Closed embedded self-shrinkers of mean curvature flow

In this article we show the existence of closed embedded self-shrinkers in $\Bbb{R}^{n+1}$ that are topologically of type $S^1\times M$, where $M\subset S^n$ is any isoparametric hypersurface in $S^n$ for which the multiplicities of the principle curvatures agree. This yields new examples of closed self-shrinkers, for example self-shrinkers of topological type $S^1\times S^k\times S^k\subset \Bbb R^{2k+2}$ for any $k$. If the number of distinct principle curvatures of $M$ is one the resulting self-shrinker is topologically $S^1\times S^{n-1}$ and the construction recovers Angenent's shrinking doughnut.


Introduction
Consider a compact n-dimensional manifold Σ that is smoothly immersed in R n+1 via a map F 0 : Σ → R n+1 . A mean curvature flow of F 0 (Σ) is a family of smooth immersions F t : Σ → R n+1 where t ∈ R varies over some interval and for which holds for all t. Here H t (x) is the mean curvature of F t (Σ) at F t (x). In other words F t (Σ) flows along its mean curvature vector in R n+1 . Due to compactness of Σ such a flow necessarily becomes singular in finite time, see e.g. [13].
By the work of Huisken [13], Ilmanen [14] and White [21] rescaling F t (M ) near the singular time in an appropriate way leads to weak limits that are so called self-shrinkers, that is immersed manifolds whose mean curvature flow is given by dilations. These self-shrinkers then take a special role in the singularity theory of the mean curvature flow.
In this paper we use the theory of isoparametric foliations of the sphere S n to construct new examples of closed embedded self-shrinkers. Concretely we show: Theorem A. For any isoparametric hypersurface M in S n , n ≥ 2, for which the multiplicities m 1 and m 2 of the principal curvatures agree there is a closed embedded selfshrinker of topological type S 1 × M in R n+1 . This hypersurface is a union of homothetic copies of the leaves of the isoparametric foliation of S n associated to M .
The theory of isoparametric hypersurfaces of the sphere S n is very rich, so the above theorem can be used to produce self-shrinkers of novel topology (for example S 1 ×S k ×S k ⊂ R 2k+2 for k ∈ N or S 1 × SO(3)/(Z 2 × Z 2 ) ⊂ R 5 ). These hypersurfaces have previously been applied to the problem of mean curvature self-shrinkers by Chang and Spruck [6], who constructed for any isoparametric hypersurface M of the sphere S n a self-shrinking end that is asymptotic to the cone C(M ).
The terminology of isoparametric hypersurfaces will be recalled in Section 2.1. The proof of Theorem A works via a reduction of the shrinker condition to a geodesic equation in a two-dimensional manifold. Simple periodic solutions of this ordinary differential equation are then established by shooting methods very similar to [2], although the equation itself is quite different.
Denoting with g the number of principal curvatures of a regular leaf of the isoparametric foliation one has in the case g = 1 that the leaves become the latitudes of a sphere. The hypersurfaces found by Theorem A are then rotationally invariant under the O(n) action on R n+1 and topologically of type S 1 × S n−1 , the same topological type as the "shrinking donut" found by Angenent [2]. It is currently an open question whether there exist embedded rotationally invariant self-shrinkers of type S 1 × S n−1 in R n+1 other than Angenent's example, so we also remark: Proposition B. In the case g = 1 the construction of Theorem A gives Angenent's shrinking doughnut [2].
The structure of this article is as follows: In section 2 we recall the necessary facts about isoparametric foliations, explain the reduction of the self-shrinker problem to an ordinary differential equation, and remark on some elementary properties of the resulting equation. In section 3 the shooting argument is presented and Theorem A is shown with the exception of a technical proposition. Proposition B is also proved in section 3. The aforementioned technical proposition is shown in section 4.
The author would like to thank Peter McGrath for interesting and helpful discussions. The author gratefully acknowledges the support of Germany's Excellence Strategy EXC 2044 390685587, Mathematics Münster: Dynamics-Geometry-Structure.

Reduction and geodesic equation
In subsection 2.1 we recall some basic definitions and facts about isoparametric foliations. Subsection 2.2 explains the reduction procedure: a result due to Angenent [2] gives that a hypersurface is a self-shrinker if and only if it is minimal in some metric g Ang . The reduction theorem of Palais and Terng [17] is then applied in order to reduce the shrinker property to a geodesic equation on an open subset of R 2 equipped with a special metric. In subsection 2.3 we present this geodesic equation and simplify its form.

Isoparametric foliations on spheres
The geometric meaning of condition (1) is that the fibres of f form a (singular) transnormal system 1 , in particular they are all equidistant to each other. Condition (2) implies that the regular fibres of the foliation are of constant mean curvature in M (cf. [7]). If M is a space-form one even has that the individual principal curvatures of such a fibre are constant along the fibre. Foliations that arise from the fibres of an isoparametric function are called isoparametric foliations. A hypersurface is called an isoparametric hypersurface if it is a regular leaf of an isoparametric foliation.
The classification of isoparametric foliations in spheres was initiated by Cartan in [3][4][5]. This has proven to be a difficult problem and, despite a long and active history of research, it is in part still open. A significant part of the structure theory of these foliations was developed by Münzner in two seminal papers [15,16].
We review now some structural facts of isoparametric foliations of S n , cf. [7,11,15,16,20] for proofs and further information: (i) The principal curvatures of any regular fibre are constant along the fibre.
(ii) The number of distinct principal curvatures of a regular fibre is the same for any two regular fibres. Denoting this number by g one has that g ∈ {1, 2, 3, 4, 6}.
(iii) There are precisely two singular fibres V 1 and V 2 . One has dist(V 1 , V 2 ) = π g . These singular fibres are closed and minimal submanifolds of S n .
(vii) The volume of a regular fibre is given by where c a positive constant that does not vary in ϕ (but will be different for different foliations).
(viii) There is a homogenous polynomial F : R n+1 → R (called the Cartan-Münzner polynomial ) of degree g so that F | S n = cos(gϕ) and for which one has: Example. The cases g ∈ {1, 2, 3} were first classified by Cartan. The list of homogenous examples was completed by Takagi and Takahashi [18] based on previous work by Hsiang and Lawson [12], here an example is called homogenous if the the fibres of the foliation arise as the orbits of an isometric action on S n . The homogenous cases always arise as the principal orbit of the isotropy representation of a Riemannian symmetric space of rank 2, see [7] for more detailed remarks and references also for the other cases.
(ii) When g = 2 the isoparametric foliation is congruent to the foliation by Clifford tori S m 1 × S m 2 . One has . The integers m 1 , m 2 are arbitrary so long as m 1 + m 2 = n − 1, in particular m 1 = m 2 is possible. The fibres are the orbits of an isometric O(m 1 + 1) × O(m 2 + 1) action on S n .
(iii) When g = 3 one has m 1 = m 2 ∈ {1, 2, 4, 8}. The fibres of the foliation arise as the distance tubes of certain embeddings of the projective planes (iv) For g = 4 there is an infinite family, introduced by Ferus, Karcher, and Münzner in [11], which contains both homogenous as well as inhomogenous examples. Two additional homogenous cases beyond this family exist, else all examples belong to this family. Here m 1 = m 2 is possible.
(v) For g = 6 one has m 1 = m 2 ∈ {1, 2}, as was shown by Abresch [1]. For both cases there exist homogenous examples. If m 1 = m 2 = 1 it was shown by Dorfmeister and Neher [8] that the homogenous example is the only one.

Reduction for self-shrinkers
Definition 2.2. Let F be the Cartan-Münzner polynomial of an isoparametric foliation of S n . Define: Compare with the notion of F -invariant in [20].
Note that the f -invariant sets are precisely those sets that are unions of homothetic copies of the regular fibres of the isoparametric foliation -that is unions of sets of the form r · M ϕ for (r, ϕ) ∈ (0, ∞) × (0, π g ). (Recall that for x = 1 one has F (x) = cos(gϕ), where ϕ is the distance to the singular fibre Recall (cf. [2,9]) that a closed submanifold X ⊂ R n+1 is a self-shrinker under mean curvature flow (short: self-shrinker ) if and only if there is an τ > 0 such that X is a minimal hypersurface in R n+1 equipped with the metric (which we refer to as the shrinker metric): The parameter τ is related to the extinction time of X. By rescaling X if necessary we take τ = 1 in what follows.
(ii) The mean curvature vector of a fiber f −1 (r, ϕ) is given by where H(ϕ) is the mean curvature of M ϕ ⊂ S n , ν ϕ is the unit normal of rM ϕ in rS n equipped with g sh , and ν r is the unit normal of rS n in R n+1 equipped with g sh .
The proof is a standard calculation and from (ii) one sees that that the mean curvature of the fibres of f form a basic field of the Riemannian submersion, meaning that it is the horizontal lift of a vector field on the base manifold. Riemannian submersions with this property are the key ingredient in the reduction theory developed by Palais and Terng in [17], recall: Theorem (Palais-Terng, cf. Theorem 4 in [17]). Let π : (E, g E ) → (B, g B ) be a Riemannian submersion for which the mean curvatures of the fibres form a basic field, then for a k-dimensional submanifold X ⊂ B one has that π −1 (X) is minimal in E if and only if X is minimal in (B, V 2/k g B ). Here V 2/k g B is the metric given by Using (3) one gets (up to a constant factor): The problem of finding f -invariant hypersurfaces that are minimal with respect to g sh is then reduced to finding geodesic segments in (0, ∞) × (0, π g ) equipped with the metric We conclude: if and only if N := f (X) is a closed geodesic in (0, ∞) × (0, π g ) with respect to the metric (4).

Geodesic equation
For the metric (4) one gets the following geodesic equation, where α denotes the angle between dr dt and dϕ dt : Here Since we are not directly interested in the parametrisation of the geodesic but rather in its orbit we perform a substitution dt new dt old = 1 r G(r, ϕ) to simplify the equation: We simplify once more by letting θ(t) := g 2 ϕ(t), substituting dt new dt old = g 2 1 sin(2θ) , letting ξ(t) := g 2 ln( 2 g r(t)) and m := 2 g n = m 1 + m 2 + 2 g to get: Here If θ (t) = 0 then ξ may be (locally) given the form of a graph over θ. This graph obeys the following ODE: Here The ODE ( * ) can be formulated for all initial conditions (ξ, θ, α) ∈ R 3 . But in coordinates (ξ, θ) the domain (0, ∞) × (0, π g ) has been transformed to R × (0, π 2 ); so we are only interested in solutions where the ξ and θ components remain in that domain. We set The ODE ( * ) admits two trivial families of solutions in D, namely for any k ∈ Z: The first of these solutions lifts to the cone R >0 · M ϕ * over the minimal hypersurface of the isoparametric folation, which is a minimal submanifold of R n+1 . The second lifts to the round sphere, albeit with the singular fibres V 1 , V 2 removed.

Elementary properties of ( * ) and symmetry
We briefly note some elementary properties of solutions of ( * ), proofs are standard and are thus omitted.
(iii) Solutions of ( * ) with initial condition in D remain in D for all times.
For our investigation we are interested in periodic solutions of ( * ). These will be found with the help of a discrete symmetry of the ODE ( * ).
Definition 2.6. Define θ * := arctan( m 1 /m 2 ) and let Remark. Note that θ * is the solution in (0, π 2 ) of l(θ) = 0. Additionally the map S is an involution that reflects θ at θ * while sending cos α → − cos α and sin α → sin α. In the event that m 1 = m 2 one has θ * = π 4 and l(2θ Proposition 2.7. If m 1 = m 2 then for any x ∈ D one has S(γ 1 (−t)) = γ 2 (t) for all t ∈ R, where γ 1 , γ 2 are the solutions to ( * ) with initial conditions x and S(x), respectively. If x = S(x) then γ 1 = γ 2 in the above proposition and one gets S(γ 1 (t)) = γ 1 (−t). Noting that ( * ) is further invariant under transformations of the form α → α + 2πk for k ∈ Z then immediately gives a criterium for finding periodic solutions: Corollary 2.8. Let m 1 = m 2 and x ∈ D with S(x) = x, let γ be solution of ( * ) with initial condition x. If there are T = 0 and k ∈ Z so that then the ξ and θ components of γ are periodic and 2T is a period.

Existence of periodic curves
In light of Corollary 2.8 we wish to find geodesic segments that begin and end on the line θ = θ * , with both intersections being orthogonal. We begin with the following defintion: ) denote the solution of ( * ) with initial condition ξ(0) = ξ 0 , θ(0) = θ * and α(0) = π 2 . Then: Note that type 1 and type 2 are not exclusive, whereas a point is type 3 precisely if it is not type 1 or type 2. In fact a point that is both of type 1 and and type 2 corresponds to a curve segment that orthogonally meets the θ = θ * line at its start and its end. If m 1 = m 2 this leads to a solution for which the ξ and θ components are periodic, which corresponds a closed embedded self-shrinker in R n+1 of topological type S 1 × M , here M is diffeomorphic to the leaves of the isoparametric foliation. The following argument then finds a value ξ * 0 that is both type 1 and type 2.
Remark. For m 1 = m 2 one can see that ξ 0 = g 4 ln m is the only type 3 point, as in this case type 3 points correspond to embedded mean-curvature convex self-shrinkers that are topologically a sphere. By [13] the only closed embedded mean-curvature convex self-shrinkers are round spheres, which in this setting are given by the line ξ = g 4 ln m. Define: Proposition 3.2. We have: This proposition will be proven in Section 4. For now we make use of the following elementary lemma: For (ξ, θ) (t) ∈ D one has the following characterisation of extrema: θ ξ Figure 1: Examples of curves of different type. The blue curve is type 1 but not type 2, the green curve is both type 1 and type 2, the orange curve is type 2 but not type 1, the red curve is type 3.
This now gives: Proof. Since ξ * 0 is not type 3 by Proposition 3.2 (iii) it must be at least one of type 1 or type 2. We first assume that ξ * 0 is type 1 but not type 2, then we have a T > 0 so that for all > 0 small enough one gets: Since solutions of the ODE ( * ) vary continuously in the initial conditions (with respect to the topology of uniform convergence on compacta) one finds a neighbourhood U ( ) of ξ * 0 so that for all ξ 0 ∈ U ( ) one has can only have maxima when ξ(t) > g 4 ln m, which implies that cos(α ξ 0 (t)) = 0 for all t ∈ (0, ] and ξ 0 ∈ V . The above shows that for ξ 0 ∈ V ∩ U ( ) one has that ξ 0 is type 1, contradicting the definition of ξ * 0 .
The assumption that ξ * 0 is type 2 but not type 1 leads to a contradiction via similar argument, we carry this out: Note first that ξ ξ * 0 (0) < 0, so the next extremum must be a minimum (Lemma 3.3 implies that whenever ξ (t) = 0 one has either a maximum, a minimum, or ξ is the trivial solution ξ = g 4 m). This gives a T > 0 such that for all > 0 small enough: As before one gets a neighbourhood U ( ) of ξ * 0 so that this extends to all initial conditions ξ 0 ∈ U ( ), i.e. for all ξ 0 ∈ U ( ): cos(α ξ 0 (T + )) > 0 Since θ ξ 0 (t) = 1 one again gets for small enough a neighbourhood V of ξ * 0 with θ ξ 0 (t) = θ * for all t ∈ (0, ] and ξ 0 ∈ V . This shows that all points ξ 0 ∈ V ∩ U ( ) are of type 2 but not type 1, again contradicting the definition of ξ * 0 .
As discussed, this yields a periodic geodesic via Corollary 2.8 in the case m 1 = m 2 .
Together with Proposition 2.4 this proves the main theorem of the paper: Theorem A. For any isoparametric hypersurface M in S n , n ≥ 2, for which the multiplicities m 1 and m 2 of the principal curvatures agree there is a closed embedded selfshrinker of topological type S 1 × M in R n+1 . This hypersurface is a union of homothetic copies of the leaves of the isoparametric foliation of S n associated to M .
In the case m 1 = m 2 Proposition 3.4 remains true and yields a simple geodesic segment that starts and ends on the θ = θ * , meeting this line orthogonally in both places. In between the two ends one has θ > θ * and the same arguments give another geodesic arc with the same properties, except now θ < θ * .
It may be useful to connect the end-points of these two arcs by line segments θ = θ * (which are geodesics). Doing so gives a simple closed curve consisting piecewise geodesic segments and having external angle sum equal to 0. By the Theorem of Gauß-Bonnet this curve then encloses a total Gauß curvature of 2π. Such a curve is an essential ingredient in [10], where an adapted curve shortening flow is used to generated closed geodesics.

The case studied by Angenent
The case g = 1 yields an embedded self-shrinker in R n+1 of topological type S 1 × S n−1 , which is invariant under an isometric O(n) action on R n+1 . This is the case investigated by Angenent in [2]. In this subsection we relate Angenent's construction to ours and show that they give the same self-shrinker.
Here τ > 0 is related to the extinction time and in [2] one has τ = 1 4 . For ease of comparison we take τ = 1 and then up to a constant conformal factor the transformation implicit in (6) gives an isometry to the metric (4) on R >0 × (0, π), as is easy to see (recall g = 1). It follows that any geodesic of (7) is a reparametrisation of a geodesic in (4). Carrying out the additional coordinate changes of subsection 2.3 one sees that the following map sends the orbits of solutions of ( * ) to the orbits of geodesics of (7): g ξ cos(2θ), e 2 g ξ sin(2θ)).
In order to show Proposition B we start with the following lemma. It follows from elementary arguments using continuity of solutions of the relevant ODEs in initial conditions, similar to Proposition 3.4.
Proposition B. In the case g = 1 the construction of Theorem A gives Angenent's shrinking doughnut [2].
These points are not type 1, contradicting the definition of ξ * 0 . The contradiction for R * < e

Crossings in finite time
In this subsection we prove two useful lemmas that expand on the analysis of extrema in Lemma 3.3. The lemmas state that if ξ (t) points towards the g 4 ln m line then we reach this line in finite time, the same holding true for θ if θ (t) points toward θ * . The proof of Proposition 3.2 (i) uses Lemma 4.2 below, and Proposition 3.2 (iii) and Lemma 4.2 use Lemma 4.1.
Proof. If this were not true then ξ(t) > g 4 ln m for all t > t 0 . By Lemma 3.3 we would then have that any extremum of ξ is a maximum when t > t 0 . Since ξ (t 0 ) < 0 it follows that ξ has no extrema for times > t 0 and then ξ(t) is monotonically decreasing in t and bounded below by g 4 ln m by assumption, hence it must converge.
The condition α(t) → π 2 + kπ for some k ∈ Z gives sin α → (−1) k . For large times the dynamics of θ(t) are then given by This gives that θ(t) converges to either 0 or π 2 as t → ∞, depending on whether k is even or odd.
Assuming now θ(t) → 0 as t → ∞ one gets from Lemma 3.3 that θ(t) admits no extrema for t large enough, so sin α < 0 for t large enough. This gives α(t) ∈ 2πZ + (π, 3π 2 ) for t large enough. However for such t α (t) = sin α sin(2θ)(e 4 g ξ − m) + 2 cos α l(θ) is a sum of two strictly negative terms. α then decreases for large times, so there is an > 0 with | cos α| > for large enough t. In particular α (t) < −2 l(θ) for large t, where l(θ) converges to m 1 . This means that in finite time α exits the interval 2πk + (π, 3π 2 ) from the bottom, contradicting that sin α < 0 for all t large enough.
The case θ(t) → π 2 can be treated in the same way. This contradiction then implies the statement for ξ(t 0 ) > g 4 ln m and ξ (t 0 ) < 0. The case ξ(t 0 ) < g 4 ln m and ξ (t 0 ) > 0 is also completely analogous.
We first assume that e 4 g ξ(t) remains bounded as t → ∞, this implies that α (t) remains bounded and then from convergence of θ one gets that θ (t) = sin α sin(2θ) converges to 0. Since sin(2θ) remains bounded away from 0 one gets that α converges to some element of πZ. Since ξ(t) is not allowed to go to +∞ one gets that lim t→∞ α(t) ∈ π + 2πZ.
Performing a coordinate transform R = ln ξ the system of ODEs ( * ) becomes: From α(t) → π + 2πk for some k ∈ Z one gets R(t) → 0 and θ(t) → θ * . But the fixpoint (θ, R, α) = (θ * , 0, π + 2πk) is hyperbolic and at this point the above ODE has as linearization: The system then has a one dimensional stable manifold -this is the line Since we are assuming θ(t) < θ * for all t > t 0 the solution cannot lie on the stable manifold, yielding a contradiction.
To complete the proof of the lemma we must show that e 4 g ξ(t) cannot be unbounded under the hypothesis θ(t) < θ * for all t > t 0 and θ (t 0 ) > 0. First recall the graph form ( * * ): Whence if θ < θ * , ξ > g 4 ln m and dξ dθ > 0 one has d 2 ξ dθ 2 < 0, even becoming arbitrarily negative if ξ becomes arbitrarily large. So if ξ(t) is unbounded from above it cannot eventually be monotonic in θ (and hence in t by θ (t) > 0) and must admit maxima, in fact infintely many such maxima. Between two maxima there must be a minimum, which can only happen for values of ξ(t) less than g 4 ln m.
However at each minimum one has θ (t) = sin(2θ), which may be bounded from below since θ stays away from {0, π 2 }. Since the system ( * ) admits a Lipschitz constant on {(θ, ξ, α) | ξ < g 4 ln(m) + 1} one finds that at each minimum of ξ the parameter θ increases by some positive number admitting a bound from below. This contradicts the assumption that θ(t) < θ * for all t > t 0 .

Proof of Proposition 3.2 (i)
The proof of Proposition 3.2 (i) is divided into two parts. First we show that for ξ 0 large enough there is a T 2 > 0 so that the solution to ( * ) with initial value (ξ, θ, α) (t = 0) = (ξ 0 , θ * , π 2 ) has the property: So θ has an extremum at T 2 , which by Lemma 3.3 is a maximum and θ (T 2 + ) < 0 for small > 0. Then by Lemma 4.2 one has that θ reaches θ * in finite time and so ξ 0 cannot be of type 3. The proof then continues by contradiction, assuming that ξ 0 is not of type 1 means it must be of type 2. Being of type 2 means that ξ must travel all the way to some value < g 4 ln(m) where we have an extremum of ξ -all the while θ is not allowed to cross the line θ * .
The proof by contradiction is carried out in Lemma 4.6, here one assumes that conditions of this scenario have been set: there is some time T 3 at which ξ(T 3 ) = 4 g ln(m) all the while θ(t) > θ * and ξ (t) < 0 for t ∈ (0, T 3 ]. Using bounds for the value of θ(T 3 ) one is however able to show that even in this worst-case-scenario θ crosses the value θ * before any extremum of ξ is possible, contradicting the assumption that ξ 0 is type 2. Hence, since it cannot be type 3, it must have been type 1.
Proof. Note that one initially has d dt cos α| t=0 < 0 whence one gets ξ (t) < 0 for small t. By Lemma 4.1 ξ(t) then descends to g 4 ln m and does not have any extrema until after this value is reached, meaning cos α(t) < 0 for all t ∈ (0, t m ] and some t m ∈ R at which ξ(t m ) = g 4 ln m. For ξ 0 large enough there will then be some intermediate time T 1 < t m for which d dθ ξ(θ) = ξ (t) θ (t) = −1 holds, since either θ (t) = 0 for some t ∈ [0, t m ] or the graph ξ(θ) must descend from ξ 0 at θ * to g 4 ln m at some value θ < π 2 . In the second case the mean-value theorem implies that the graph achieves slope −1 at some point. Figure 2: A sketch of the argument for Proposition 3.2(i). The black curve gives the evolution of (ξ, θ) up until the extremum of θ. The dashed blue line describes the worst-case scenario for the evolution of (ξ, θ) (t) after this extremum. The red line, which crosses the line θ = θ * without any extrema of ξ, is an estimate of actual evolution starting on a certain point of the worst-case scenario.
Lemma 4.4. There are constants c 1 , c 2 , c 3 ∈ R >0 so that if ξ 0 is large enough one has Proof. Before beginning with the proper analysis one notes that by well definedness one has θ(T 1 ) ≤ π 2 , whence by the mean value theorem ξ 0 − ξ(T 1 ) ≤ π 2 − θ * , some finite value bounded from above.
We will now prove that ξ 0 is of type 1 by contradiction. Lemma 4.6. If ξ 0 is large enough then it is of type 1.
Proof. If ξ 0 is large enough by Lemma 4.5 there is a T 2 > 0 for which θ (T 2 ) = 0, which corresponds to a local maximum of θ. This means θ reaches θ * in finite time by Lemma 4.2 and so ξ 0 is not type 3. We now assume it is not type 1, so it must be type 2.
Hence ξ(t) has an extremum before θ(t) reaches θ * . We let T 3 > T 2 denote the time of this extremum and note first that θ(t) ∈ (θ * , θ * + 1 is a maximum and any further extremum of θ must take place behind the line θ = θ * ).

Proof of Proposition 3.2 (ii)
We consider the solution curve with initial condition ξ 0 = g 4 ln m + and show that this is not of type 1 for sufficiently small. To do this we assume that it is of type 1 -only to later arrive at a contradiction. If it were of type 1 then there is a T > 0 with θ(T ) = θ * and ξ (t) < 0 for all t ∈ (0, T ). Since θ can only have maxima when θ > θ * , we find that the trajectory {(θ, ξ) (t) | t ∈ [0, T ]} must be the union of two graphs of ξ over θ. The maximum of θ occurs at the point denoted by (ξ 2 , θ 2 ) in Figure 3.
In the upper graph one has that the slope dξ dθ starts at 0 and must go to −∞ (which occurs when θ (t) = 0). Along the way ξ has been negative and one verifies that ξ 2 has decreased to a value far enough below g 4 ln m (c.f. Lemma 4.10). When we then switch to the lower graph the e 4 g ξ − m term in the ODE for d 2 ξ dθ 2 (θ) will be large enough to push dξ dθ over the value 0 before θ reaches θ * , contradicting the assumption that ξ 0 was type 1.
Note that this does not prove that g 4 ln m+ is of type 2, because the proof by contradiction assumes that θ(t) has a maximum.
In what follows ξ, θ and α will denote the components of the solution of ( * ) with initial condition (ξ, θ, α) (0) = ( 4 g ln m + , θ * , π 2 ). The proof begins by investigating an auxilliary value θ 1 , which is defined to be the least (and for small only) value of θ for which one has dξ dθ (θ 1 ) = −1 in the upper graph.
Proof. Assuming that ξ 0 is of type 1 means that there is a time T > 0 for which θ(T ) = θ * and ξ (t) = 0, θ(t) = θ * for all t ∈ (0, T ). Since θ (0) = sin(2θ * ) > 0 one Figure 3: The figure sketches the argument for Proposition 3.2(ii). The dashed blue line denotes the form that (ξ, θ) (t) must be if the initial condition were type 1. The red line, which has an extremum of ξ, is an estimate of the actual evolution starting at (ξ 2 , θ 2 ).
finds that θ (t) > 0 for small t, whence θ(t) must go through an extremum before it can go back to θ * and there is a T 2 < T so that θ (T 2 ) = 0. On the other hand one has ξ (0) = − sin(2θ * )m(e satisfying Lemma 4.7 there is no pair (T 1 , T 2 ) satisfying the lemma with T 1 > T 1 .
This allows us to introduce the following notation: Lemma 4.9. For small enough one has that ξ(T 1 ) ≤ g 4 ln m.
Lemma 4.11. If is small enough then ξ 0 = g 4 ln m + is not type 1.
Proof. As noted before the assumption that ξ 0 = g 4 ln(m) + is type 1 leads to (ξ, θ) being the union of two graphs of ξ over θ. In the previous lemmas we investigated the upper graph and found that it ends at the turning point (ξ 2 , θ 2 ).