On the structure of singularities of weak mean curvature flows with mean curvature bounds
-
Maxwell Stolarski
Abstract
This paper studies singularities of mean curvature flows with integral mean curvature bounds
1 Introduction
A time-dependent family of embeddings
Given Huisken’s result [10], it is natural to ask if the trace of the second fundamental form, namely the mean curvature
Given that there exist smooth mean curvature flows which develop singularities with bounded mean curvature [30, 26, 2], the focus of this current article is to instead study the singularities of any mean curvature flow
Let
For any
Note that the stationary cones 𝐂 in Theorem 1.1 are generally integer rectifiable 𝑛-varifolds that are dilation invariant and have zero first variation (see Lemma 2.7 and Theorem 2.13 for more precise statements of Theorem 1.1).
Given a Brakke flow
Theorem 1.1 in particular shows that, for flows with mean curvature
uniqueness of the backward tangent flow of
Let
It is worth mentioning some related uniqueness results.
Huisken’s monotonicity formula ensures tangent flows are always self-similarly shrinking
In [4], uniqueness of the backward tangent flow is proved when Σ is smooth and asymptotically conical (see also a generalization due to [16]).
Importantly, the uniqueness results in [4, 16] require Σ to be smooth and do not apply when
Under additional hypotheses to Theorem 1.2, we can also describe an accompanying limit flow that arises as a more general blow-up limit around a singularity.
Let
We refer the reader to Section 4 for the definitions relevant to Theorem 1.3.
For now, we simply note that the mean curvature flow solutions constructed in [30] provide examples of mean curvature flows satisfying Theorem 1.3.
Theorem 1.3 states that general mean curvature flows with
The paper is organized as follows.
In Section 2, we establish notation, specify definitions, and obtain some general results used throughout the paper.
In particular, Theorem 1.1 is proven here.
Section 3 proves the uniqueness of backward tangent flows given by regular stationary cones, Theorem 1.2.
In Section 4, we obtain refined dynamics of mean curvature flows near regular stationary cones and prove Theorem 1.3.
Finally, Appendix A reviews some well-known results about integral varifolds with generalized mean curvature
2 Preliminaries
2.1 Brakke flows with 𝐻 bounds
Let
Observe that, by the definition given in Definition 2.1, if 𝑉 has generalized mean curvature
Throughout, we use the following notion of Brakke flow as in [34, Definition 5.1 with empty boundary
Let
for almost every
for any
(where, for almost every
for any
(2.2)
We refer to inequality (2.2) as Brakke’s inequality.
Let
Let
For simplicity, we will abbreviate this assumption as “
We will also often assume that, for some
The assumption that the flow has locally uniformly bounded areas is quite mild.
For example, it holds for Brakke flows
On the other hand, the assumption that the flow has generalized mean curvature
is much more restrictive.
Nonetheless, [26] shows that even smooth mean curvature flows
with mild singularities and small singular sets, informally speaking.
The next lemma shows that Brakke flows with
Let
for all
for all
Proof
Let
By compactness (Lemma A.1), there exist a subsequence (still denoted
This defines the varifold
By [12, Theorem 7.2 (ii)], for any
This proves 1.
For 2, simply note that [12, Theorem 7.2 (iii)] implies the first inequality in (2.3) is an equality for all but countably many times
To prove 4, let
Taking
Taking
Convergence as varifolds then follows from Lemma A.2 and completes the proof of 4.
To prove 5, it suffices to check Brakke’s inequality.
Let
2.2 Huisken’s monotonicity formula and Gaussian density
Let
denote the backwards heat kernel based at
denote the spherically shrinking localization function based at
Huisken’s monotonicity formula and its localized analogue for Brakke flows states
for all
is well-defined for
For an 𝑛-dimensional Brakke flow
Often, we omit the basepoint
2.3 Densities and blow-up limits
Recall the density of a varifold 𝑉 at
when the limit exists and where
The map
When
Lemma 2.8 (Existence of tangent cones of time-slices)
Let
Any 𝐂 that arises as such a limit is called a tangent cone of
Proof
By Lemma 2.7,
have generalized mean curvature
for all
As
is well-defined and the blow-up sequence
It now follows from compactness (Lemma A.1) that there are a subsequence (still denoted
which implies 𝐂 is stationary.
Moreover, the convergence
The monotonicity formula (A.3) for stationary varifolds finally implies 𝐂 is dilation invariant, that is,
We will show in Lemma 2.10 below that Huisken’s monotonicity formula (2.4) similarly allows us to extract tangent flows from space-time blow-ups. First, we show local uniform area bounds imply local uniform area ratio bounds. These area ratio bounds are necessary for the blow-up argument in the proof of Lemma 2.10 below.
Note that, for general Brakke flows defined for
Lemma 2.9 (Area ratio bounds)
Suppose that
Proof
The proof will proceed by combining the monotonicity formula (A.3) with the local uniform area bounds.
Given
Let
For any
where
Taking the supremum over
Lemma 2.10 (Existence of tangent flows)
Let
For any
Moreover, 𝐂 is a stationary, dilation invariant varifold in
Any 𝐂 that arises as such a limit is called a (backward) tangent flow of
Proof
Let us fix
(see [11, Lemma 8] or [31]).
Moreover,
and, for any
Additionally,
Because
for all
Since
2.4 Remarks on (backward) tangent flows
Lemma 2.10 ensures subsequential limits converge
Let
An integral 𝑛-Brakke flow
An integral 𝑛-Brakke flow
A backward tangent flow
If an integral 𝑛-Brakke flow
Clearly, the restriction of a full tangent flow to negative times
Consider the 𝑛-plane
where
Take
Clearly, the full tangent flow
Observe from Definition 2.11 that tangent flows of
Observe that uniqueness of full tangent flows implies uniqueness of backward tangent flows.
For compact or cylindrical tangent flows as in [19, 5, 35], uniqueness of backward tangent flows is equivalent to uniqueness of full tangent flows.
Indeed, suppose a cylinder is the unique backward tangent flow to
Often in the literature, authors study the uniqueness of tangent flows to smooth mean curvature flows
2.5 Equivalent densities and blow-up limits
A priori, the tangent cones from Lemma 2.8 could be entirely unrelated to the backward tangent flows from Lemma 2.10. Indeed, the two limiting objects arise from entirely different blow-up sequences. The next theorem, however, proves that the tangent cones from Lemma 2.8 exactly correspond to backward tangent flows from Lemma 2.10.
Let
For any
For any sequence
Proof
By translation, we can assume without loss of generality that
For any sequence
Lemmas 2.8 and 2.10 imply there exists a subsequence (still denoted by index 𝑖) such that
where
We first claim that
For any
By Lemma 2.9, the
term is uniformly bounded above by say
Thus
Note that, since 𝐂 is dilation invariant,
and similarly for
We claim that the reverse inequality
also holds (for some or equivalently all
where as usual 𝐻 denotes the mean curvature of the flow
Let
as in the monotonicity formula (A.3).
Note
Thus
Taking
In summary, we have shown that
In particular,
a contradiction.
Thus
As an immediate consequence of Theorem 2.13, we deduce that the uniqueness of tangent cones of time-slices is equivalent to the uniqueness of backward tangent flows.
Let
In particular, the tangent cone
We emphasize that Corollary 2.14 applies for backward tangent flows only, not full tangent flows.
It may be possible that there exist an integral 𝑛-Brakke flow
For integral 𝑛-Brakke flows
Let
In particular, if
Combining this fact with [12, Theorem 7.2 (ii)] and Lemma 2.7 gives
Hence
We refer the interested reader to [3, 6, 14, 18] for precise definitions.
3 Uniqueness of tangent flows given by regular cones
Define the link
Then 𝐂 is said to be a regular cone if
If
The main result of this section is the following uniqueness result which restates Theorem 1.2.
Let
The proof essentially follows from [22, §7, Theorem 5] applied to the time slice
Since
show that
apply interior estimates to improve the
In what follows, we use
and
For
denote
Note that if 𝐂 is a regular cone and
For 𝐂 a regular cone, let
For
For functions
and define scale-invariant
Since we use the
where also the suprema above are restricted to points in
While the
(Parabolic scaling invariance) If
is given by
(Time translation invariance) If
is given by
(Equivalent to standard Hölder norms) There exists
The choice of using radius
Lemma 3.7 (Graphicality propagates out and back)
Let
for some
for some extension
Proof
By translation, assume without loss of generality that
Parabolically dilate
as
it follows that, in fact,
Now the stationary flow
since 𝐂 has smooth link.
The upper semi-continuity of Gaussian density then implies that, for any
for all
By White’s local regularity theorem [33] (and its generalization to integral Brakke flows in [27]), it follows that, for
Interior regularity for mean curvature flow then implies that the convergence
is smooth on
for some
Since
and derivatives of
for all
for
which contradicts the choice of the
Let
for some
and
on
Proof
Throughout, we assume
for some
By Lemma 3.7, for any
for some extension
Consider the parabolically rescaled flow
where
If
In this region
(see [24, Corollary 3.8]).
The bounds (3.1) imply (3.2) is a linear parabolic PDE system for
In terms of
for all
The conclusion of Lemma 3.8 is effectively the same statement as [22, (7.23)]. We can now prove Theorem 3.2 by adapting the argument [22, §7] used for tangent cones of stationary varifolds.
Proof of Theorem 3.2
Throughout, we use
By Theorem 2.13, there exists some sequence
and
so long as
is smooth.
In particular, for
Lemma 3.8 implies
where
and thus
The uniqueness of the tangent cone now follows. ∎
As a corollary, we note that Theorem 3.2 implies the flow may be written as a graph over the cone in certain space-time regions near the singularity.
Let
for some function
Proof
For any
4 Pinching Hardt–Simon minimal surfaces
Throughout this section, we restrict to the case where
The goal of this section is to prove Theorem 1.3, which says that if ℳ has bounded 𝐻 and develops a singularity with backward tangent flow given by an area-minimizing quadratic cone, then there is a type II blow-up limit given by a Hardt–Simon minimal surface (see Section 4.2 for the relevant definitions).
The example of the static flow of an area-minimizing quadratic cone shows Theorem 1.3 is false when
We restrict to the case of smooth mean curvature flows here because the arguments in this section and the proof of Theorem 1.3 make reference to the second fundamental form and rely on a pseudolocality theorem for mean curvature flows (see Lemma 4.4 and Lemma 4.5 below in particular). The pseudolocality theorem fails for general integral 𝑛-Brakke flows because Brakke flows may suddenly vanish.
4.1 Flows near a regular cone
We begin with general results for mean curvature flows locally close to a regular cone
We say ℳ is 𝜖-close to
with
for some
We say ℳ is 𝜖-close to
Throughout the remainder of this section, 𝜖 is always assumed to be less than some small constant
Suppose ℳ is 𝜖-close to
to be the largest interval such that, for all
Occasionally, we may write
We note the following consequence of pseudolocality for mean curvature flow in our setting.
Define constants
There exists
Proof
Let
Let
for some
Interior estimates for mean curvature flow [7] then imply 𝑢 satisfies
of the form
for some constant
If
Estimate (4.1) now follows from undoing the dilation
There exists
Proof
If not, (4.1) implies
which contradicts the smoothness of the flow at
Note that Lemma 4.5 crucially relies on smoothness of the flow and fails when the smooth mean curvature flow
Next, we show that
If
Proof
Fix
Let
if
Observe also that the sphere
and an analogous argument applies to show
which contradicts estimate (4.2).
This completes the proof after relabelling
4.2 Finding Hardt–Simon minimal surfaces
Assume additionally throughout this subsection that ℳ has
Since
Denote the constant
Denote
Moreover, Lemma 4.7 implies that the minimizer
Define
By construction, for all 𝑖,
In what follows, we deviate from the notation of Section 2.2 and write Huisken’s monotonic quantity as
where
Along some subsequence
(entropy bound) for all
for all
for all
Proof
For any
by the limiting behavior of
The fact that ℳ has mean curvature
Thus the limiting Brakke flow
Let
Let
is 𝜖-close to some
For
Definition 4.12 (Hardt–Simon foliation)
Let
For any
We refer to this family of minimal hypersurfaces as the Hardt–Simon foliation.
Let
be a generalized Simons cone, and let
Proof
It follows from the entropy bound Lemma 4.9 1, compactness of Brakke flows with uniform local area bounds, and Huisken’s monotonicity formula that there exists a limiting shrinker
Consider the Hardt–Simon foliation (4.4) rotated so that
Since
Because
Proof of claim
Consider some index 𝑖 with
Theorem 2.13 then gives that
Thus we have equality throughout and
It follows from Huisken’s monotonicity formula (2.4) that
Write
Let
On the other hand, Solomon–White’s strong maximum principle [25] applied to the smooth manifold
In particular, either
By the entropy bounds (Lemma 4.9 1) and the constancy theorem [23, Chapter 8, §4], it follows that, in fact,
for any index 𝑖 such that
Suppose for the sake of contradiction that there exist arbitrarily large indices 𝑖 with
In particular, White’s local regularity theorem [33]
implies
With Claim 4.14 in hand,
Let
Because
However, the second case is impossible, since it would imply (together with
Additionally, 𝑀 is minimal (
since
Using the strong maximum principle [25] and the fact that
Since
Finally, suppose for the sake of contradiction that there exist
Theorem 4.13 and the results of this section complete the proof of Theorem 1.3.
Funding source: Leverhulme Trust
Award Identifier / Grant number: ECF-2023-182
Funding statement: The author is supported by a Leverhulme Trust Early Career Fellowship (ECF-2023-182).
A Varifolds with generalized mean curvature
H
∈
L
loc
p
In this appendix, we collect some standard results for varifolds with generalized mean curvature
Lemma A.1 (Compactness)
Let
the
the
Proof
Allard’s compactness theorem [1] gives compactness under the weaker assumption where (2) is replaced by the bound
Therefore, there exists a subsequential limit
However, convergence as varifolds
The Radon–Nikodym theorem then implies that
Let
If
then
Proof
We first show
where
Let
Then
where the last inequality follows by (A.2).
This completes the proof that
Next, we prove the varifold convergence
Since the
Because
Proposition A.3 (Monotonicity formula)
Let
where
and
Proof
The proof follows [23, Chapter 4, §4].
We provide the proof for
For any
where
The integral on the right-hand side of (A.4) can be estimated by
where in the last step we used
To simplify the notation, we write
and using the fact that
Integrating from 𝜎 to 𝜌 then gives
Using that
Letting
In particular, we have monotonicity of
Hence its limit as
Acknowledgements
I would like to thank Felix Schulze for many helpful conversations, particularly regarding the results in [17]. I appreciate the anonymous referee’s comments that improved the presentation of this paper.
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