Boundary Singularities in Mean Curvature Flow and Total Curvature of Minimal Surface Boundaries

For hypersurfaces moving by standard mean curvature flow with boundary, we show that if a tangent flow at a boundary singularity is given by a smoothly embedded shrinker, then the shrinker must be non-orientable. We also show that there is an initially smooth surface in 3-space that develops a boundary singularity for which the shrinker is smoothly embedded (and therefore non-orientable). Indeed, we show that there is a nonempty open set of such initial surfaces. Let k be the largest number with the following property: if M is a minimal surface in 3-space bounded by a smooth simple closed curve of total curvature less than k, then M is a disk. Examples show that $k<4\pi$. In this paper, we use mean curvature flow to show that $k>3\pi$. We get a slightly larger lower bound for orientable surfaces.

with (fixed) boundary Γ and with initial surface M (0) = M . Furthermore, if Γ lies on the boundary of the convex hull of M , then the flow is regular at the boundary for all times.
(See [Whi21, 5.1 and 13.2] for the definitions of Brakke flow with boundary and standard Brakke flow with boundary. Briefly, standard Brakke flows are those that are unit-regular -which prevents certain gratuitous vanishing -and that take their boundaries as mod 2 chains. In particular, triple junction singularities do not occur in standard Brakke flow.) In this paper, we explore boundary regularity of such a flow M (·) without assuming that Γ lies on the boundary of the convex hull of M .
In particular, we show that if a shrinker corresponding to a boundary singularity is smooth and embedded, then it must be non-orientable. We also show that smooth, non-orientable shrinkers do arise as boundary singularities of certain smooth initial surfaces in R 3 . Indeed, we show that there is a nonempty open set of such initial surfaces. See Theorem 16.
We also apply mean curvature flow to questions in minimal surface theory: (1) What is the largest number κ such that if M is a smooth minimal surface in R 3 bounded by a smooth simple closed curve of total curvature < κ, then M must be a disk? (2) What is the largest number κ ′ such that if M is an orientable smooth minimal surface in R 3 bounded by a smooth simple closed curve of total curvature < κ ′ , then M must be a disk? Examples show that κ < 4π. Here, we show (Theorem 17) that κ > 3π. Examples of Almgren-Thurston show that κ ′ ≤ 4π. It is conjectured that κ ′ = 4π. We show (Theorem 19) that κ ′ > (2π) 3/2 e −1/2 ∼ 3π(1.014).
For the existence results in this paper, we work with standard Brakke flows of two-dimensional surfaces with entropy less than two. Such flows are rather wellbehaved: they are smooth at almost all times, and the shrinkers corresponding to tangent flows are smoothly embedded and have multiplicity one. See Theorem 9.

A Bernstein Theorem for Orientable Boundary Shrinkers
Theorem 1. Let M ⊂ R n+1 be a smoothly embedded, oriented shrinker bounded by an (n − 1)-dimensional linear subspace L. Then M is a half-plane.
Proof. We can assume that L = {x : x 1 = x 2 = 0}. Consider the 1-form on R n+1 \ L. Let C be an oriented closed curve in M \ L. The winding number of C about L is equal to the intersection number of C and M . Since we can move C slightly in the direction of the unit normal to M to get a curve C ′ disjoint from M , we see that the winding number of C about L is equal to 0. Thus C dθ = 0.
Note that θ(·) extends continuously to L. Let M ′ be a connected component of M .
Proof of claim 1. Let g be the shrinker metric on R n+1 . Choose an R large enough that the g-mean curvature vector of ∂B(0, R) points outward. (That is, choose R > √ 2n.) Let α be the maximum of θ on {x ∈ M ′ : |x| ≤ 3R}. By rotating, we can assume that α = 0.
Choose a surface S that minimizes g-area with among all such surfaces. Then S is smooth, g-minimal, and g-stable.
(The restriction that S be rotationally invariant ensures that S is smooth; otherwise, S might have an n − 7-dimensional singular set. If the smoothness is not clear, note that the rotational symmetry implies that the quotient set is a curve; that curve must be a geodesic with respect to a certain Riemannian metric on (0, ∞) × R.) According to [Bre16,Proposition 5], S is flat with respect to the Euclidean metric. (One lets k → ∞ in the statement of that proposition.) But that is impossible since S \ ∂S ⊂ {x 2 > 0} and since ∂S is an (n − 1)-sphere in {x 2 = 0}. This completes the proof of the claim.
By Claim 1 and the strong maximum principle, θ is constant on M ′ . Thus M ′ is a half-plane with boundary L. Consequently, M is a union of such half-planes. Since M is embedded, it is a single half-plane.
Remark. The proof of Theorem 1 was inspired by the proof of Theorem 11.1 in the Hardt-Simon boundary regularity paper [HS79].

A Bernstein Theorem for Orientable Boundary Shrinkers with Singularities
In this section, we extend Theorem 1 to possibly singular shrinkers. The proof is not longer, but it does use more machinery. This section is not used in the rest of the paper. (By definition, reg M is a smooth, properly embedded manifold-with-boundary in R n+1 \ (sing M ), the boundary being L \ sing M .) Proof. As in the proof of Theorem 1, we assume that L is the subspace {x 1 = x 2 = 0}. As in that proof, there is a smooth function First, we claim that I has length at most π. For if not, by rotating we can assume that inf M ′ θ < 0 and that sup But then {x ∈ M ′ : θ(x) ∈ (−π, 0)} and {x ∈ M ′ : θ(x) ∈ (π, 2π)} are nonempty and lie in different components of M ′ ∩ {x 2 < 0}, contradicting Claim 2. Next, we claim that M ′ is a half-plane. For suppose not. Then θ|M ′ does not attain a maximum or a minimum by the strong maximum principle. Thus the interval I = θ(M ′ ) is an open interval of length at most π. By rotating, we can assume that θ(M ′ ) = (−α, α) for some α with 0 < α ≤ π/2. Now rotate M ′ by π about L to get M ′′ . Then M * := M ′ ∪ M ′′ is an embedded shrinker without boundary. Furthermore, reg(M * ) ∩ {x 2 > 0} is not connected, which is impossible by Claim 2 (applied to M * ).
We have shown that each component of reg(M ) is a half-plane bounded by L. By (1), there can be at most one such half-plane.

Basic Properties of Entropy and Total Curvature
Suppose that Γ is an (m − 1)-dimensional submanifold of R n and that v ∈ R n . We define the cone C Γ,v and the exterior cone E Γ,v over Γ with vertex v by We will also use C Γ,v and E Γ,v to denote the corresponding Radon measures on R n (counting multiplicity). Thus Now C Γ,v and E Γ,v depend continuously on v for v ∈ R n \ Γ. However, the dependence is not continuous at v ∈ Γ. For suppose v i / ∈ Γ converges to v ∈ Γ. Then, after passing to a subsquence, C Γ,vi converges to the union of C Γ,v and a half-plane bounded by Tan(Γ, v), and similarly for E Γ,vi . Consequently, Here 1 Γ (·) is the characteristic function (or indicator function) of Γ. We define the vision number vis(Γ) of Γ to be the supremum of the quantity (2) over v ∈ R n . If Γ is compact, then the supremum is attained by a v in the convex hull of Γ.
Remark 3. Because the quantity (2) depends continuously on v, Since the first and last expressions are the same, in fact equality holds. Thus in the definition of vision number, it does not matter whether or not we include the term 1 2 1 Γ (p). Definition 4. If M is a Radon measure on R n and if Γ is a smooth (m − 1)dimensional manifold in R n , then the entropy of the pair (M, Γ) is is a Brakke flow with boundary Γ in R n . Then e(M (t); Γ) is a decreasing function of t.
where tc(Γ) is the total curvature of Γ. Equality holds if and only if Γ is a convex planar curve.

Then
(1) At each spacetime point, each tangent flow is given by a smooth, multiplicity 1 shrinker.
(2) The flow M (·) is regular at almost all times.
(3) If n = 3, then for each tangent flow at a boundary singularity, the corresponding shrinker is non-orientable. (4) Every sequence of times tending to ∞ has a subsequence t(i) for which M (t(i)) converges smoothly to an embedded minimal surface. Proof. See Appendix A for the proofs of Assertions 1 and 2. In particular, Lemmas 20 and 21 give Assertion 1, and Proposition 23 gives Assertion 2. In Assertion 3, the non-orientability of Σ follows immediately from Assertion 1 and Theorem 1.
To prove Assertion 4, suppose t(i) → ∞. By passing to a subsequence, we may assume that the flows t → M (t(i) + t) converge to a standard limit Brakke flow with boundary Γ. (This is by the compactness theory for standard Brakke flows with boundary: see Theorems 10.2, 10.2, and 13.1 and Definition 13.2 in [Whi21].) Since the area of M ′ (t) is independent of t (it is equal to lim t→∞ area(M (t))), it follows that M ′ (t) is a stationary integral varifold V independent of t. By Assertion 2 applied to the flow M ′ (·), the surface M ′ (t) is smoothly embedded for almost all t. Thus V is smoothly embedded. The uniqueness in Assertion 5 follows from Assertion 4 and from the Lojasiewicz-Simon inequality [Sim83, Theorem 3]. See Theorem 29 below.
The proof of Assertion 6 is the same as the proof of Assertion 5.

A general existence theorem for eternal flows
In the following two theorems, F is a set of C 1 compact manifolds-with-boundary in R 3 with the following property: if M ∈ F and if M ′ is isotopic to M , then M ′ ∈ F . For example, F might be the set of non-orientable surfaces, or the set of genus-one orientable surfaces, etc.
Theorem 10. Suppose that Γ is a smooth, simple closed curve in R 3 of total curvature at most 4π. Suppose that Γ bounds a minimal surface in F and that Γ is a smooth limit of curves Γ i such that Γ i does not bound a minimal surface in F .
Then there is an eternal standard Brakke flow Proof. The set of minimal surfaces in F bounded by Γ is compact (by Theorem 25), so there is a surface M that attains the least area A among all such surfaces.
by Corollary 7 and Theorem 8. For each i, let M i be a surface diffeomorphic to M and bounded by Γ i such that M i converges smoothly to M as i → ∞. Then e(M i ; Γ i ) → e(M ; Γ), so by passing to a subsequence, we can assume that be a standard Brakke flow with boundary Γ i and with initial surface As t → ∞, we can assume, by passing to a subsequence, that the flow M i (·) converges to a standard Brakke flow M (·) with M (0) = M . Since M is smooth and minimal, the only such flow is the constant flow M (t) ≡ M .
As t → ∞, M i (t) converges smoothly to an embedded minimal surface M i (∞) (by (3) and Theorem 9). By hypothesis on Γ we see that there must be singularities in the flow. Let T i be the first singular time. Since the flow t → M i (t) converges to the constant flow t → M , we see that By passing to a subsequence, we can assume that the time-shifted flows with boundary Γ. By Corollary 7 and Theorem 8, the flow satisfies the entropy bound (i).
By choice of T i , M ′ i (t) in F for t < 0. By local regularity [Whi05], Since 0 is a singular time of the flow M ′ i (·), it follows (again by local regularity [Whi05]) that 0 is a singular time of the flow M ′ (·). Hence the flow M ′ (·) is not constant, so Since A is the least area of any minimal F -type surface bounded by Γ, it follows that area M ′ (−∞) = A and that M ′ (∞) is not of type F .
Theorem 11. Suppose that the family F does not include disk-type surfaces. Suppose there is a smooth simple closed curve Γ 0 of total curvature < 4π that bounds a minimal surface of type F . Then there is a curve Γ such that tc(Γ) ≤ tc(Γ 0 ) and such that Γ satisfies the hypotheses of Theorem 10.
Proof. By Theorem 28, there is a smooth one-parameter family s ∈ [0, 1] → Γ s of simple closed curves (starting from the given curve Γ 0 ) for which Γ 1 is a round circle and for which each curve has total curvature ≤ tc(Γ 0 ).
Let S be the set of s ∈ [0, 1] such that Γ s bounds a minimal surface of type F . By Theorem 25, S is closed. And S is nonempty since 0 ∈ S. Thus S has a maximumŝ. Note thatŝ < 1. Hence Γŝ bounds a minimal surface of type F , and Γŝ is a smooth limit of the curves Γ s , s >ŝ, that do not bound minimal surfaces of type F .

Generalized Möbius Strips
Suppose that M ⊂ R 3 is a smoothly embedded, compact surface with exactly one boundary component. More generally, we can allow M to have self-intersections and singularities, provided they occur away from the boundary. That is, we only require that ∂M is a smooth, simple closed curve and that M is a smoothly embedded manifold-with-boundary (the boundary being ∂M ) near ∂M .
Although M may not be orientable, we can choose an orientation for ∂M . Now push ∂M slightly into M to get another smooth embedded curve C. For example, if ǫ > 0 is sufficiently small, we can let We let λ(M ) be the linking number of ∂M and C. This can be defined in various (equivalent) ways. For example, let Σ be a compact oriented surface (not necessarily embedded) with boundary ∂M . By perturbing Σ slightly, we can assume that C intersects Σ transversely. Then λ(M ) is the intersection number (in R 3 \ ∂M ) of C and Σ Alternatively, we can let Σ be a compact oriented surface with boundary C. Then λ(M ) is the intersection number of ∂M and Σ in R 3 \ C.
(We began by choosing an orientation for ∂M , but the resulting value of λ(M ) does not depend on that choice, since reversing the orientation of ∂M also reverses the orientations of C and of Σ, thus leaving the value of λ(M ) the same.) Of course if M is smoothly embedded and orientable, then λ(M ) = 0, since we can let Σ be the portion of M bounded by C. (Note that Σ is an oriented surface with boundary C and that Σ is disjoint from ∂M , so the linking number of C and ∂M is 0.) Proof. Let S be an embedded, orientation-reversing path in the interior of M . Note that we can perturb S slightly to get a curve S ′ that intersects M transversely and in exactly one point. Thus the mod 2 linking number of ∂M and S is 1, and therefore the integer linking number of ∂M and S (whichever way we orient those curves) is an odd integer. Now push ∂M into M to get an embedded curve C as in the definition of λ(M ). Then C is homologous in M \ ∂M (and therefore in R 3 \ ∂M ) to S traversed twice, so λ(M ), the linking number of ∂M and C, is equal to the twice the linking number of ∂M and S.
Definition 13. A generalized Möbius strip in R 3 is a smoothly embedded, compact (not necessarily connected) surface M in R 3 such that ∂M has exactly one component, and such that λ(M ) is nonzero.
Every generalized Möbius strip is non-orientable, but not every non-orientable surface in R 3 is a generalized Möbius strip. For example, if we attach a handle to a flat disk to make a non-orientable surface M (topologically a Klien bottle with a disk removed), then λ(M ) = 0.
Note that if a generalized Möbius strip in Euclidean space is a minimal surface, then it has no components without boundary and thus must be connected. Theorem 15. Let κ gm be the infimum of the total curvature tc(Γ) among smooth, simple closed curves Γ in R 3 that bound generalized minimal Möbius strips. Then If κ gm < κ < 4π, then there exists a smooth, simple closed curve Γ in R 3 of total curvature < κ and an eternal standard Brakke flow t ∈ R → M (t) with boundary Γ such that (1) sup t e(M (t); Γ) < tc(Γ) 2π . By (11), (12), and Lemma 14, M i (·) must have a boundary singularity. By (10) and Theorem 9, the corresponding shrinker must be smoothly embedded and nonorientable, and must have multiplicity one.

The Three Pi Theorem
Theorem 17. Suppose that Γ ′ is a smoothly embedded, simple closed curve in R 3 and that Γ ′ bounds a smooth minimal surface that is not a disk. Then tc(Γ ′ ) > 3π.
By Theorem 11 (applied to the family F of non-disk surfaces), there is a smooth, simple closed curve Γ with tc(Γ) ≤ tc(Γ ′ ) and a there is a standard Brakke flow t ∈ R → M (t) with boundary Γ for which M (−∞) is not a disk and M (∞) is a disk.
Thus the flow must have at least one singularity. Consider the shrinker Σ corresponding to a tangent flow at the first singular time T .
If the singularity is a boundary singularity, e(M (·); Γ) > 3 2 by Theorem 18 below. Now suppose the singularity is an interior singularity. Since M (t) is connected and with nonempty boundary for t < T , we see that Σ is noncompact. By [BW17, Corollary 1.2], the entropy of Σ is greater than or equal to the entropy σ 1 = (2π/e) 1/2 ∼ = 1.52 of a round cylinder.
Thus in either case (boundary singularity or interior singularity), On the other hand, Thus Theorem 18. Let Σ ⊂ R n+1 be a smooth, non-orientable shrinker whose boundary is an (n − 1)-dimensional linear subspace L. Then e(Σ; L) > 3 2 .
Proof. Rotate Σ by π about L to get Σ ′ . Then M := Σ∪Σ ′ is a smoothly immersed shrinker without boundary. Since it is non-orientable, it must have a point p of self-intersection. Thus the entropy of M is ≥ 2. In fact, the entropy must be > 2, since otherwise M would be a cone centered at p. Since M is smooth, that means M would be planar, which is impossible since it is nonorientable. Thus

Oriented Surfaces
We can improve Theorem 17 slightly in the case of oriented surfaces.
Theorem 19. Suppose that Γ ′ is a smoothly embedded, simple closed curve in R 3 and that Γ ′ bounds a smooth, oriented minimal surface that is not a disk. Then where σ 1 = (2π/e) 1/2 is the entropy of S 1 × R.
This is a slight improvement over the 3π in Theorem 17 because It is conjectured that Theorem 19 holds with 4π in place of 2πσ 1 . The constant 4π would be sharp since (by work of Almgren-Thurston [AT77] or the simplified version by Hubbard [Hub80]), for every ǫ > 0 and g, there is a smooth simple closed curve of total curvature < 4π+ǫ that bounds no embedded minimal surface of genus ≤ g. Such a curve bounds immersed minimal surfaces (the Douglas solutions) of each genus ≤ g, and an embedded minimal surface (the least area integral current) of genus > g. See the discussion in the introduction of [EWW02].
Let F be the family of connected, oriented surfaces of genus ≥ 1. By Theorem 11, there is smooth simple closed curve Γ with tc(Γ) ≤ tc(Γ ′ ) and a standard Brakke flow t ∈ R → M (t) with boundary Γ such that M (∞) is a smooth disk and such that M (−∞) is smooth, orientable minimal surface that is not a disk. Thus the flow must have a singularity. Consider the shrinker Σ corresponding to a singularity at the first singular time T . Since M (t) is orientable for t < T , Σ must be orientable. Thus the singularity is an interior singularity (by Theorem 9.) As in the proof of Theorem 17, the entropy of Σ is greater than or equal to the entropy σ 1 of a cylinder, and thus σ 1 ≤ e(M (T ); Γ) < tc(Γ) 2π .
Proof. First we prove: If Σ is a cone, then it is a multiplicity-one plane.
If Σ is a cone, it must be a stationary cone, so its intersection with the unit sphere is a geodesic network. Since the entropy is < 2, each geodesic arc occurs with multiplicity 1. Also, at each vertex of the network, 3 or fewer arcs meet. But 3 arcs cannot meet at a point because the flow is standard and therefore has no triple junctions.
Thus there are no vertices. That is, Σ ∩ ∂B consists of disjoint, multiplicity-one geodesics. Because the entropy is < 2, there can only be one such geodesic. Thus Σ is a multiplicity-one plane. This proves (14).
In the general case, note that Σ is a stationary integral varifold for the shrinker metric on R n . Let C be a tangent cone to Σ at a point p. Then t ∈ (−∞, 0) → C = |t| 1/2 C is a tangent flow to the flow (13) at the spacetime point (p, −1). By (14), C is a multiplicity-one plane. Hence (by Allard regularity) p is a regular point of Σ. Proof. First we prove: (16) If Σ is a cone, then it is a multiplicity-one half-plane.
Note that if Σ is a cone, then Σ ∩ ∂B is a geodesic network. Each geodesic arc occurs with multiplicity 1. Exactly as in the proof of Lemma 20, there can be no vertices of the network, except the two points of L ∩ ∂B. Thus Σ ∩ ∂B consists of j geodesic semicircles with endpoints L ∩ ∂B, together with k geodesic circles (for some integers j and k). Consequently Σ consists of j halfplanes (each with boundary L) together with k planes. The extended entropy of Σ is (j/2) + k + 1/2, so (17) j + 1 2 + k < 2 and therefore j < 3. By standardness, the mod 2 boundary of Σ is L, so j is odd. Thus j = 1. By (17), k = 0. This proves (16). Now we consider the general case. Exactly as in Lemma 20, Σ is smooth and embedded except perhaps along L. Let C be a tangent cone to Σ at a point p ∈ L. Then t ∈ (−∞, 0) → C = |t| 1/2 C is a tangent flow to the flow (15) at the spacetime point (p, −1), so C is a multiplicityone half-plane by (16). Thus p is a regular point of Σ by Allard regularity.
Corollary 22. Let Σ be as in Lemma 21. If Σ is invariant under translations in some direction, then Σ is a half-plane.
Proof. The direction of translational invariance would have to be L. Let P be the tangent half-plane to Σ at 0. Then P and Σ are g-minimal surfaces that are tangent along L (where g is the shrinker metric.) Thus P and Σ coincide.
Proposition 23. Suppose that Γ is a smoothly embedded curve in R n and suppose that is a standard Brakke flow with boundary Γ satisfying the entropy bound e(M (t); Γ) < 2 for all t.
Then the set of singular times has measure 0.
Proof. Let Σ be the shrinker for the tangent flow at a singularity. Then Σ is smoothly embedded, so if it had a direction of translational invariance, then by Corollary 22, the singularity would not be at a boundary point. Thus Σ would be a cylinder. Regularity at almost all times follows from the standard stratification theory [Whi97].
Remark 24. From the proof, we see that we do not really need to assume that the surfaces have entropy < 2; it is enough to assume that all the tangent flows have entropy < 2. Theorem 26. Let α < 4π. Let C(α) be the collection of simple closed, polygonal curves in R n with total curvature at most α. Then C(α) is connected.
Proof. It suffices to show that any curve in C(α) can be deformed through curves in C(α) to a triangle. Let Γ be a curve in C(α). According to Milnor's Theorem [Mil50], we can assume (by rotating and scaling) that the image of height function h : x ∈ Γ → x · e n is [0, 1], and that for each y ∈ (0, 1), there are exactly two points γ 1 (y) and γ 2 (y) of Γ at which h = y.
The total curvature of Γ(t) is a decreasing function of t, so Γ(t) ∈ C(α) for all t ∈ [0, 1). Note that for t close to 1, Γ(t) is a triangle.
(3) Each γ t is a piecewise-smooth simple closed curve.
(4) The total curvature of γ t is a decreasing function of t. Then γ 0 = γ, the total curvature of γ t is a decreasing function of t (by [Mil50]), and γ 1 is polygonal.
Note also that if N is sufficiently large, then each γ t will be an embedding.
Theorem 28. Let Γ be a smooth, simple closed curve of total curvature ≤ α < 4π. Then Γ can be deformed among such curves to a planar convex curve.
Proof. If C is a simple closed curve in R n , let φ t (C) be the result of flowing (1) Γ 0 = Γ.
(5) The total curvature of Γ s is a decreasing function of s.
Note that we have deformed Γ to the plane convex curve φ ǫ (Γ) through smooth, simple closed curves, each of total curvature ≤ α. (By [Hät15, Lemma 3.4], curve shortening in R n reduces total curvature.)