Singularities of mean convex level set flow in general ambient manifolds
We prove two new estimates for the level set flow of mean convex domains in Riemannian manifolds. Our estimates give control - exponential in time - for the infimum of the mean curvature, and the ratio between the norm of the second fundamental form and the mean curvature. In particular, the estimates remove a stumbling block that has been left after the work of White [Whi00, Whi03, Whi11], and Haslhofer-Kleiner [HK13], and thus allow us to extend the structure theory for mean convex level set flow to general ambient manifolds of arbitrary dimension.
Let be a Riemannian manifold. For any mean convex domain we consider the level set flow starting at , i.e. the maximal family of closed sets starting at that satisfies the avoidance principle when compared with any smooth mean curvature flow [ES91, CGG91, Ilm94]. The level set flow of coincides with the smooth mean curvature flow of for as long as the latter is defined, but provides a canonical way to continue the evolution beyond the first singular time. Mean convexity is preserved also beyond the first singular time in the sense that whenever .
In the last 15 years, Brian White developed a deep regularity and structure theory for mean convex level set flow [Whi00, Whi03, Whi11], and recently the first author and Kleiner gave a new treatment of this theory [HK13]. Concerning the size of the singular set, White proved that the singular set of any mean convex flow has parabolic Hausdorff dimension at most [Whi00, Thm. 1.1], see also [HK13, Thm. 1.15]. Concerning the structure of the singular set, the main assertion one wants to prove is that all blowup limits of a mean convex flow are smooth and convex until they become extinct. In particular, one wants to conclude that all tangent flows of a mean convex flow are round shrinking spheres, round shrinking cylinders, or static planes of multiplicity one. While the theorem about the size of the singular set is known in full generality, the structure theorem has been proved up to now only under some additional assumptions [Whi03, Thm. 1], [Whi11, Thm. 3] and [HK13, Thm. 1.14]. Namely one has to restrict either to blowups at the first singular time, or to low dimensions, or to the case where the ambient manifold is Euclidean space.
As explained in [Whi11, Appendix B], the missing step to extend the structure theorem to general ambient manifolds of arbitrary dimension is to prove that the ratio between the smallest principal curvature and the mean curvature has a finite lower bound on the regular points contained in any compact subset of space-time.
The purpose of this work is to remove this stumbling block. To this end, we prove two new estimates for the level set flow of mean convex domains in Riemannian manifolds.
To state our estimates, we denote by the set of regular boundary points at time . Our first main estimate gives a lower bound for the mean curvature.
Theorem 1.1 (Lower bound for ).
There exist constants and such that
Our estimate from Theorem 1.1, as well as our second main estimate below, depends exponentially on time. It is clear from simple examples (e.g. flows in hyperbolic space), that this exponential behavior in time is the best one can possibly get.
Our second main estimate controls the ratio between the norm of the second fundamental form and the mean curvature.
Theorem 1.3 (Upper bound for ).
There exist constants and such that
Theorem 1.3 shows that all principal curvatures are controlled by the mean curvature, and thus in particular provides a (two-sided) bound for the ratio . As explained above, this exactly fills in the missing piece that is needed to extend the structure theorem for mean convex level set flow to the general case without restrictions on subsequent singularities, the ambient manifold, and the dimension. We thus obtain:
Theorem 1.5 (Structure theorem).
Let be a mean convex domain in a Riemannian manifold. Then all blowup limits of its level set flow are smooth and convex until they become extinct. In particular, all backwardly selfsimilar blowup limits are round shrinking spheres, round shrinking cylinders, or static planes of multiplicity one.
Theorem 1.5 gives a general description of the nature of singularities of mean convex level set flow in arbitrary ambient manifolds. As mentioned above, this generalizes the structure theorems from [Whi03, Thm. 1], [Whi11, Thm. 3] and [HK13, Thm. 1.14].
Applications. Let us now discuss some applications of the above theorems.
Our first application concerns topological changes in mean convex mean curvature flow. In [Whi13], White proved that under mean convex level set flow elements of the -th homotopy group of the complementary region can die only if there is a shrinking singularity for some , assuming that or that the ambient manifold is Euclidean. Thanks to Theorem 1.5 we can remove the assumption on the dimension and the ambient manifold, and thus obtain:
Corollary 1.6 (Topological change).
Let be a mean convex domain in a Riemannian manifold. If for some there is a map of the -sphere into that is homotopically trivial in but not in , then at some there is a singularity of the flow at which the tangent flow is a shrinking for some .
Our second application concerns the estimates for mean convex level set flow in the setting of Haslhofer-Kleiner [HK13]. These estimates are based on the noncollapsing condition that each boundary point admits interior and exterior balls of radius comparable to the reciprocal of the mean curvature at that point [Whi00, SW09, And12]. It has been unknown up to now if this noncollapsing condition holds for mean convex level set flow in general ambient manifolds of arbitrary dimension. Combining Theorem 1.1 and Theorem 1.5 we can answer this in the affirmative:
Corollary 1.7 (Noncollapsing).
Let be a mean convex domain in a Riemannian manifold. Then there exists a positive nonincreasing function such that each admits an interior and exterior ball tangent at of radius at least . In particular, all estimates from [HK13] apply in the setting of mean convex level set flow in general ambient manifolds of arbitrary dimension.
Our third application concerns a sharp estimate for the inscribed and outer radius for mean convex level set flow in Riemannian manifolds. In [Bre15] and [Bre13], Brendle proved sharp bounds for the inscribed radius and outer radius at points in a smooth mean convex mean curvature flow where the mean curvature is large. The first author and Kleiner [HK14] found a shorter proof of Brendle’s estimate, which also works in the nonsmooth setting provided that one has some noncollapsing parameter to get started. Thanks to Corollary 1.7 the argument from [HK14] is applicable for mean convex level set flow in general ambient manifolds, and we thus obtain:
Corollary 1.9 (Sharp estimate for inscribed and outer radius).
Let be a mean convex domain in a Riemannian manifold. Then for any positive nonincreasing function , there exists a positive nonincreasing function depending only on and such that every with admits an interior ball of radius at least and an exterior ball of radius at least .
The estimates are very easy to prove for smooth flows, so let us start by explaining this: First, from the evolution equation for the mean curvature [Hui86, Cor. 3.5],
and the maximum principle, one sees that the minimum of the mean curvature can deteriorate at most exponentially in time. Second, combining the evolution equation for the square norm of the second fundamental form [Hui86, Cor. 3.5],
and the evolution equation for the mean curvature, one sees that the maximum of increases at most exponentially in time. We emphasize that the above estimates crucially rely on one another. Namely, to control the reaction terms in the evolution for we need the lower bound for from the first step.
Having sketched the argument in the smooth case, the main difficulty is to generalize this argument to the level set flow beyond the first singular time. As in White [Whi11], a natural first approach to try would be to use elliptic regularization. Recall that the time of arrival function of a mean convex flow is defined by if and only if . For mean convex flows in Euclidean space, the time of arrival function is a bounded real valued function with domain , and can be approximated by solutions of the Dirichlet problem
The elliptic regularization technique has been known for a long time [ES91, CGG91], see also [Ilm94], and arguing as in [Whi11, HK13] can be used to prove that the two main estimates (with ) hold for the level set flow in Euclidean space. However, extending these arguments to level set flow in Riemannian manifolds is not straightforward.
The key difference between level set flow in Euclidean space and level set in general ambient manifolds, is that in the latter case the flow generally does not become extinct in finite time, but converges to a nonempty limit for . Consequently, the time of arrival function is only defined on the set . Thus, it is (a) not clear a priori how to approximate by smooth solutions, and (b) even if one succeeds in approximating by smooth solutions it is not obvious how to prove our main estimates using the approximators, since one would have to somehow bring in the exponential in time factor and would have to cut off all quantities under consideration for .
To overcome the above difficulties, we consider a new double-approximation scheme. Namely, we consider functions solving the Dirichlet problem
The idea, inspired in part by the Schoen-Yau proof of the positive mass theorem [SY81], is that for the maximum principle gives the a-priori sup bound . Thus, as we will see in Section 2, for positive the Dirichlet problem (1) can be solved using a standard continuity argument. We then argue that for we have convergence in an appropriate sense to functions , which in turn for converge to the time of arrival function , see Section 5. This solves the above difficulty (a).
More fundamentally, we use our double approximation to also solve the difficulty (b). Namely, in Section 3 and Section 4 we prove two estimates for carefully chosen quantities at the level of the double approximators . We choose our quantities in such a way, that on the one hand they satisfy the maximum principle and on the other hand taking the limits and of the estimates for the double approximators yields the two main estimates for the actual level set flow. There is obviously quite some tension between these two desired properties, and we thus have to design our quantities for the double approximate estimates very carefully. For example, to estimate we consider the quantity
which turns out to indeed satisfy the maximum principle after taking in account also an improved Kato inequality at points where is large, see Section 4. Finally, in Section 5 we show that taking the limits and of our double approximate estimates indeed yields Theorem 1.1 and Theorem 1.3, and thus Theorem 1.5.
Acknowledgements. We thank Brian White for bringing the problem of subsequent singularities in Riemannian manifolds to our attention. This work has been partially supported by the NSF grants DMS-1406394 and DMS-1406407. The second author wishes to thank Jeff Cheeger for his generous support during the work on this project.
2 Existence of double approximators
The goal of this section is to prove the existence of double approximators.
If is a mean convex domain in a Riemannian manifold, then the Dirichlet problem (1) has a unique smooth solution for every .
For the problem has the obvious solution . We will now derive the needed a priori estimates for . Note first that we have the sup-bound
which follows directly from the maximum principle. To proceed further, we consider the graph . We write for the unit vector in direction, and for the upward pointing unit normal of (here and in the following we drop the dependence on in the notation when there is no risk of confusion). Written more geometrically, equation (2) takes the form
where is the mean curvature of , and . We write for the product metric on , and for the covariant derivative on . We will frequently use the following general lemma about graphs.
On any graph we have
Moreover, the weight function , where is a constant, satisfies
where denotes the tangential part of .
Let be an orthonormal frame with at the point in consideration, and let be the components of the second fundamental form. Note that
where here and in the following repeated indices are summed over. Using this, we compute
The Codazzi identity gives . Since there is no curvature in -direction we have , and equation (2.6) follows.
Arguing similarly, we compute , and
This proves the lemma. ∎
On we have
Recall that and that . Using this and the formula
the claim follows from a short computation. ∎
Choosing the function satisfies
If attains its minimum on we are done. Suppose now attains its minimum at an interior point . If there is nothing to prove. Suppose now . Since is the graph of , we have , and since is a critical point of , we have , and thus . Using this, and dropping some terms with the good sign, Corollary 2.11 implies that
at . On the other hand, recalling that , we have . Together with the bound and the estimate (2.3) we thus obtain
a contradiction. This proves the proposition. ∎
Recalling that , we see that the lower bound for from Proposition 2.14 is equivalent to an upper bound for .
Lemma 2.19 (c.f. [Es91, Thm. 7.4]).
There exists a constant , such that
Let be the distance function to , and let , to be chosen later, be such that is smooth on . By estimate (2.3), for any , the quantity satisfies on . We will now show that, for large enough, is a supersolution of equation (2). To this end we compute
Note that where . Since , by the smoothness of and the Riccati equation, there exists a such that is smooth on and there. Thus, for , the function is a supersolution of (2). Since , this implies that . ∎
We can now prove the main theorem of this section.
Proof of Theorem 2.1.
Note first that equation (1) is of the form
If and are two solutions of the Dirichlet problem, then at an interior minimum of we have and thus
which implies . Changing the roles of and , this proves uniqueness.
To prove existence, fix , and let
We want to show that . Since , it sufficies to show that is open and closed.
where is independent of . By DeGiorgi-Nash-Moser and Schauder estimates we get -independent higher derivative bounds up to the boundary for solutions of the -problem if . If and , it follows that a subsequence of converges to a solution of the -problem, which implies that .
To show that is open, consider the operator given by
Assuming , its linearization at is given by
Note that at a positive maximum of ,
and similarly at a negative minimum point, . Hence, is the unique solution to with zero boundary. Thus, by standard elliptic theory, the map is invertible, and by the inverse function theorem, the map given by is locally invertible. Taking also into account the higher derivative estimates we conclude that is open, and we are done. ∎
3 Double approximate estimate for
The goal of this section is to derive a lower bound for the mean curvature. As explained in the introduction, we will work at the level of the double approximators , where is a solution of (1) with . The task is then to find a suitable quantity that on the one hand satisfies the maximum principle and on the other hand gives the desired mean curvature bound in the limit . It turns out that for the mean curvature estimate the quantity does the job.
There exist constants and such that
for every , whenever is a solution of (1) with .
In view of the equation , proving Theorem 3.1 amounts to improving the lower bound for from Section 2 in two ways. Namely, we will argue that in the case the factor in Proposition 2.14 can be replaced by the better factor , and we will replace Lemma 2.19 by a boundary estimate which is uniform in and .
Choosing the function satisfies
Consider the function where . As in the proof of Proposition 2.14 we can assume that attains its minimum at an interior point and that (otherwise there is nothing to prove). The estimate (2.16) with and reads
Combining this with the inequalities , , and yields
which contradicts our choice of . This proves the proposition. ∎
Lemma 3.8 (Uniform boundary estimate).
There exists a constant such that
By mean convexity, for small enough the restricted time of arrival function is smooth and satisfies the estimates
for some . Recall also that satisfies the equation
For , let , . We will now show that for small enough the function is a supersolution of equation (1). To this end, we compute
where we used equation (3.11) in the last step. Now observe that
Thus, taking also into account (3.10) we conclude that
which is negative if is sufficiently small. Thus, for such , the function is a supersolution of equation (1) with on and on . Therefore,
This proves the lemma. ∎
Remark 3.16 (Uniform lower bound).
Similarly, considering the function we see that there is a constant such that .
4 Double approximate estimate for
The purpose of this section is to prove the following estimate.
There exist constants , , and , such that
for all , whenever is a solution of (1) with and .
We will prove Theorem 4.1 by applying the maximum principle to the function
where , , and where and will be specified later. As will become clear below, the extra term is crucial for the maximum principle. We begin by computing the Laplacian of the norm of the second fundamental form.
At any interior point with we have
To make use of the gradient term, we prove the following improved Kato inequality.
There exist constants and such that
For any unit vector , we will derive an estimate for the quantity
Let be the totally symmetric part of the 3-tensor , i.e.
Using the Codazzi identity and the bound we see that
Next, observe that any totally symmetric 3-tensor can be decomposed as , where
is the trace-part, and is the totally traceless part.
Using again the Codazzi identity and the bound we see that
Observing that , the remaining task is to estimate the norm of . This can be done by a straightforward computation:
Putting everything together, the proposition follows. ∎
We will apply Proposition 4.12 in combination with the following lemma.
At any critical point of we have the estimate
The equation can be written in the form
Observing that and , and solving for we obtain
The claim follows. ∎
We are now ready to prove the main theorem of this section.
Proof of Theorem 4.1.
Throughout the proof we write for a constant that can change from line to line. This should not be confused with , which is a fixed dimensional constant given by Proposition 4.12.
Consider the function defined in (4.4). The parameters and will be specified in the last line of the proof (depending only on the dimension and geometry of ). For now, we only impose the condition that , where is the constant from Theorem 3.1. We will choose . Thus, tacitly assuming that and , we have inequalities like and at our disposal.
Theorem 3.1, Remark 3.16 and DeGiorgi-Nash-Moser and Schauder estimates up to the boundary give a uniform upper bound for . Thus, if the maximum of occurs at the boundary we are done. Suppose now the maximum of is attained at an interior point . If at , then Theorem 3.1 together with the constraint yields and we are done. Suppose now
for some . This will be an important ingredient for the estimate below.