Abstract
We study solutions of high codimension mean curvature flow defined for all negative times, usually referred to as ancient solutions. We show that any compact ancient solution whose second fundamental form satisfies a certain natural pinching condition must be a family of shrinking spheres. Andrews and Baker (J Differ Geom 85(3):357–395, 2010) have shown that initial submanifolds satisfying this pinching condition, which generalises the notion of convexity, converge to round points under the flow. As an application, we use our result to simplify their proof.
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1 Introduction
In this paper, we consider ancient solutions to the mean curvature flow with pinched second fundamental form. A family of smooth immersions \( F : {\mathcal {M}}^n\times ( t_0 ,t_1) \rightarrow \mathbb {R}^{n+ k}\) is a solution to the mean curvature flow if
where H(x, t) denotes the mean curvature vector. We will always assume that \( n\ge 2\), \(k \ge 1\), and that \( {\mathcal {M}}^n\) is a complete smooth manifold. A solution is referred to as ancient if \(t_0 = \infty \).
The mean curvature flow is (weakly) parabolic and hence illposed backwards in time, however ancient solutions are interesting for several reasons. They arise naturally as tangent flows near singularities [13, 17, 18, 25, 26], and are therefore models for singularity profiles of the flow [17].
Solutions that shrink homothetically provide the first of many examples of ancient solutions. Writing \(\mathcal M_t^n := F(\mathcal M^n,t)\), we have the family of round spheres \(\mathcal M_t^n = \mathbb S^n_{R(t) }\) with \( R(t) = \sqrt{ 2 n t } \) and the shrinking cylinders \(\mathcal M_t^n = \mathbb S^{nm}_{R(t) }\times \mathbb {R}^m\), \(R(t) = \sqrt{ 2 (nm) t}\). The Angenent oval [2] (also known as the paperclip solution [21]) is an example of a nonhomothetically shrinking compact ancient solution to the curve shortening flow, obtained by gluing together two copies of the grim reaper (or the hairpin solution [3]). The grim reaper itself, the bowl solitons and other translating solutions are not only ancient but eternal (\(t_1 = \infty \)). In [14], Haslhofer and Hershkovits construct convex ancient solutions which flow from each of the cylinders \(\mathbb S^{nm} \times \mathbb R^m\) at \(t_0 = \infty \) to a round \(\mathbb S^n\) as \(t \rightarrow 0\). Bourni, Langford and Tinaglia recently constructed the first example of a compact ancient solution which is interior collapsing [6], see also [24].
In codimension one, a great deal is known about the mean curvature flow under natural curvature conditions such as convexity. In the present work we build mainly on a recent paper by Huisken and Sinestrari [19], where estimates proven earlier by Huisken [15] are used to give several characterisations of the shrinking sphere amongst convex ancient solutions (similar results were proven in [14] by other methods). In particular, Huisken and Sinestrari show that any mean convex ancient solution with uniformly pinched second fundamental form,
must be a family of shrinking spheres. Similar results also hold for the Ricci flow [7, 9], and for a large class of fully nonlinear flows of hypersurfaces [20].
In higher codimensions, far less is known about the mean curvature flow in general. This is due to the presence of the normal curvature, which complicates the structure of evolution equations for geometric quantities along the flow, and the fact that the second fundamental form is now a normal bundlevalued tensor, so that no obvious notion of convexity is available. We continue here the study of a natural pinching condition, \(h^2 \le c H^2\), which Andrews and Baker [1] (cf. [16]) have successfully employed as an alternative to convexity in higher codimensions. They demonstrated that for values \(c \le \min \{\frac{4}{3n}, \frac{1}{n1}\}\), this condition is preserved by the flow, and solutions satisfying it flow to round spheres. To motivate this condition, we note that in Euclidean space, a mean convex hypersurface satisfying \(h^2 \le \frac{1}{n1}H^2\) is automatically weakly convex (see Lemma 3.1), while a general submanifold satisfying this condition has nonnegative sectional curvature [8], and in fact, nonnegative curvature operator (this we prove in Lemma 3.2 below).
Our main result is a high codimension analogue of the sphere characterisation of Huisken and Sinestrari [19], assuming the AndrewsBaker condition in place of (2).
Theorem 1.1
Let \(F: {\mathcal {M}}^n \times (\infty ,0) \rightarrow {\mathbb {R}}^{n+k}\), \(n \ge 2\), be a compact ancient solution to mean curvature flow with nonvanishing mean curvature vector. Suppose there is a constant \(\varepsilon >0\) such that for each \(t <0\), the second fundamental form of \(\mathcal M_t^n\) satisfies
where
Then \({\mathcal {M}}^n_t\) is a family of shrinking spheres.
The constant \(c_n\) is optimal for \(n \ge 4\), since, for \(k=1\) and every \(n\ge 2\), Haslhofer and Hershkovits [14] have constructed an ancient solution other than the shrinking sphere which satisfies \(h^2 \le \frac{1}{n1}H^2\). The values \(c_{2}\) and \(c_3\) come out of the analysis in [1], rather than geometric considerations, and might be improved. There is however an immersion of the Veronese surface into \(\mathbb R^5\) which shrinks homothetically under the mean curvature flow and satisfies \(h^2 = \frac{5}{6} H^2\), so one cannot hope to do better than \(c_2 = \frac{5}{6}\). We note that in [23], a similar theorem was proven assuming the extra condition that the second fundamental form is uniformly bounded.
The paper is arranged as follows. In Sect. 3, we show that any complete submanifold of Euclidean space with bounded, nonvanishing mean curvature, and which satisfies the pinching condition (3), must be compact. This is a natural high codimension generalisation of a theorem of Hamilton [11], which asserts the compactness of complete hypersurfaces satisfying (2). In Sect. 4 we use this compactness result to prove Theorem 1.1, as well as a further characterisation of the shrinking sphere as the only weakly pinched ancient solution with typeI curvature growth. We then apply our results to provide an alternate proof of the convergence theorem of Andrews and Baker which does not make use of Stampacchia iteration or a gradient estimate. Finally, in Sect. 5, we prove a classification analogous to Theorem 1.1 for high codimension ancient solutions in the sphere.
We would like to thank Mat Langford for many helpful discussions which have benefited this work.
2 Preliminaries
Let \({\mathcal {M}}^n\) be a smooth, immersed submanifold of a Riemannian manifold \({\mathcal {N}}^{n+k}\). Denote the curvature operator of \({\mathcal {N}}^{n+k}\) by \({\bar{R}}\). We will usually take \(\mathcal N^{n+k}\) to be Euclidean space—only in the final section do we consider \({\mathcal {N}}^{n+k} = {\mathbb {S}}^{n+k}\). We work in local orthonormal frames for the tangent bundle \( T \mathcal M \) and normal bundle \( N \mathcal M\), denoted by \(\{e_i\}\) and \( \{ \nu _\alpha \} \) respectively. When working in such bases, unless otherwise specified, we will sum over repeated indices whether they are raised or lowered. For example, we may write the mean curvature vector as
We can then write familiar equations such as Codazzi’s equation as
Gauss’ equation is given by
and for the normal curvature we have
In fact, the normal curvature depends only on the traceless second fundamental form. Writing
we have
2.1 Evolution equations
Equations for the evolution of all relevant geometric quantities along the flow are computed in detail in [1]. For the second fundamental form we have
and taking the trace with respect to i and j,
The equations for \( h^2\) and \(H^ 2 \) are then
and
The last term in the evolution equation for \(h^2\) can be expressed purely in terms of the normal curvature,
We will find it convenient to denote the reaction terms above by
2.2 Preservation of pinching
We consider the quadratic quantity
where c and a are positive constants. Combining the evolution equations for \( h ^ 2 \) and \( H^ 2\) yields
The gradient estimate
which is proven as in Hamilton [10] and Huisken [15], shows that the gradient terms in (9) are strictly negative if \( c < \frac{ 3}{n+2}\). Careful estimating shows that for \(c < \frac{4}{3n}\) we also have \(R_1  cR_2 < 0\) (see [1]), so by the maximum principle:
Lemma 2.1
Let \(F:\mathcal M ^ n \times [0,T)\rightarrow \mathbb {R}^{n+k}\) be a solution to the mean curvature flow such that \(\mathcal M_0\) satisfies
for some \(a > 0\) and \( c \le \frac{4}{3 n }\). Then this condition is preserved by the mean curvature flow.
As a consequence, we see that the flow preserves both \(H >0\) and \(h^2  c_n H^2 \le  \varepsilon H^2\).
3 Curvature pinching
We begin with a purely algebraic calculation, which in particular shows that sufficiently pinched symmetric matrices are positive/negative definite.
Lemma 3.1
Let B be a symmetric matrix. Fo any two eigenvalues \(\kappa _1, \kappa _2 \) of B, there holds
In particular, we have
Proof
We expand the righthand side
and note that
This gives
\(\square \)
Hence, we see that if \(B^2  \frac{1}{n1} ({{\,\mathrm{tr}\,}}B) ^2 \le 0\), then all the eigenvalues of B have the same sign. In particular, if \({{\,\mathrm{tr}\,}}B > 0\) then B is positive definite. This allows us to pull the pinching condition (3) back to an intrinsic curvature condition on \(\mathcal M^n\).
Lemma 3.2
An immersed submanifold \(\mathcal M^n \subset {\mathbb {R}}^{n+k}\), \(n\ge 2\), whose second fundamental form satisfies
for some \(\varepsilon > 0 \) has curvature operator pinched by
Proof
It suffices to work over a single point \(p \in \mathcal M^n\). Using the Gauss equation, we split the curvature operator into components
For any fixed index \(\alpha \) we can choose an orthonormal frame \(\{e_i\}\) for \(T_p \mathcal M\) which diagonalises \(h_\alpha \), in which case the bivectors \(e_i \wedge e_j\) with \(i\not = j\) diagonalise \(\mathcal R_\alpha \). The corresponding eigenvalues are (no summation)
so that by Lemma 3.1,
for any \(\omega \in \bigwedge ^2 T_p \mathcal M\). Taking the sum, we obtain
\(\square \)
As a direct application of the above estimate and a theorem of NiWu [22], we obtain a high codimension version of the main theorem in [11].
Corollary 3.3
Any complete immersed submanifold \(\mathcal M^n\) of \({\mathbb {R}}^{n+k}\) with bounded, nonvanishing mean curvature vector, and which has second fundamental form pinched by
for some \(\varepsilon > 0\), is compact.
Proof
Since H is bounded, the pinching ensures that h, and therefore the full curvature operator, are also bounded. Taking traces of the Gauss equation shows that the scalar curvature \(Sc\) is given by
so applying Lemma 3.2, we see that the curvature operator is pinched by
By a result of NiWu [22], \(\mathcal M^n\) must then be compact. \(\square \)
4 Ancient solutions in Euclidean space
The following theorem is due to Huisken and Sinestrari [19] in case \(k=1\). With the estimates of Andrews and Baker in place, the proof in higher codimensions is the same.
Theorem 4.1
Let \(F: \mathcal M^n \times (\infty , 0) \rightarrow {\mathbb {R}}^{n+k}\) be a compact ancient solution to mean curvature flow satisfying the pinching condition \(h^2  c_n H^2 \le  \varepsilon H^2\) for some \(\varepsilon > 0\). If, in addition, the area \(\mu (\mathcal M_t^n) = \int _{\mathcal M_t^n} d\mu _g\) satisfies the decay condition
for some positive constants c, r and T independent of t, then \(\mathcal M_t^n\) is a family of shrinking spheres.
Proof
We show that for small enough \(\sigma >0\), the function
vanishes identically along the flow. It was shown in [1, Lemma 5] that for pinched solutions, there are constants \(p \gg 1\) and \(\sigma \sim \frac{1}{\sqrt{p}}\) depending only on n and \(\varepsilon \) such that
(Note that here in Proposition 12 of [1], we retain the term \(\int H^2 f_\sigma ^p \, d\mu _g\) in the first line of equation (41L)). Setting \(\gamma := 1 + \frac{2}{\sigma p}\), we have \(f_\sigma ^{\gamma p} \le H^2 f_\sigma ^p\), and Hölder’s inequality implies that
and in turn,
Let \(\varphi \,= \int f_\sigma ^p \, d\mu _g\) and suppose that \(\varphi (s) >0\) for some \(s \in (\infty , T]\). Since \(\varphi \) may not increase in time, this implies that \(\varphi (t) > 0\) for all \( t\in (\infty , s]\), and we have
Integrating in time then yields
Since \(\sigma p \sim \sqrt{p}\) we may choose p so large that \(\sigma p >2r\), in which case the righthand side of the last inequality becomes unbounded as \(t \rightarrow \infty \). This is a contradiction, so it must be the case that \(\varphi (s) = 0\) for all \(s \in (\infty , T]\), and hence for all \(s <0\). \(\square \)
In other words, a pinched ancient solution with sufficiently slow area decay must be a family of shrinking spheres. To control the area of pinched, codimension one solutions, Huisken and Sinestrari [19] use a Gauss–Bonnettype result. A similar approach works in all codimensions for flows of surfaces.
Proof of Theorem 1.1 (\(n=2\)) Lemma 3.2 says that the Gauss curvature \(\kappa \) of \(\mathcal M_t^n\) satisfies
and since \(\mathcal M_t^n\) evolves smoothly and shrinks to a round point as \(t \rightarrow 0\) by [1], it is diffeomorphic to \(\mathbb S^2\) (and not \(\mathbb RP^2\)) at every fixed time. We may therefore apply the Gauss–Bonnet theorem to conclude that
Substituting into the area decay formula then yields
which we integrate in time to obtain
Theorem 4.1 can then be applied to finish. \(\square \)
It is not clear that this type of argument generalises to higher dimensions and codimensions, however we are still able to prove an area decay estimate in this setting by applying Corollary 3.3. We will find it convenient to introduce a dichotomy analogous to the one used when classifying finitetime singularities  we say that an ancient solution is of type I if there are positive constants C and T such that
and of type II in case
Proof of Theorem 1.1 (\(n \ge 3\)) We proceed by ruling out typeII blowdowns, and then showing that any typeI solution has slow enough area decay to apply Theorem 4.1.
Suppose that F is of type II. To derive a contradiction, we choose \((x_j,t_j) \in {\mathcal {M}}^n \times [j, 0)\), \(j \in {\mathbb {N}}\), so that
We set \(L_j = H(x_j,t_j)^2\), and note that the typeII condition implies
Following Huisken–Sinestrari [19] (cf. [12, 17]), we consider the sequence of rescaled and translated flows
Let \(H_j\) denote the mean curvature vector corresponding to \(F_j\) and observe that
Our definition of \((x_j,t_j)\) ensures that for \(\tau \in (0, t_jL_j)\),
so we have the bound
This implies that \(H_j(\cdot , \tau )^2 \le 2\) for all \(\tau \in (0, \frac{1}{2}t_j L_j)\), which combined with the pinching assumption provides a uniform bound for the full second fundamental form. Thus the sequence of rescalings subconverges smoothly to a nonflat limiting solution of mean curvature flow that is defined for all positive times, and yet has compact timeslices by Corollary 3.3; this is a contradiction.
We are left with the possibility that F has typeI curvature growth on (without loss of generality) the time interval \((\infty , 1]\). That is,
for some \(C>0\). The area decay formula then yields
which we integrate to obtain
Hence \(\mathcal M_t^n\) is totally umbilic for all times by Theorem 4.1. \(\square \)
Theorem 1.1 implies the following further characterisation of the shrinking sphere. Huisken and Sinestrari prove an analogous result for \(k =1\) [19], and we adapt their argument.
Theorem 4.2
Let \(F: {\mathcal {M}}^n \times (\infty ,0) \rightarrow {\mathbb {R}}^{n+k}\), \(n \ge 2\), be a compact ancient solution to mean curvature flow with nonvanishing mean curvature vector which is weakly pinched,
and has typeI curvature growth. Then \(\mathcal M_t^n\) is a family of shrinking spheres.
Proof
Recall that the typeI condition says
This implies the bound
for any \(p \in \mathcal M\), so for any pair of points \(p, q \in \mathcal M\) there holds
as long as \(t \le C^{1} \left( \sup _{p, q \in \mathcal M^n} F(p, T)  F(q,T)\right) ^2\).
Suppose now for a contradiction that F is not uniformly pinched. That is, we assume there is a sequence of times \(t_j \rightarrow \infty \) and points \(x_j \in \mathcal M^n\) such that
We define a sequence of rescaled flows,
which, as a consequence of (13), all take values in a single compact subset of \({\mathbb {R}}^{n+k}\). The typeI condition provides a uniform upper bound for the functions \(H_j\), which translates to an upper bound for the sequence \(h_j\) via pinching, so the sequence \(F_j\) converges to a compact, weakly pinched solution \(F_\infty \) defined on \(\mathcal M^n \times [\frac{3}{2},1]\). The mean curvature of the limit \(H_\infty \) satisfies \(H_\infty  \ge 0\), so (9) and the strong maximum principle imply that \(H_\infty  > 0\) for \(t > \frac{1}{2}\). Our choice of sequence then implies the existence of an \(x_\infty \in \mathcal M\) such that
in which case another application of the strong maximum principle to (9) yields
This forces the gradient and reaction terms in (9) to vanish on all of \(\mathcal M \times [\frac{1}{2}, 1]\), which implies that F is a shrinking sphere solution (see [5]), contradicting (14). \(\square \)
4.1 Convergence to round points
For dimensions \(n \ge 3\) our results give an alternate proof of the convergence theorem due to Andrews and Baker, which says that pinched solutions shrink to round points. Consider a compact solution \(F : \mathcal M^n \times [0,T) \rightarrow {\mathbb {R}}^{n+k}\) such that T is maximal and the pinching condition (3) is satisfied at \(t =0\). For convenience we assume F is scaled so that \(T >1\). That the pinching condition is preserved by the flow follows from the maximum principle applied to (9). If F undergoes a typeII singularity as \(t \rightarrow T\), then we perform a Hamilton blowup, similar to that in the proof of Theorem 1.1. By assumption the quantity
blows up as \(j \rightarrow \infty \). Thus, setting \(L_j = H(x_j,t_j)^2\), the sequence of rescalings
subconverges to a complete solution defined for times \(\tau \in [1,\infty )\), which is uniformly pinched and has bounded, nonvanishing mean curvature. This is a contradiction, since Corollary 3.3 implies that the solution is compact on every timeslice, and must therefore become singular in finite time. We conclude that F undergoes a typeI singularity, that is, there is a constant \(C > 0\) such that
In this case, we define for \(j \in {\mathbb {N}}\) the sequence of rescalings
As in the proof of Theorem 1.1, the type I assumption provides a uniform radius bound, as well as the curvature bound,
so the sequence subconverges to a uniformly pinched ancient solution on the time interval \(\tau \in (\infty , 2]\). Corollary 3.3 again ensures compactness on timeslices, so we may apply Theorem 1.1 to conclude that the limit is a shrinking sphere.
5 Pinched ancient solutions in the sphere
In this section we prove an analogue of Theorem 1.1 for the mean curvature flow in a spherical background. We consider solutions \(F: \mathcal M^n \times (\infty , 0 ) \rightarrow \mathbb S ^{n+k}_{R}\), where \( \mathbb S ^{n+k}_R\) is the \((n+k)\)sphere of radius \(R>0\) with sectional curvature \( K = R^{2}\).
Unlike Euclidean space, the sphere contains compact minimal surfaces. The simplest examples are the totally geodesic spheres, such as the equators. Minimal surfaces generate static solutions to the mean curvature flow, which are not only ancient but eternal. Further examples of ancient solutions are the shrinking spherical caps, which flow out of geodesic spheres on the equator and shrink to round points at a pole. The following theorem says that sufficiently pinched ancient solutions must be of one of these two forms (cf. [19] Theorem 6.1).
Theorem 5.1
Let \(F : \mathcal M^n \times (\infty , 0) \rightarrow {\mathbb {S}}_R^{n+k}\) be a compact ancient solution to the mean curvature flow satisfying \(H > 0\) for all times.

1)
If there is a \(\delta >0 \) such that for every \(t \in (\infty , 0)\) there holds
$$\begin{aligned} h^2  \frac{1}{3} H^2 \le (2\delta ) K, \qquad n =4, \end{aligned}$$or
$$\begin{aligned} h^2  \frac{1}{n1} H^2 \le 2K, \qquad n \ge 5, \end{aligned}$$then \(\mathcal M_t^n\) is either a shrinking spherical cap or a totally geodesic sphere.

2)
If \( h^2 \le \frac{4}{3n} H^2\) for every \( t \in (\infty , 0)\) then \( \mathcal M_t^n\) is a shrinking spherical cap.
Proof
1). Consider the auxiliary function with
and \(\varepsilon \in (0,1)\) to be fixed later. The evolution of this function was computed in [5], and is given by
with \(R_1\) and \(R_2\) as above. In all dimensions, the pinching condition implies that
and we still have the gradient estimate
so for \(\varepsilon \) sufficiently small,
The righthand side is nonpositive for all \(n \ge 4\) and this term can be discarded. Before estimating the reaction terms, we introduce some notation. Around any point in \(\mathcal M^n\) we choose a local orthonormal frame \(\{ \nu _\alpha \}\) for \(N \mathcal M\) such that \(\nu _1 = \frac{H}{H}\) and a local orthonormal frame \(\{e_i\}\) for \(T \mathcal M\) which diagonalises \(h_1\). We then write
so that . Baker computes (we set \(\sigma =0\) in Proposition 5.2 of [4]),
and derive the estimate
so we have
for any constant \(\vartheta \), which we take to be small and positive. In case \(n=4\), we now use the pinching to bound
and observe that the righthand side can be made nonpositive by taking \(\vartheta = 2\varepsilon \) sufficiently small. When \(n > 4\), this is possible even for \(\delta =0\), so in all dimensions we obtain
Assume now that there is a time \(t_1 \in (\infty , 0)\) such that \(\mathcal M_{t_1}^n \) is a sphere. This implies that \( f \not \equiv 0\) on \( \mathcal M_{t_{1}}^n\), so by (16),
for all \( t< t_i\). It follows that \( \max _{ \mathcal M_t^n} f \rightarrow \infty \) as \(t \rightarrow \infty \), but our pinching condition implies
so that \( f \le 1\) for all times. This is a contradiction, so \(\mathcal M_t^n\) must be totally umbilic for all \(t < 0\), and is therefore either a shrinking spherical cap or totally geodesic sphere.
2) We now set \(b = 0\) in the definition of f. Proceeding exactly as before, we find that the pinching condition \(h^2 \le \frac{4}{3n}H^2\) is exactly what is required to ensure that \({{\,\mathrm{II}\,}}\le 4nkf\), and also implies that \({{\,\mathrm{I}\,}}\le 0\). The same contradiction argument used above then shows that \(\mathcal M_t^n\) is totally umbilic for all times, and since the pinching rules out geodesic spheres, \(\mathcal M_t^n\) must be a shrinking spherical cap. \(\square \)
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Lynch, S., Nguyen, H.T. Pinched ancient solutions to the high codimension mean curvature flow. Calc. Var. 60, 29 (2021). https://doi.org/10.1007/s00526020018881
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DOI: https://doi.org/10.1007/s00526020018881