dH H = ∆H + |A|2H − (3.7) dt 2 UNKNOTTEDNESS 8

X⊥ d X⊥ Now we compute ∆ 2 and dt 2 : X⊥ H X⊥ X ∆ = − − |A|2 − h , ∇Hi (3.8) 2 2 2 2

d X⊥ X H X⊥ X⊥ X⊥ X⊥ = −h , ∇Hi − + = ∆ + |A|2 + (3.9) dt 2 2 2 4 2 2 4 X⊥ Putting this together we get the evolution equation for H − 2 : d X⊥ X⊥ X⊥ 1 X⊥ (H − ) = ∆(H − ) + |A|2(H − ) − (H − ) (3.10) dt 2 2 2 2 2

t/2 X⊥ df 2 Hence f = e (H − 2 ) satisfies the dt = ∆f + |A| f. Now following [4] consider the following function:

η(X, t) = 1 + |X|2 + 2n(t + 1) (3.11)

It solves the heat equation, and d ( − ∆)ηα = −α(α − 1)|∇ log η|2ηα (3.12) dt Setting u = ηαf, we get that d ( − ∆)u + 2α∇ log η · ∇u = |A|2u + α(α + 1)|∇ log η|2u (3.13) dt Note since the flow is smooth by pseudolocality (lemma 3.5 below) there certainly exists C such that

C = sup sup |A| + |∇A| + |∇2A| < ∞ (3.14) t∈[0,T ] Mt

Psuedolocality applied to the corresponding regular MCF implies the same for X⊥ as well because the surface will stay asymptotically conical. Since asymptotically conical surfaces have finite entropy, the result follows by the Ecker Huisken noncompact maximum principle [20] as modified/specialized in Theorem A.1 of [4]. 

To recap, we have shown the following for elements of Σ so far:

Lemma 3.3. For M ∈ Σ, a smooth RMCF exists for some short time and its shrinker mean convexity is preserved with the lower bound 3.6. UNKNOTTEDNESS 9

3.1. The renormalized mean curvature flow with (localized) surgery. Our goal is to construct an inward level set flow out of M ∈ Σ by the RMCF. To do that we will start by constructing a mean curvature flow with surgery out of M. Giving a brief account of the surgery flow, recall that in the mean curvature flow with surgery one finds for a mean convex surface M (in higher dimensions, 2-convex) curvature scales Hth < Hneck < Htrig so that when H = Htrig at some point p and time t, the flow is stoped and suitable points where H ∼ Hneck are found to do surgery where “necks” (at these points the surface will be approximately cylindrical) are cut and caps are glued in. The high curvature regions are then topologically identified and discarded and the low curvature regions will have mean curvature bounded on the order of Hth (we give a more detailed qualitative description which meets our needs in lemma 4.4 below). The flow is then restarted and the process repeated. There are a couple different approaches on the construction of the mean curvature flow with surgery (see the work [41] of Huisken and Sinestrari for the original paper on MCF with surgery and the paper of Brendle and Huisken [7] for its extension to n = 2); here we will follow Haslhofer as their results are local in nature. There the curvature thresholds are in turn determined by the parameters αp = (α, β, γ). Here α is a noncollapsing constant: we say a surface is α noncollapsed if there are inner and outer osculating balls of radius (at least) α/H; Andrews and Sheng and Wang [1, 64] independently showed this is preserved under the MCF. β is a 2-convexity assumption which for our case is set to 1 (since we are only involved with surfaces in R3), and γ is an initial bound on mean curvature. X⊥ For our purposes, we will replace the role of H with F = H − 2 and say surfaces which are noncollapsed with respect to F are F α-noncollapsed; recall from above that convexity of F with respect to the renormalized flow is preserved. We discuss now for the sake of exposition F -noncollapsing under the RMCF in just the compact case: Lemma 3.4. Suppose M is a compact manifold which is F α-noncollapsed and consider Mt, the flow of M under the renormalized mean curvature flow. Then there is a function C(t) > 0 depending on α only with C(0) = α for which Mt will be F α-noncollapsed with with constant C(t). Proof: In remark (7) of [1] Andrews notes that noncollapsing is preserved under the df 2 (regular) mean curvature flow for positive functions f satisfying dt = ∆f +|A| f (see also [2] for more general homogeneous flows and [53] for general Haslhofer-Kleiner X⊥ type curvature estimates). Noting that f = −tH + 2 is such a function in our df 2 setting (with respect to the regular MCF, dt = ∆f + |A| f due to Smoczyk [63]), noncollapsing with respect to f is preserved under the MCF on [−1,T ). Using the UNKNOTTEDNESS 10 transformation 3.5 and that for any interval [−1, c), c < 0, that the distortion in the reparameteriztion 3.3 is bounded, so that balls will not be mapped to points and the regions they bound will have curvature controlled by that of the original balls. Within these regions then we can find osculating balls with diameter bounded below only in terms of the original ones (i.e. depending on the noncollpasing constant) and t giving us the statement. 

We will localize the mean curvature flow with surgery much as in the spirit of the authors previous work [59]; we first remark that a version of the pseudolocality theorem holds for the RMCF via the reparameteriztion 3.3:

Lemma 3.5. Let M ∈ Σ and consider its RMCF Mt. For any , T > 0 finite there c exists R1 such that for any ball B(p, r) ⊂ B(0,R1) , |A| <  on Mt ∩ B(p, r) × [0,T ]. Below we will refer to an application of lemma 3.5 by abuse as pseudolocality. With this in hand we now discuss how to define a mean curvature flow with surgery on elements in Σ:

Theorem 3.6. For any M ∈ Σ, there is a flow with surgery St starting from M, defined on [0, ∞), which agrees with the renormalized mean curvature flow except for a discrete set of times ti at which necks are cut and replaced by caps.

3 Proof: Reiterating for mean convex surfaces in R the curvature scales Hth,Hneck,Htrig depend on an α noncollapsing constant and initial bound on H. As we discussed above in the compact case α noncollapsing with respect to F is preserved with some deterioration in the constant for compact noncollapsed surfaces; we face the added difficulty of noncompactness though and, since F → 0 at the ends, there may be no choice of α for which our M ∈ Σ is α-noncollapsed as well. We will deal with this issue of noncollapsing by localizing it where it is needed. We will say a surface M is α-noncollapsed in a ball B if for any x, y ∈ M ∩ B the x (resp y) is not in either the inner or outer osculating ball at y of radius α/F (y) (resp x) Let M ∈ Σ and suppose its smooth flow exists on [0,T ). By pseudolocality, one may choose B(0,R) large enough so that the singularity at time T occurs within B(0,R). Recalling from lemma 3.2 above the decay rate of F is bounded below on the ends for finite times so that in a sufficient large annulus A = B(0, 2R) \ B(0,R) F > c on [0,T ) and hence the surface is F α-noncollapsed for some α0 in the annulus A. Swithcing momentarily to the corresponding regular MCF and denoting momen- tarily F = −tH − X⊥ , T for t−1(τ), and similarly defining M , A, and α (such e 2 e ft e f0 UNKNOTTEDNESS 11 an α exists from the analysis in the proof of lemma 3.4) we get that Mft is Fe non- collapsed in Ae on [−1, Te). By the argument of proposition 3.2 in [59], where the maximum principle is applied to the function Z(x, y, t) from Andrew’s proof [1] of noncollapsing, the noncollapsing constant extends into the inner ball bounded by the annulus. Switching back to the RMCF Mt gives that the noncollapsing constant α0 from above extends into B(0,R). This allows us to employ the mean curvature flow with surgery within B(0,R) as in section 7 of the paper of Haslhofer and Ketover [32], where the mean curvature flow with surgery is developed for curved ambient spaces; roughly speaking the point is that the mean curvature vector will dominate X when H is very large so singularities forming (in a fixed bounded ball) will be doing so approximately, with increased percision as the singularity is reached, by the regular MCF. Note that the surgery can be arranged so that if a surgery is done at a time Ts < T the noncollapsing constant obtained still holds. Picking some uniform T > 0 we get from the above argument intervals Ik = [kT, (k + 1)T ) for which surgery with constants Hth,k < Hneck,k < Htrig,k as described above may be performed, defining a surgery flow out of M for all times, with discrete surgery times. We will denote it by k St when we are interested in the surgery parameters used and St when we are not. 

3.2. An inward level set flow for elements of Σ. Recall the definition of (set-theoretic) weak and level set flows by Ilmanen [43] for the regular mean curvature flow. A weak set flow is a family which satisfies the avoidance principle: Definition 3.2 (Weak Set Flow). Let W be an open subset of a Riemannian manifold and consider K ⊂ W . Let {`t}t≥0 be a one -parameter family of closed sets with initial condition `0 = K such that the space-time track ∪(`t × {t}) ⊂ W is relatively closed in W . Then {`t}t≥0 is a weak set flow for K if for every smooth closed surface Σ ⊂ W disjoint from K with smooth MCF defined on [a, b] we have

`a ∩ Σa = ∅ =⇒ `t ∩ Σt = ∅ (3.15) for each t ∈ [a, b] In a nutshell, the set theoretic level set flow is the largest weak level set flow: Definition 3.3 (Level set flow). The level set flow of a set K ⊂ W , which we denote Lt(K), is the maximal weak set flow. That is, a one-parameter family of closed sets Lt with L0 = K such that if a weak set flow `t satisfies `0 = K then `t ⊂ Lt for each UNKNOTTEDNESS 12 t ≥ 0. The existence of a maximal weak set flow is verified by taking the closure of the union of all weak set flows with a given initial data. We warn the reader though that the level set flow of noncompact sets can be quite wild in comparison to the compact case (see section 7 of [45] for some pathological examples) so one must proceed with caution. Since the RMCF is a reparameterization of the MCF the avoidance principle still holds; hence one can use the same definitions with respect to the RMCF (in fact, a so-called super avoidance principle holds, as discussed in [11]); in fact the level set flow with respect to RMCF can be gotten from the one for the MCF via the reparameterization. From work of Ilmanen (see section 10.3 in [43]) the set theoretic formulation of level set flow is equivalent to that of Evans-Spruck and Chen-Giga- Goto [22], [9] where they define it as the level sets of viscosity solutions to  ∇w  w = |∇w|Div (3.16) t |∇w| In [43] section 10 Ilmanen shows in the compact case these notions are equivalent and its also true in our case as explained in the beginning of lemma 3.10. Denote k,i by St a sequence of surgery flows with curvature thresholds {(Hneck,k)i} → ∞ as i → ∞ for each fixed k with initial data M ∈ Σ. From the construction above we see that on any finite interval [0,T ] we may suppose these flows are F α-noncollapsed for some uniform α in a uniform bounded set B outside of which they have curvature bounded by 1 (this last assertion by barriers). The work of Laurer [51] and Head [33] suggest that we should expect the Hausdorff k,i limit of St to recover the level set flow. To show this we proceed essentially as in section 4 of the author’s previous paper [59], with some slight changes and some details/clarifications added. We start with the following small generalization of a result of Hershkovits and White to the noncompact setting; roughly speaking the idea is force the level set flow to be what one expects by using appropriate level set flows, modifying the function w below appropriately, as barriers to “squeeze” it: Theorem 3.7 (c.f. Theorem 15 in [37]). Suppose that Y and Z are open subsets of Rn+1 where Y (but not necessarily Z) is bounded. Suppose that t ∈ [0,T ] → M(t) is a weak set flow Y ∪ Z. Suppose that there is a continuous function w : Y ∪ Z → R with the following properties (1) w(x, t) = 0 if and only if x ∈ M(t). (2) For each c, t ∈ [0,T ] → {x ∈ Y | w(x, t) = c} UNKNOTTEDNESS 13

defines a weak set flow in Y . (3) w is smooth with bounded, non-vanishing gradient on Z and its level sets in Z are connected hypersurfaces. Then t ∈ [0,T ] → M(t) is the level set flow of M(0) in Rn+1 Proof: The original statement is if M(t) is compact and Y,Z are bounded so in this case without loss of generality Z is unbounded. The proof can be followed exactly as in [37] as far as defining the weak flows associated to level setsw ˜ = e−αtw for appropriately picked α (the bounded geometry is used at this step to find a constant cK for use in (12) that works within all of Z). The main detail that needs to be checked, as used at the very end of their proof, is the avoidance principle applied to the level set flow (denoted in their paper by Ft(M)) and the “squeezing” weak flows {x | |w˜(x, t)| ≥ c} defined as defined in their eq. (16) – its shown in section 10 of [43] that two initially disjoint weak set flows stay disjoint when one of them is compact but here we need a noncompact version (which we apply then in our case to both connected boundary components individually of the weak set flows {x | |w˜(x, t)| ≥ c}). The idea in the compact case is to find a C1,1 interpolating surface I between the two disjoint initial data and use that weak set flows avoid compact mean curvature flows (note below we will not require the distance between the initial data and I to be equal). From the proof in the compact case we can similarly find a C1,1 interpolating hypersurface I. However, due to its noncompactness it can’t be used immediately as a barrier. We can overcome this however by considering approximating compact flows as barriers to get the following:

Lemma 3.8. Suppose that L1 and L2 are two, possibly noncompact, connected weak set flows which are disjoint such that for every t (L1)t and (L2)t have bounded geom- etry (smooth with bounded curvature) outside some compact set K. Then they stay disjoint under the flow. To show this statement we use another fact, which is a pseudolocality theorem in a sense:

N Lemma 3.9. Suppose M1, M2 are two submanifolds of R with such that for each compact domain K there exists C(K) so that Area(Mi ∩K) < C(K) and whose mean curvature flow exists on the interval [0,T ] and |A|2 is uniformly bounded initially by 0 0 0 say C. Picking  and R, there exists R > R so that if M1 ∩B(0,R ) = M2 ∩B(0,R ) 2 then (M1)t ∩ B(0,R) is  close in C local graphical norm to (M2)t ∩ B(0,R) for all t ∈ [0,T ]. Proof. Without loss of generality R = 1. Suppose the statement isn’t true; then there is a sequence of hypersurfaces {M1i,M2i}, Ri → ∞ and times Ti ∈ [0,T ] UNKNOTTEDNESS 14

2 so that M1i = M2i on B(0,Ri) but ||MTi − MiTi ||C >  in B0(1), satisfying the curvature and area bounds. By passing to subsequences by Arzela-Ascoli via the curvature bounds and area bounds we get limits M1∞, M2∞ so that M1∞ = M2∞ (the flows of these manifolds will exist on [0,T ]) but the flows don’t agree at some time T1 ∈ [0,T ]; this is a contradiction since the MCF in this case is known to be unique by [10], where they extend classical uniqueness theorems for the flow to the noncomapct setting.  Without loss of generality, we consider the flows at time t = 0 (the statement only needs to be verified at every instance). Similarly we only need to consider the flows of the barriers we construct below for arbitrarily short periods of time. Note that by the bounded geometry assumption on L1 we can suppose by following L1 outside K (where it is given by a smooth hypersurface) and by approximating I by a smooth hypersurface in C0 norm within K that I is smooth with bounded geometry. There exists a δ > 0 for which I and its compact approximators defined below have a smooth flow on [0, δ]. Consider a smooth domain U ij(p) such that: (1) B(p, j) ∩ I = B(p, j) ∩ ∂U ij(p), where B(p, j) is a ball of radius j centered at p ij (2) U (p) is disjoint from Li For example if nearby p I is approximately planar U ij(p) could be taken to be approximately a closed up hemisphere. By the bounded geometry assumption and that level sets of w (and hence I) are connected hypersurfaces outside K the curvature ij of I, ∂DA (p) is bounded by some universal constant C no matter how large j is and, adjusting δ if necessary, will all have a smooth flow which exists on [0, δ]. By 1j applying lemma 3.9 with some  > 0 the distance between B(p, 1) ∩ ∂DA (p) and 2j B(p, 1) ∩ ∂DA (p) is bounded by 2 for large enough j on a time interval [0, δ] for which the smooth flows exist, so that the distance between L1 and L2 in B(p, 1) can’t decrease by more than 2 in this time frame. By varying p we see then that this distance between L1 and L2 can’t decrease by more than 2 on [0, δ]. Taking  → 0 then gives their distance is nondecreasing on [0, δ] and hence lemma 3.8, finally giving the theorem.



Note that assumption (3) above in theorem 3.7 seems to be essentially optimal with example 7.3 in [45] in mind (there the initial data is not connected); the full as- sumption was used above among other spots in getting curvature bounds on compact approximators to the interpolating hypersurface. UNKNOTTEDNESS 15

k,i Now, since each of the St are weak set flows, their Hausdorff limit is too, and also the Hausdorff limit is inward. Denote their limit Lt, which we will soon show to be the true level set flow. Applying the reparameterization back to regular MCF, we will use the (singular) foliation of their limit to construct w for our case. As a consequence of lemma 2.2 in [51], Lt varries continuously (that proof is with respect to Huisken and Sinestrari’s formulation of surgery but also applies to Haslhofer and Kleiner’s, see corollary 1.26 in [31]). From pseudolocality Lt will have bounded geometry far away from the origin (on a given finite time interval). −1 We then define w in terms of the time translates of Lt; in other words. w (a) = Lt−a, for a > 0. From the discussion above w is an admissable function for use with the theorem above. Note that it can only be used to study Lt when t > 0; one can show essentially as above that Lt in the case the initial data L0 = M is connected and smooth with bounded curvature will agree with the smooth mean curvature flow Mt of M as in the compact case so this causes no issue. Note similarly the surgery flows agree with Mt for the short time it is smooth as well. Hence theorem 3.7, applied to the regular MCF (and hence the RMCF after repa- rameterizing) gives convergence of the surgery flows to the level set flow when t > 0 or in other words Lt defined above truely is the level set flow of M. Notice also from the construction that Lt will be inward and nonfattening. Arguing as in section 4 of [30] one can see the convergence of the surgery flows is strongly Hausdorff (for a given time) and so weakly noncollapsed. Now we discuss measure theoretic proper- ities of Lt; we recall the following definition of mean curvature flow from the GMT perspective initiated by Brakke in his thesis [5]: Definition 3.4. A (n-dimensional integral) Brakke flow is a family of Radon mea- sures µt such that: (1) For almost every t ∈ I there exists an integral n-dimensional varifold V (t) so that V (t) has locally bounded first variation and has mean curvature vector H~ orthogonal to Tan(V (t), ·) a.e. (2) For a bounded interval [t1, t2] ⊂ I and any compact set K,

Z t2 Z 2 (1 + H )dµtdt < ∞ (3.17) t1 K (3) (Brakke inequality) For all compactly supported nonnegative test functions φ, Z Z Z t0 Z dφ φ ≥ φdµ + φH2 − Hh∇φ, νi − dµdt (3.18) V (0) V (t0) 0 V (t) dt We will say a Brakke flow has unit density a.e. if it is true for the varifolds V (t) defined above. UNKNOTTEDNESS 16

In the following discussion by abuse of notation we often conflate Lt with its reparameterization:

Lemma 3.10. Under applying the reparameterization 3.3 Lt is a unit density a.e. Brakke flow on [−1, 0). Proof: In the compact case, it then follows from Ilmanen’s arguments in [43], sections 10 and 11 that Lt is an a.e. unit density Brakke flow; in particular the properties used in the relevant estimates are that Lt doesn’t fatten, that it has finite area, and that the level set flow can be obtained by elliptic regularization (see [58] for a more in-depth recap). This in particular doesn’t seem to quite work verbatum in our case though (although perhaps the necessary estimates could be localized) – but there are easy ways to deal with this. In this argument we first give an argument(s) which are essentially closely related workarounds of this and then discuss what seems to be a somewhat less standard approach inspired by work of J. Head [34] assuming additionally the number of surgery times is uniformly bounded (which is in line with a well known conjecture of Colding and Minicozzi, see conjecture 1.6 in [16]) using the surgery flows themselves. In the following we restrict ourselves to noncompact component(s); the compact ones can be dealt with as just described.

We first discuss for the sake of exposition how Lt corresponds to a viscosity solution to 3.16 and why merely this doesn’t suffice. As a consequence of its construction and that all the surgery flows were strictly mean convex Lt1 ∩ Lt2 = ∅ for t1 6= t2, so we may use time translates of the flow to define a local foliating function f as in section 10.4 of [43] in some open U ⊂ Rn, and by the semigroup property of level set flow its level sets will stay disjoint under the level set flow. One can then proceed as Ilmanen does to see the level set flow corresponds to a viscosity solution u(x) to equation 3.16 in U as in the compact case. Just knowing this though doesn’t immediately realize u(x, t) as the limit of solutions u to the elliptic regularized version of 3.16 however, which is how one shows the level set flow is a Brakke flow – one shows the graphs of u/ give Brakke flows and then invoke Brakke compactness (as essentially discussed in the next paragraph). On the other hand using the nonfattening it is possible to “nearly” realize it as such, without directly solving the regularized level set equation, by arguing with slight modifications due to noncompactness as in [70] (and as used in [59]): first one constructs a C2 function f whose zero set is the initial data using that the initial data in such a way that Ecker-Huisken applies to its graph (bounded in C1 suffices) so it (and its scalings) stays graphical under the flow and so that the Gaussian densities of the graph are uniformly bounded. This can be arranged using the by considering the (signed) distance function and mollifying it at focal points and the boundedness needed from the previous sentence holds since M is asymptotically conical. Taking UNKNOTTEDNESS 17 also that grad f 6= 0 on M/its zero set (the signed distance function clearly works here), for k ∈ N+ the graph of kf(x) converges to M × R, so with this in mind one defines a Brakke flow of M by the flows of the graphs kf (which stay graphical by [21]) via Brakke compactness (which just needs mass bounds on compact sets). Taking k → ∞ for the flows of the graphs kf one gets a Brakke flow which splits off a line (in terms of the graph, the vertical direction), so the cross section is a Brakke flow of M. By nonfattening (the limit cross section is also a weak set flow – Brakke flows are weak set flows by plugging in an appropriate test function into the Brakke inequality, see section 10.5 of [45]), this gives that Lt is a Brakke flow. This skips over the noncompactness issues by going straight to the “graphical mean curvature flow” interpretation of solutions to the regularized level set flow equation, sidestepping involving facts that (possibly) need compactness of the initial data. Similarly, since the flow is nonfattening one could also just appeal to section 7 of [12] (specifically see proposition 7.3, lemma 7.4, and the following discussion therein) as mentioned in the introduction (the main point of showing the surgery flow gives the level set flow for us in the sequel is the Hausdorff convergence, which we already have), which roughly reduces the situation down to the compact case. Finally we point out in passing it does seem that one might be able to argue the support of Lt is a Brakke flow indirectly using its true in the compact case, as well as Lt being the limit of surgery flows, so since the surgery flows constructed above are locally equivalent to the compact ones Lt should be a Brakke motion from this. Because it could be of independent interest, next we discuss at some length an alternative approach to showing the limit is a Brakke flow more directly via the ap- proximating surgery flows assuming a uniform finite bound on surgery times over a given time interval [0,T ] (the arguments indicated above do not require this assump- tion). As shown by J. Head, the surgery flows can be used to study more refined properties on Lt besides Hausdorff convergence. First we discuss item (3) which is of most interest, the Brakke inequality: In lemma 3.4 of his paper, Head estimates the Lp norm drop of H and remarks consequently (remark 3.7 and equation 3.4 in [34]) that the error across individual surgeries in the Brakke inequality, before components are removed, is bounded by n C(n)Lr0 sup φ, where C is a constant depending only on n, L a constant related U + from the proof to surgery depending on n and p, r0 is the scale of the neck (this is phrased in terms of Huisken and Sinestrari’s approach but works fine for our setting as well), and φ is the test function inserted. There is also the error from removing high curvature components but we deal with this contribution first. Letting the −1 surgery parameters tend to infinity gives r0 → 0 since r0 ∼ Hneck, so if the number of surgeries at any given time is finitely bounded by a bound which doesn’t depend UNKNOTTEDNESS 18 on an upper bound for Hneck the discrepency in the Brakke inequality across cap placement will tend to zero as i → ∞.

Remembering we restricted ourselves to a noncompact component of the flow, consider a path γ from the region outisde B (so starting where H < 1) to a surgery spot where H ∼ Hneck. Recall the canonical neighborhood theorem of Haslhofer and Kleiner, theorem 1.22 in [31], which says that given  > 0 if H > Hcan,k(αp ) then the surface is -close to an ancient mean curvature flow at that point (as discussed in the section above this also holds true for our RMCF surgery flows). Since H is continuous along γ, if H > Hcan the constant from Haslhofer and Kleiner’s canonical neighborhood theorem, there will be a point along γ where H = Hcan and will be from the manner it was discovered modeled on a cylinder. In this way we obtain a lower bound on the volume of the surface contained in every “neck” along which a surgery will occur if Hneck > Hcan. Since the area of the surgery flows within the ball (on a finite time interval) are bounded uniformly (one can apply one sided minimization in the Gaussian metric to reparameterized Lt, which is equivalent to the Euclidean metric in a compact sets, and this works for the surgery flows as well), we obtain a finite upper bound for the number of such necks and hence the number of distinct surgery spots for a fixed surgery time which doesn’t depend on an upper bound for Hneck, because at most two surgeries could be done further along the neck (the canonical neighborhood theorem precludes the neck “branching” further along).

To show item (3) also requires us to also control the area of the high curvature components thrown out; if surgeries are performed then (as discussed in section 4) the high curvature regions thrown out will be tubular neighborhoods of embedded intervals I, with diameter approaching zero as Hneck → ∞. So it suffices to show the lengths of such I may be taken to be bounded. Indeed, by Colding and Minicozzi’s Lojasiewiscz-Simon inequalities for the MCF in [15] we may bound the length of γ arguing as Gianniotis and Haslhofer do in [28] to get a bound on the length of I in terms of the data αp and time interval [0,T ] under consideration. From the above discussion there will only be finitely many such components (with a parameter independent bound on the number), giving that the discrepency across surgery times tends to zero along the sequence of the surgery flows.

The final observation we use is that the convergence of the surgery flows may be taken to be varifold convergence (at every time slice) by applying Allard compactness theorem for integral varifolds as in corollary 5.3 of [33], because from the diameter control we also obtain local Ln−1 bounds on H again arguing as in [28] and, by one sided minimization, local area bounds. To put this all together, fixing a test function UNKNOTTEDNESS 19

i i φ and denoting by δt(φ) the discrepancy (if any) for a surgery flow St at time t we have the following:

Z Z Z t0 Z 2 dφ i φdµ ≥ φdµ + ( φH − Hh∇φ, νi − dµ) − δtdt (3.19) Si Si 0 Si dt 0 t0 t Since the convergence is varifold and hence as measures, the LHS converges to R φ L0 as i → ∞ and similarly for the first integral of φ on the RHS. For the double integral on the RHS we apply Fatou’s lemma, considering the inner (spatial) integrals as a sequence of functions fi(t): these are measurable functions and since φ is a fixed compactly supported positive smooth function and there are L1 bounds on H, the fi(t) are bounded uniformly below. By the varifold convergence in the limit we get the double integral considered on Lt which appears in Brakke inequality. Since i we assume there is a finite bound on the set of times for which δt(φ) 6= 0 and the discrepancy at each such time tends to zero as i → ∞ we get the additional contribuitions tend to zero so that Brakke’s inequality holds for Lt. Note that without the finiteness assumption on number of surgeries, it could be i i the case δt(φ) → 0 but the “total” discrepancy along the sequence St stays bounded below by a positive number (for example, a sequence on which for each i there are i i surgeries within some fixed interval at which times δt(φ) = 1/i), so in the limit the Brakke inequality is invalid – given that this isn’t the case as the alternative approaches show this somehow isn’t an issue in practice so the argument above is probably nonoptimal.

Now we discuss properties (1) and (2) of a Brakke flow. The local L1 bounds on H along the sequence of approximating surgery flows combined with the varifold con- vergence give (1). By one sided minimization we will always have finite area within any bounded region U giving (1) and, by plugging in appropriate bump functions φ into (3), we get (2). From its construction as a limit of surgery flows, and hence as a boundary of a set, it can be seen to have unit density almost everywhere directly or by De Giorgi’s theorem (cutting the region swept out into bounded pieces). All put together this gives that Lt corresponds to/is the support of a.e. unit density Brakke flow via the surgeries with the natural finiteness assumption on surgery times. 

In particular, Lt (applying 3.3 back and considering it on [0, ∞)) satisfies the Brakke regularity theorem (see White [70], sections 6 and 7 apply) and limit flows will be mean convex Brakke flows if all the base points are a finite distance away from the origin (which they are for any given bounded time). The asymptotic behavior of Lt can then be used to find a stable self shrinker as follows. As mentioned a UNKNOTTEDNESS 20 shrinker mean convex/inward RMCF level set flow gives a mean convex foliation in the Gaussian metric by 3.4 so, amongst other results of White’s work holding for our flow, therefore satisfies the one sided minimization property of White (section 3 of [69]) with respect ot it. In particular by sections 7, 8 and 11 of [69]:

Theorem 3.11. If lim Lt is nonempty, then the limit of its boundary is a stable self t→∞ shrinker. Compare with Theorem 11.1 of [69] – the compactness assumption there is unim- portant since stability can be verified by checking on a compact exhuastion. We do note that by using instead a Frankel theorem as discussed in the concluding remarks, stability is not completely vital and in this way one can seem to skip appealing to this consequence of White’s mean convex MCF theory. Also we point out one should indeed expect singularities should occur; for example an outward perturbation of the Angenent torus will develop under RMCF a neckpinch about its axis of rotation (in this case, the RMCF will then flow outwards to spatial infinity, a consequence of the Angenent torus being unknotted). Hence it seems necessary to consider a weak flow as above – see also the recent work of of Lin and Sun [54] which says this is the case for all compact nongeneric shrinkers.

4. Proof of Theorem 1.1 Our goal as in [61] and [52] is ultimately to appeal to a Waldhausen type theorem, the original result shown by its namesake in [69]: Theorem 4.1. Suppose M is a Heegaard splitting of S3 of genus g. Then it is isotopic to the standard genus g surface of S3. A Heegaard splitting is a surface in a 3 dimensional (for now, take it to be comapct) manifold N which splits N into two handlebodies: regions homeomorphic to topo- logically closed regular neighborhoods of properly embedded, one-dimensional CW- complexes in N. We define standard embeddedness for compact closed surfaces as surfaces isotopic to any of the following. The standardly embedded torus we take to be the embedding T 2 ,→ R3 ∼ S3 \ {∞} ,→ S3 given by rotating the unit circle S(2, 1) in the xy plane about the z axis. The standardly embedded genus g surface can be constructed by taking g standardly embedded tori, arranging so that their centers fall along a line and so that their convex hulls are pairwise disjoint, and tak- ing a connect sum of adjaicent tori using straight cylinder segments at two closest points. In our noncompact case, we will say that a one ended surface is standardly em- bedded if there is a diffeomorphism of R3 which takes it to the connect sum with a standardly embedded genus g surface attached to a plane, in agreement with Frohman UNKNOTTEDNESS 21 and Meeks [26] (in particular see figure 1 and the surrounding discussion in [26]) - one can alternately think of define these in terms of isotopies allowing for “infinite speed” (for a simple example why this is necessary consider defining an isotopy from a conical to a cylindrical end; on the other hand it seems plausible by replacing plane with cone or cylinder that “regular” isotopies could be used). In order to use this fact one needs conditions that guarantee it; Lawson in [52] gives the following criteria for a surface being a Heegaard splitting in S3; (2) is particularly useful for verfication using ideas from geometric analysis. 3 Lemma 4.2. Let M be a closed hypersurface in S and denote by Rin and Rout the inner and outer regions bounded by M. Then M is a Heegaard splitting exactly when either (and hence both) of the two statements in the following is true:

(1) The inclusion maps ι : M → Rout, ι : M → Rin both induce surjections of fundamental groups ι∗ : π1(M) → π1(Rout), π1(Rin). (2) Rdout and Rdin, where Rb denotes the universal cover, have path connected boundary. To be more specific, the outer region is the region the (outward) normal points into and the inner region is the one it points away from. Throughout we will refer to Rout,in, et cetera when we want to discuss the pairs Rout,Rin simultaneously in the fashion that the argument would apply either using Rout or Rin, which is often (but not always) the case. We will also often refer to the first criterion as “π1 surjectivity” with respect to a given domain. In [61] we compactify R3 to consider their self shrinker as a hypersurface in S3 so as to apply 4.1 but in the present case there are ends which makes the state of affairs for Waldhausen type theorems much more subtle. To see this note that cutting off a noncompact surface (with well controlled ends) by a large ball the problem is closely connected topological uniqueness problems for Heegaard splitings with boundary of balls (defined appropriately), and incredibly there are examples of knotted minimal surfaces with boundary constructed by P. Hall in [29] which give in turn topologically nonstandard Heegaard splitings of the three ball. Note that these have multiple boundary components; if there is just one boundary component then a Waldhausen theorem holds though, see section 2 of [55]) – we will use this below. On the other hand as we discuss in the concluding remarks the unknottedness result [56] of Meeks and Yau shows the relationship between the noncompact and boundary case isn’t perfect, but nonetheless we should proceed with caution. However, there are positive results in the noncompact case. In [26], Frohman and Meeks define surfaces to be Heegaard splittings, following the defintion in the compact case, as two sided surfaces which bound closed regular neighborhoods of one- dimensional CW complexes, and they subsequently prove a Waldhausen theorem: UNKNOTTEDNESS 22

Theorem 4.3 (Theorem 1.2 in [26]). Heegaard surfaces of the same genus in R3 are ambiently isotopic. Equivalently, given two diffeomorphic Heegaard surfaces in R3, there exists a diffeomorphism of R3 that takes one surface to another surface. The noncompactness of the problem introduces some extra delicacy to the problem of discerning whether a given surface is a Heegaard splitting however. In particular, Frohman and Meeks are very careful in their paper to distinguish between surfaces which bound open and closed handlebodies; a significant portion of their paper is showing the bounded handlebodies (they show their minimal surfaces are open han- dlebodies by the π1-surjectivity criterion) may be taken to be closed. In our setting however the geometry of our ends is well controlled (in contrast to the more general situation they consider) so this is not an issue. In fact for our case, we can also appeal to the Waldhausen theorem for splittings of a 3-ball mentioned above if in a large enough ball B (large enough so that M in the complement is ∼ R2 \ D(0, 1)) we know apriori M ∩ B will give a Heegaard spliting. Its easy to see the validity of π1 surjectivity of M with respect to either the inner or outer component will be inherited by M ∩ B for such a ball B, so in summary: Remark 4.1. Lawson’s criteria given in lemma 4.2 above can be used to show an asymptotically conical surface with one end is topologically standard. From here on out denote by M a self shrinker which either has one asymptotically conical end or is compact. First we consider the asymptotically conical case and afterwards we discuss the case it is compact.

4.1. M has an asymptotically conical end. Suppose that R\out,in has discon- nected boundary (which one in particular is unimportant), so that it has at least two path components A and B. Since M is not Euclidean mean convex (since it is not a cylinder, by [13]) and hence entropy unstable, so by lemmas 4.1 and 4.2 of Bernstein and L. Wang [4] one may find, switching choice of normal depending on which domain Rout,in is in question, a shrinker mean convex perturbation of M, M  ∈ Σ the set defined in section 3. Moreover it will be entropy decreasing. In short it is a perturbation which, on each end, asymptotes to the original asymptotic cone using the first eigenfunction (which has a sign) of the Jacobi operator. Theorem 1.2 then gives us a renormalized mean convex/inward level set flow with intial data M  which exists for all time, and if it is nonempty the limit will be a stable self shrinker. In the following, note one may lift a (flow/perturbation of a) submanifold/varifold N of Rout,in to a submanifold Nb of R\out,in by, for every open set U in a covering of R\out,in by open sets on which the projection map π is locally UNKNOTTEDNESS 23 a diffeomorphism, defining U ∩ Nb = U ∩ π−1(π(U) ∩ N). This is well defined and by similarly lifting the outward normal gives a way to lift perturbations and flows – if the lift of N has multiple components Ci we can also define the flow out of a single component C by lifting the flow of N and throwing out the components which did not originally emanate from C. Note that by our definitions if N is embedded and boundaryless so is Nb. With that in mind to show the flow of M  will be nonempty we consider the lifts of  the perturbation M to the universal cover of R\out,in to get a graphical perturbation A of A and the other boundary components; note that since A is a covering of M which is asymptotically conical (and hence has a uniform tubular neighborhood) the perturbation M  can be taken to be embedded and so will the lift A. Furthermore, we may considering the lifted approximating surgery flows Sbt and Lbt of Lt in R\out,in which flow “out” of A (we do not lift the flow to the other boundary components). We now discuss some properties of these lifted flows:

Lemma 4.4. Any lifted approximating flow Sbt with fine enough surgery parameters Hneck,k large enough (for each k) and hence the lifted level set flow Lbt satisfy the following properties:

(1) The flow will never collide with a boundary component of R\out,in (2) The flow of St is nonempty for all t ∈ [0, ∞)

(3) Supposing R\out,in has (at least) two boundary components A, B and that Lt flows out of A. Then for any curve γ betewen A and B which has nonvan-  ishing mod 2 intersection number with A , Sbt ∩ γ 6= ∅ for all t ∈ [0, ∞) so that Lbt (and hence Lt) will have a nonempty limit as t → ∞.

Proof: We focus our discussion on a fixed surgery flow St which hence implies the same facts for Lt by theorem 1.2. Item (1) is by the avoidance principle as follows: first note that by passing down to the base that it suffices to show St never collides with M  since A and B both are lifts of M. We also see it suffices to consider the flow on a fixed time interval [0,T ]. Considering an annulus A(R, r) = B(0,R) \ B(0, r), by shrinker mean convexity and that M is asymptotically conical, so the flow is can  be written as an upwards moving graph in A(R, r), St and M (and hence M) plainly must have distance bounded below by a positive amount within A for r sufficiently large (depending on T ) and a fixed R > r. By the classical avoidance principle (under reparameterization) then St and (the flow of) M must stay disjoint within (the image of) B(0, r) on [0,T ] as well. We also see from the same argument one 3 may take R → ∞ to see that St and M stay disjoint in R \ B(0, r) in turn, showing (1). (2) is clearly a consequence of (3) but we highlight it because of its importance. UNKNOTTEDNESS 24

To see item (3) we first note that when Sbt flows by the (lifted) smooth mean cur- vature flow that the mod 2 intersection number is preserved (considering throughout generic times when the intersection with γ is transverse, or alternately slightly de- forming γ as long as one always stays in a fixed neighborhood of the original curve), following the same proof that it is preserved under isotopy for two compact closed surfaces, noting that how the lifted flow was defined that Sbt will always be embedded because St is, this is because γ is compact (intuitively, so that intersection points are not “lost” to spatial infinity), by (1) that Sbt is isolated from the endpoints of γ, and that Sbt is boundaryless (again from how it was defined). These facts force the spacetime track of the intersection points to be spatially bounded intervals or closed loops so that the mod 2 intersection number is preserved. It then remains to consider how the intersection number may change during surgery times. We remark here that alternately it seems such situations could possibly be avoided altogether by perturbing the curve γ slightly because the singular set of Lt is negiligble but we deal with this possibility anyway.

To proceed, we must first describe in more detail what could unfold during a surgery time t∗. This discussion applies (almost - a case related to (iii) below is added) equally to St and Sbt as discussed shortly after:

If St∗ has high curvature everywhere, it is either i) convex, ii) close (in appropriate norm, see after remark 1.18 [30]) to a tubular neighborhood of some open curve with convex caps, or iii) close to a tubular neighborhood of a closed curve. In these cases the surface is either a sphere or a torus. If there are low curvature regions on (a connected component of) St∗ , then there are couple cases for the high curvature regions it may border. Considering a given high curvature region bordering a low curvature one, there will be a neck (a region where at every point after appropriate rescaling the surface is nearly cylindrical) which following along the direction of its axis away from the original low curvature region, one will find either a) a convex cap or b) another low curvature component of surface. In the former, there will be one surgery spot and in the later there will be two on either side of the neck region and hence four caps will be placed (so that the capped off neck is topologically a sphere). This discussion is encapsulated in the figure below: UNKNOTTEDNESS 25

Figure 1. A diagram displaying the possibilities one may encounter at a surgery time for a compact flow in R3. High curvature regions are in red and surgery spots are in green. Note in (b) there are two spots along the neck where surgery will be done giving two pairs of caps.

When the surgery parameters are sufficiently fine most of the above discussion applies to the lift of the surgery flows as well, because surgeries which occur at any given finite time take place in compact regions about the origin where as such points within will have a uniform lower bound on the diamter of their evenly covered neighborhood, so if the surgery parameters are fine enough (i.e. Hneck,k large enough, k depending on the time interval under consideration) all local models – necks, caps, high curvature convex regions discussed above will be lifted to necks, caps, and convex regions in the universal cover using the local models of each will be completely contained in such a neighborhood. Going to the cover there is possibly one extra case though: note that if case (iii) occurs along the surgery flow in the base then the bounded loop may concievably be lifted to an embedded cylinder in the cover, which we refer to below as case (iv). No other new cases can be “added” by the covering map since, following the central curves from one side of the neck region in the base in the other cases ((a), (ii), (b)), and considering then the lift, if a cap or low surgery region is arrived at in the base this must also be the case following the lift of the central curve in the cover. In other words, the lifts of these types of high curvature regions in the base correspond to the same cases passing to the cover (although of course in the cover there may be many copies). With this in mind, consider the very first time a surgery is performed. If γ does not intersect any regions where surgeries are performed then there is nothing to do. In the following we will consider γ restricted to open sets containing surgeries, where it UNKNOTTEDNESS 26 may have multiple connected components; we will always implicitly restrict ourselves to one component.

Suppose now that γ does intersect some surgery regions and denote by U an open set which contains all surgery regions and such that γ intersects Sbt in U only in points affected by surgery; we see it will suffice to show the intersection number of γ with Sbt in U will be preserved across surgeries (note: this number may be odd or even, it doesn’t reflect on the global count). First suppose γ intersects no future surviving caps: points where a cap will intersect once a surgery is done and is part of a low curvature region (hence the adjective surviving), and so only intersects high curvature regions Di which are immediately discarded. Denoting by Di the high curvature regions with necks cut and caps placed (if necessary to do so), since no future caps are intersected and the caps are topologically discs the γ intersects Di in the same parity of points as Di (as a side remark, note that by transversality its safe to assume γ always intersects Sbt transversely, even after surgeries). Since for cases (i)-(iii) the Di are closed surfaces γ intersects each of them in an even number of points. In the possible new case encountered in the cover, case (iv), this is also true because γ is an embedded compact interval (if this weren’t the case, it could enter the neck and stay in the core, so the its intersection number was 1). On the other hand in this case γ will not intersect Sbt in U at all post surgery. Since the number of intersection points went down in U by an even number, we are done in this case.

Now suppose γ did intersect some future surviving caps. First we consider those it intersects an odd number of times and focus on one of them, which we’ll call C. In this case, the other side of the future cap is either a high curvature region diffeomorphic to a sphere or another low curvature region, and we will refer to the high curvature regions discarded in these cases as the “horn” and “neck” respectively (in pictures, (a) and (b) respectively in the figure above). We will denote by V ⊂ U an open set containing precisely the horn or neck in accordance with focusing on C.

In the horn case the intersection number of γ with Sbt in V is odd. To see this note since γ’s intersection number with the future cap itself is odd eventually γ goes through the core of the surgery neck and doesn’t return to the cap (parameterizing the curve appropriately) – but its endpoint must not lay in V so γ eventually leaves V through the horn and does not reenter. Denoting the horn region by D and the post cap placement horn by D, γ will intersect D an even number of times since it is closed. But since C is the cap placed opposite the horn and the intersection number of γ with it is odd the number of intersection points in V stays odd, preserving mod 2 intersection number in V (and hence U). Applying the same argument at every UNKNOTTEDNESS 27 such future surviving cap which came from a neck bordering a horn covers this case.

The neck case is the same if γ intersects it at only one future surviving cap, but there is also the case γ intersects both bordering caps. In this case there will be two pairs of surgery caps placed, a pair associated to C (where the other cap in the pair is part of a neck which is thrown out) and a “far pair” on the opposing low curvature region: the cap on the far pair which survives surgery we’ll call the far opposing cap C0. In (b) in the figure above labeling the four caps 1 through 4 from left to right if C were “1” then C0 would be “4.” In the case γ intersects both C and C0 there are two cases: it intersects the neck an odd number of times and C0 an even number of times, or vice versa: it can’t intersect both an odd number of times or else γ would have a triple junction (i.e. a “Y”) and it can’t intersect both an even number of times because it has an odd intersection number with C and it is boundaryless in U: in this case an enpoint would lay within the core of the neck but the flow is strictly separated from A or B as used before. In the first case the mod 2 intersection number of γ with Sbt in V presurgery is odd and in the first case even (the caps aren’t placed yet before surgery!). In the first case the intersection number across C postsurgery is odd, across C0 even, so the mod 2 intersection number is preserved. In the second case γ will intersect both C and C0 an odd number of times, so the total intersection number in V postsurgery is even again preserving intersection parity. Again one applies this same argument at every such cap C. Now consider those future surviving caps which γ intersects an even number of times, again focusing on one of them and denoting it again by C. In the horn case, γ must intersect D an even number of times because it intersects C an even number of times (so every time γ goes past C into D, it must come back), so as before the mod 2 intersection number is preserved. In the neck case, one can see arguing as above that γ much intersect the far opposing cap and the neck both an even or odd number of times since it intersects C an even number of times, giving that the mod 2 intersection number is again preserved under surgery like before. As above then one applies this at every such cap C. Repeating the argument for subsequent surgery times gives us that for any such γ

Sbt will always intersect γ in an odd number of points. Since the domains R\out,in are simply connected and A is a graph over A, such a γ certainly exists giving us the conclusion of item (3).  UNKNOTTEDNESS 28

Note that in the lift any curve from A to itself will intersect A an even number of times so (3) is indeed particular to our case, in that we needed two different boudnary components in the lift. For a concrete example, the universal cover of R3 \ B3 is simply connected and, taking the boundary of the ball to be the shrinking sphere, any outward perturbation will flow away to infinity under the RMCF. Denote by N the limit surface we obtain from theorem 3.11. Since M was proper, by Ding and Xin (theorem 2.1 above) it had polynomial volume growth and hence finite entropy, so that the entropy decreasing perturbation M  does as well (more elementary but more specific to our setting, it has polynomial volume growth because it has a single conical end). By the monotonicity of entropy for nonfattening level set flow this implies N does as well, which in turn gives N has polynomial volume growth. In particular it must not be a stable shrinker, giving a contradiction and showing π1 surjectivity holds with respect to both the inner and outer components of M. Since M is asymptotically conical we obtain theorem 1.1 in this case as indicated in remark 4.1.

4.2. Revisiting the compact case. To conclude we discuss how to reproduce the unknottedness theorem for compact self shrinkers more in line with the technique above. First note in this case we may appeal straight to Waldhausen’s theorem for Heegaard splitings of S3 after one point compactifying R3 as discussed in [61], note the isotopy can be arranged to avoid {∞} and hence gives rise to a bounded isotopy in R3. Using the first eigenfunction of the Jacobi operator as above to get a shrinker mean convex perturbation (for the compact case, see lemma 1.2 in [18]) to then flow; the corresponding level set flow Lt is then constructed exactly as above (surgery in the compact case is easier from a technical viewpoint because the surgery need not be “localized”). Siimilarly the convergence to level set flow (and that it is a Brakke flow) is easy. We see at no point did we use the noncompactness of M in the proof of lemma 4.4, so we get a nonempty (note: possibly noncompact) stable shrinker N from which we derive a contradiction as before. Alternately (using the compactness assumption) one may also derive a contradiction by theorem 7 in [35], which says the flow of the perturbation must clear out.

Remark 4.2. Note that in some cases the π1 surjectiveness is evident for Rin; for example when M is a torus. In this case if M merely has nonpositive shrinker curvature it must be unknotted from the above argument. On the other hand, it is easy to construct knotted shrinker mean convex tori by taking very thin tubular 1 neighborhoods of knotted S . In fact it seems that Rout (or perhaps better said one of Rin or Rout) should typically play the more important role but we are unsure of what UNKNOTTEDNESS 29 can be said of it in general. For example, in the Kapouleas, Kleene, and Møller [50] examples Rin and Rout are essentially reflections of each other. 5. Concluding remarks We begin our discussion with the current state of affairs of unknottedness theorems for classical minimal surface, which will lead naturally into the other topics mentioned in the introduction. Theorem 1.1 is very much in the spirit of the various works by Freedman, Frohman, Meeks, and Yau on classical minimal surfaces in R3 – see the papers [24–27, 55, 56]; their papers give an essentially complete answer to the type of question under consideration here for minimal surfaces, although the arguments in these papers do not seem to obviously carry over to our setting as we explain. The paper most relevent to our present situation is that of Meeks, where he shows the following on page 408 of [55]: Theorem 5.1. Suppose h·, ·i is a complete metric on R3 with non-positive sectional curvature. Let M be a complete proper embedded minimal surface in R3 which is diffeomorphic to a compact surface punctured in a finite number of points. Then (1) If M has one end, then M is standardly embedded in R3. In particular, two such simply connected examples are isotopic. (2) If M is diffeomorphic to an annulus, then M is isotopic to the catenoid. Note that item (2) in the shrinker context is essentially covered by Brendle in Theorem 2 of [6], mentioned already in the introduction. Item (1) has a Morse theoretic proof, where the nonpositive sectional curvature enters via Gauss formula to see that the Gaussian curvature of a minimal surface at any point on M must be negative; this implies the height functions involved have no critical points of index 2 which allows Meeks to show minimal surfaces must bound handlebodies in many situations, allowing him to reduce again to a Waldhausen-type theorem in the case of one end as above. Even ignoring the incompleteness of the Gaussian metric, by calculations of Cold- ing and Minicozzi in [13] the scalar curvature of the Gaussian metric does not have a sign so neither do the sectional curvatures in this metric, as discussed in section 2 although the regions where the scalar curvature is positive and negative are clearly q n2+n “simple” in that the region where it is positive is a ball (of radius 2 n−1 ). However the aforementioned examples of P. Hall [29] seem to rule out decomposing the surface into different ambient curvature regimes and applying different arguments in each because these boundaries may have multiple boundary components. On the other hand, in contrast to the examples of P. Hall, we do note the subse- quent paper of Meeks and Yau [56], on complete minimal surfaces with finite topology UNKNOTTEDNESS 30 and multiple ends, reduces to the one ended case in a way which sidesteps any possible pathological behavior – the fact that the minimal surfaces considered are complete is vital. This suggests our result could possibly be extended to the case of shrinkers with multiple ends, or that perhaps a decomposition indicated in the above para- graph was actually workable. We caution the reader though that many arguments in this paper use the solution to the Plateau problem (in particular Meeks–Simon–Yau minimization in an isotopy class; this is discussed more below) which would require the incompleteness of the Gaussian metric to be dealt with, and perhaps more impor- tantly they also invoke other deep results in the classical theory of minimal surfaces in R3 (such as the annular end theorem [39]) which would need to be checked to prove the exactly analogous statement of their theorem for shrinkers, at least if their approach was followed closely.

When the metric is Euclidean the main result is also a corollary of Theorem 2 in the same paper of Meeks, where it is shown that minimal surfaces of the same genus in a mean convex ball sharing the same connected boundary component are isotopic to each other and furthermore standard; see also Theorem 3.1 in his paper with Frohman [26] for a “noncompact” analogue. This proofs in either go by showing, after making the same reduction to boundary components as above, that nonflat minimal surfaces in a domain with mean convex boundary must intersect by a moving plane argument in the first or the argueing as in the proof of the halfspace theorem of Hoffman and Meeks [40] in the second. The moving plane argument doesn’t apply in our setting; but the Frankel theorem for f-Ricci positive metrics (which the Gaussian metric is) could provided the boundary of the ball under consideration is mean convex (see theorem 6.4 in [68]). There might be concerns dues to possible noncompactness in the lift – although one might hope this plays out better than in the Frenkel theorem below in terms of conditions needed since the Gaussian metric restricted on finite domains has bounded curvature. The main catch to the seems to be that spheres of large (Euclidean) radius (or more general convex sets of large in-radius) are not mean convex in the Gaussian metric: the ball considered might have to in fact be quite large in the proof because it is picked so that the minimal surface under consideration consists of k annuli in its complement (here, k = 1). And indeed, one can see suitably large domains (those strictly containing the round shrinking sphere) will never be shrinker mean convex at all points of their boundary in the Gaussian metric (for instance by using the RMCF and comparison principle), ruling out more clever design of domains. A possible way to deal with this might be, upon intersecting with the ball, also perturb the metric (or more precisely, f) to make the boundary mean convex (in a natural sense along the nonsmooth points): even granting this such a perturbation would also require positivity in the Hessian (by the definition of f-Ricci positive) and it was not immediately clear this could be UNKNOTTEDNESS 31 guaranteed to the author at the time of this writing, although perhaps this can be arranged by a more careful examination of the perturbation function one would use.

Reducing down to surfaces with boundary is not strictly necessary of course, and more in line with Lawson’s original argument in [52] one might ask if a Frenkel theorem could be applied directly in the noncompact setting to rule out the two boundary components A and B of R\out,in discussed in the proof of theorem 1.1. Wei and Wylie’s Frankel property for general f-Ricci positive metrics doesn’t apply in this case because it requires f be bounded, which it isn’t in our setting if we don’t consider subdomains with boundary. This leaves to the authors knowledge the following two statements to consider applying, the first due to Impera, Pigola, and Rimoldi and the second very recent one due to Chodosh, Choi, Mantoulidis, and Schulze (the author thanks A. Sun for this reference):

m m Theorem 5.2 (Theorem B in [48]). Let Σ1 and Σ2 be properly embedded connected m+1 self-shrinkers in the Euclidean space R . Assume that Σ2 has a uniform regular normal neighborhood T (Σ2). If

dist m+1 (z, Σ ) lim inf R 1 > 0 (5.1) −b|z|2 −1 |z|→∞,z∈Σ2 e P(|z|) 1 for some polynomial P ∈ R[t] and some constant 0 ≤ b < 2 , then Σ1 ∩ Σ2 6= ∅. In the following, F -stationary means stationary with respect to Gaussian area: Theorem 5.3 (Corollary C.4 in [12]). If V,V 0 are F -stationary varifolds, then supp V ∩ supp V 0 6= ∅.

In the first statment above properness enters because for self shrinkers in Rm it guarantees polynomial volume growth by the result of Ding and Xin [19]. The issue though is that there are cases (hyperbolic metrics on genus g surface) where the the universal cover of a bounded (and hence polynomial volume growth) surface has exponential volume growth, so it is not obvious that the first statement can be applied in the lift. In fact, the boundary components could be stable for the same reason; in general it is known that the spectrum of the Laplacian (and imaginably more general elliptic operators, such as Jacobi operators) may decrease upon lifting to universal cover, unless the fundamental group is amenable: see Brooks [8] (the author thanks R. Unger for bringing this paper to his attention). This was a detail overlooked in [61] (particularly claim 2.1) which can be fixed as above by lifting a perturbation by eigenfunction of the Jacobi operator to the universal cover – note this will also give a strictly positive distance between the two self shrinkers found in the argument of that paper because the shrinkers in question are compact. UNKNOTTEDNESS 32

The proof of the second statement does not seem to require a polynomial volume growth assumption but uses the fact that shrinkers “collapse” onto the origin in Rm. In the case of two smooth self shrinkers where one is compact its a simple consequence of the avoidance principle: the distance between them must not decrease but on the other hand they both shrink to the origin after one “second” under the flow. However, this argument doesn’t seem to apply when passing to coverings (for one, the origin could be the preimage of many points in the universal cover) so does not seem to apply to the lifts A and B of M discussed in section 4. Indeed it is true they should never intersect, the issue is that there seems to be no good reason that their flows should approach a common point (in contrast to shrinkers in R3). For example if M is a self shrinker where the origin lays within the region bounded by M but the lift of Rout has two connected components A and B, it seems that (the lifted convex hulls of M corresponding to) A and B should retreat from each other, and in so doing not giving a contradiction, because their corresponding (lifts of) origin(s) are “behind” A and B. Instead of working entirely in the universal cover, the next thing one might try, inspired by the ideas above, could be to find a minimal surface in the universal cover proceeding as in the author’s joint paper with S. Wang [61] and then project it back down to R3 and use a Frankel theorem there. In the compact case one can deal with the incompleteness of the Gaussian metric by an intermediate perturbation argument (as in [61] or Brendle’s paper [6] to then find a stable minimal surface Nb (in the Gaussian metric) in R\out,in by solving a sequence of Plateau problems over domains exhausting a connected component (say A) – note since it might not have polynomial volume growth this itself does not give a contradiction (as falsely claimed by the author as an aside in his thesis [60]). The noncompact case is more delicate since M will always have points which lay in the “perturbed region” but by chosing the perturbations correctly as Brendle does in proposition 12 of his paper the same construction seems to work in this case as well. At any rate, note that a Nb found by such means is concievably not equivariant under deck transformations (in the exhaustion of A, none of the domains would be equivariant – although perhaps if one cound pick the exhaustion to be comprised of whole fundamnetal domains/”tiles” of A it seems plausible the limit minimal surface obtained might be equivariant) so in fact might pushed down by the covering map to something that is at least intrinsically smooth but nonproper. This possible nonproperness seems to give technical issues, at least in the noncompact case where the distance between the two shrinkers could possibly be zero (in the compact case, or where there is positive distance between the two shrinkers, it seems to be fine arguing as in the author’s thesis [60] by slightly “tilting” the compact one to use the classical avoidance principle – this is another issue overlooked in [61]). This certainly rules out immediately invoking the result of UNKNOTTEDNESS 33

Impera, Pigola, and Rimoldi, and it also seems to rule out invoking at least as a black box the Frankel property of Chodosh, Choi, Mantoulidis, and Schulze because there is an implicit properness assumption in most of the literature on Brakke and level set flows (see section 2.4 of their paper [12]) and it is not immediately clear where to the author the assumption might be used in the background facts and theorems quoted in its proof. (Of course, once we obtain the self shrinker we do from the flow, then their Frankel theorem can be applied alternate to how we finish above.)

To avoid these issues of potentional large volume growth, nonproperness, and nonequivariance then, it seems appropriate to work in the base as much as possible when searching for an “impossible” self shrinker. With the elementary intersection number argument in lemma 4.4, minimizing within an isotopy class (with respect to Gaussian area) to find such a self shrinker perhaps is the most natural next step because if the minimization can be done purely in Rin or Rout it should be nonempty by essentially the same argument in the aforementioned lemma. As is well known, in fact Meeks, Simon, and Yau developed a Plateau problem type approach to this exact type of problem in [57] (note that the minimizer found in their paper need not be in the same isotopy class or even connected – degeneration could occur) – specifically see sections 4 and 6 of their paper. The issue of incompleteness of the Gaussian metric still remains though and bounded geometry is assumed in their work. Of course, then essentially as above one would want to consider a family of perturbations of the Gaussian metric “at infinity” which “regularize” it and limit to the original one and pass a limit although some more care needs to be taken as follows. If our original 3 shrinker M, the boundary of the regions Rout,in it splits R into, can always be taken to be outwardly mean convex in the Gaussian metric along the sequence, this seems to work since its can be used as a barrier (and indeed, this is the case when M is closed – this gives a slightly different proof of the result in this case as that described above). However, when M isn’t compact as in our present situation it isn’t clear this can be arranged because the perturbations necessary to regularize the Gaussian metric would also affect the geometry of M, making it concievable that the lift of each minimizer found by Meeks–Simon–Yau for each perturbation doesn’t intersect the curve γ discussed in lemma 4.4 (in the base, this might correspond the the “knotted” part of the surface degenerating away for each perturbation). Perhaps barriers could be constructed to force the potential intersection points of the minimizers with M to tend to infinity as the limit is taken in the perturbations of the metric. There is also the issue that there are no compact representative in the isotopy class of M but their argument allows for the isotopy class to have elements with boundary so an intermediate exhaustion argument could be used to avoid this as utilized in the argument in Meeks and Yau [56]. UNKNOTTEDNESS 34

Perhaps the best way to argue with this technique is, because the incompleteness of the Gaussian metric is “at infinity,” one might instead apply Meeks–Simon–Yau in compact exhaustions of Rout,in, with the metric perturbed near the boundary of these sets as in Brendle. Since the Gaussian metric has bounded curvature on any compact subset, standard curvature estimates should then allow one to pass to a stable limit (indeed, this is essentially how the noncompact case is handled in section 6 in their paper). The limit, which we repeat should be nonempty by an intersection number argument in the universal cover, should be stable as a limit of stable minimal surfaces and should have polynomial volume growth by a comparison argument (provided the limit is nondegenerate, which could concievably happen). Of course, merely knowing the limit is disjoint from M and satisfies the properness conditions discussed above suffices to garner a contradiction by the Frenkel theorem. It seems perturbations by eigenfunctions wouldn’t be necessary in this approach, so the full statement in the case of cylindrical ends might also be attained this way. Of course, the reader should keep in mind in this sketch some important detail may have been overlooked. The renormalized mean curvature flow deals with the incompleteness of the Gauss- ian metric in an entirely different way, where by instead of flowing by the mean curva- ture flow in the Gaussian metric the “worst” part of the speed function is discarded. By this we refer to the exponential term in the Gaussian mean curvature as discussed in section 3 above, which is clearly an artifact of the incompleteness of the metric (or perhaps better said, artifact of the reason for incompleteness). It also gives an instance (of course, not the first, considering for example the papers of Bernstein and Wang studying noncompact self shrinkers of low entropy, which clearly inspired this one) where the parabolic theory can be used fruitfully in a problem, in this case find- ing a stable critical point, which is “bad” from an analytical perspective, involving both incompleteness of the ambient metric and noncompactness of the submanifolds in question.

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