Curve flows with a global forcing term

We consider embedded, smooth curves in the plane which are either closed or asymptotic to two lines. We study their behaviour under curve shortening flow with a global forcing term. Firstly, we prove an analogue to Huisken's distance comparison principle for curve shortening flow for initial curves whose local total curvature does not lie below $-\pi$, and show that this condition is sharp. Secondly, for bounded forcing terms, we exclude singularities in finite time. Thirdly, for immortal flows of closed curves whose forcing terms provide non-vanishing enclosed area and bounded length, we show convexity in finite time and smooth and exponential convergence to a circle.


Introduction
Let Σ 0 ⊂ R 2 be an embedded, smooth curve, parametrised by the embedding X 0 : I → R 2 , where I ∈ {S 1 , R}. We seek a one-parameter family of maps X : I × [0, T ) → R 2 with X( · , 0) = X 0 satisfying the evolution equation for (p, t) ∈ I × (0, T ), where the vector ν is the outward pointing unit normal to the curve Σ t := X(I, t), κ is the curvature function and T is th maximal time of existence. The global term h is smooth and smoothly bounded whenever the curvature is bounded. For the curve shortening flow (CSF), h ≡ 0. For closed curves, the enclosed area preserving curve shortening flow (APCSF) has the global term where L t = L(Σ t ) is the length of the curve. The length preserving curve flow (LPCF) has the global term The total curvature of a curve Σ t = X(I, t) is given by where α = 2π if the curve Σ = X(S 1 ) is embedded, closed and positively oriented. For I = R, we assume that Σ t = X 0 (R, t) is, up to translation, smoothly asymptotic to two distinct time-independent lines for p → −∞ and p → ∞, where we also assume that α(t) ≡ α 0 ∈ (−π, π) (1.5) as well as Σ0 |κ| ds < ∞ . (1.6) Note that (1.5) follows from the evolution equation of α(t), see Lemma 3.1, and the asymptotic behaviour of the curve.
The APCSF was first studied by Gage [8]. He proved that initially embedded, closed, convex curves stay embedded, smooth and convex, and converge smoothly to a circle of radius A 0 /π, where A 0 = A(Σ 0 ) is the enclosed area of the initial curve. In [21], Maeder-Baumdicker studied APCSF for convex curves with Neumann boundary on a convex support curve and showed smooth convergence to an arc for sufficiently short, convex, embedded initial curves. She proved a monotonicity formula and excluded type-I singularities for embedded, convex curves under the APCSF. For the LPCF, Pihan [25] showed that initially embedded, closed, convex curves stay embedded, smooth and convex, and converge smoothly and exponentially to a circle of radius L 0 /2π.
In this paper, we will adapt theory from CSF. For CSF in the plane, Gage-Hamilton and Grayson [9,11] showed that all embedded, closed initial curves stay embedded until they smoothly and exponentially shrink to a round point. In [18], Huisken gave a different proof for this result by bounding the ratio of the extrinsic distance d(p, q, t) := X(q, t) − X(p, t) R 2 and the intrinsic distance l(p, q, t) := q p ds t for curves Σ t = X(R, t) with asymptotic ends, respectively the extrinsic distance and the function ψ(p, q, t) := L t π sin π l(p, q, t) L t for curves Σ t = X(S 1 , t), below away from zero, and by applying singularity theory for CSF. In [3], Andrews and Bryan found an explicit function to proof curvature bounds via the distance comparison principle. To analyse curvature blow-ups, one distinguishes between type-I and type-II singularities and rescales the curve near a point of highest curvature. Using his famous monotonicity formula in [17], Huisken showed that if an immersed curve develops a type-I singularity under CSF, the curves Σ t have to be asymptotic to a homothetically shrinking solution around the singular point. Abresch and Langer [1] had previously classified all embedded, homothetically shrinking solutions of CSF as circles. One concludes, in case of a type-I singularity, that the curves shrink to a round point. For the type-II singularities, Hamilton [12] and Altschuler [2] showed that each rescaling sequence converges to a translating solution. For curves in the plane, the only solution of this kind is the so-called grim reaper which is, for all τ ∈ R, given by the graph of the function u(σ, τ ) = τ − log cos(σ), where σ ∈ (−π/2, π/2). On the grim reaper inf(d/l) = 0, so that type-II singularities can be excluded. Since T < ∞ and a singularity has to form, it has to be of type I. This paper is structured as follows. In Section 2, we state evolution equations for the geometric quantities under (1.1) and draw first conclusions. In Section 3, we consider angles of tangent vectors and derive a strong maximum principle for the local total curvature θ(p, q, t) := q p κ ds t . (1.7) In the subsequent sections, we study the flow (1.1) for embedded, positively oriented, smooth initial curves Σ 0 = X 0 (I) with θ 0 (p, q) = q p κ ds ≥ −π (1.8) for all p, q ∈ I. Note that for convex curves θ 0 ≥ 0. Figure 1 is an example for condition (1.8), where all the angles lay between −π and 3π, e. g. θ(p, q) = −π, θ(q, p) = 3π, θ(q, r) = 2π, θ(r, q) = 0, θ(r, p) = π.
In Sections 4 and 5, we modify the distance comparison principles from [18] and prove that, for for I = R 0, 1 2π Σt κ 2 ds t + 2π L t for I = S 1 (1.9) and if the initial embedding Σ 0 satisfies (1.8), the ratio d/l for I = R and d/ψ for I = S 1 is bounded from below away from zero uniformly in time. We conclude that the curves Σ t stay embedded for all t ∈ [0, T ). We also show that the condition (1.8) is sharp, that is, one can construct initial curves which violate (1.8) arbitrarily mildly and for which the resulting flow self-intersects in finite time. An example is the initial curve in Figure 2 with length sufficiently large compared to the C 3,α -norm of its embedding and for which min I×I θ 0 < −π, e. g. θ(p 1 , p 2 ) < −π.
In Section 6, we assume that T < ∞ and there exist constants 0 < c, C < ∞ so that h satisfies (1.9) and additionally for t ∈ [0, T ) and study curvature blow ups via parabolic rescaling. We use the distance comparison principles from Sections 4 and 5 in the same fashion as for CSF in [18] to exclude type-II singularities and conclude that the flow exists for all positive times.
In Section 7, we assume that a solution is immortal, that is, it exists for all positive times and the global term satisfies the following. Let δ ∈ (0, ∞) be given so that δA 0 is the desired limit area and where γ = (δ − 1)A 0 / L 2 0 /4π − A 0 . We prove that the curves become convex in finite time. The global term above ensures that the enclosed area is bounded away from zero and the length is bounded away from infinity throughout the flow. In Section 8, we assume that an immortal solution of (1.1) with h satisfying (1.11) is convex. We expand Gage's and Pihan's results and show smooth and exponential convergence to a round circle. parameter s : I → [0, L] is given by s(p) := p p0 v(r) dr, so that ds = vdp and d ds = 1 v d dp . For Σ = X(S 1 ), the arc length parameter is given by s : andX := X • s −1 : S 1 L/2π → R 2 parametrises Σ by arc length. For Σ = X(R), the arc length parameter is given by s : R → R. The unit tangent vector field τ to Σ in direction of the arc length parametrisation is given by τ := d dsX . The outward unit normal is given by ν := (τ 2 , −τ 1 ). We define the curvature by Let X : I × [0, T ) → R 2 be a one parameter family of maps. For fixed t ∈ [0, T ), we can parametrise Σ t = X(I, t) by arc length via the arc length parameter s( · , t), where s(I, t) ∈ {S 1 Lt/2π , R} and the arc length parametrisation is given byX( · , t) = X( · , t) • s −1 ( · , t) : s(I, t) → R 2 . The evolution equation (1.1) applied to the arc length parametrisation reads for s ∈ s(I, t), whereν(s, t) =ν(s(p, t), t) = ν(p, t) and we used the identity ∆ ΣX = d2 ds 2X = κ for the curvature vector. Whenever we will calculate via the arc length parametrisation, we will do so at a fixed time. Since the images X(I, t) = X(s(I, t), t) are the same and X andX only differ by a tangential diffeomorphism, we will omit the "∼" in the following above geometric quantities related toX if these depend on s rather than p. Lemma 2.1 (Gage [8]). Let X : I × (0, T ) → R 2 be a solution of (1.1). Then, for t ∈ (0, T ), Corollary 2.2 (Huisken [16,Thm. 1.3]). Let X : S 1 × [0, T ) → R 2 be a solution of (1.1) and let κ ≥ 0 on Σ 0 . Then κ > 0 on Σ t for all t ∈ (0, T ).
Proof. Like in [25,Section 6.3] (see also [5,Chapter 4]), we can bound the derivatives of the curvature in terms of the curvature as long as the curvature is bounded. The proposition then follows like in [15,Thm. 8.1].

Angles and local total curvature
We want to exploit the relationship between angles of tangent vectors and local total curvatures and prove a strong maximum principle for the latter.
Define ϑ : I × [0, T ) → S 1 to be the angle between the x 1 -axis and the tangent vector, so that if e 2 , τ (p, t) < 0 .
For a fixed time t ∈ [0, T ), we can define the angleθ via the arc length parameter byθ : s(I, t) → [0, 2π). As explained earlier, we can omit the "∼" for simplicity. Like in (1.7), we define the total local curvature θ : where we integrate in direction of the parametrisation. The total curvature α(t) is given by the full integral over the curvature as stated in (1.4). For I = S 1 and p, q ∈ [0, 2π), we set for all p, q ∈ S 1 . For I = R and p < q, we set θ(q, p, t) = −θ(p, q, t). By Lemma 3.1, where ω ∈ Z is the local winding number. Hence, θ is the angle between the tangent vectors at two points on the curve modulo the local winding number. If a curve Σ = X(I) is embedded and convex, then 0 ≤ θ(p, q) < α for all p, q ∈ I. For I = S 1 and fixed p ∈ S 1 , Hence, θ is discontinuous along the diagonal {p = q} ⊂ S 1 × S 1 .

Distance comparison principle for noncompact curves
We adapt the methods from Huisken [18] to obtain estimates that imply a certain non-collapsing behaviour of the evolving curves.
The intrinsic distance l : I × I × [0, T ) → R is given by . Embedded curves satisfy (d/l)(p, q) > 0 for all p, q ∈ R. If a curve is not a line, then there exist p, q ∈ R so that d(p, q) < l(p, q) and thus inf R×R (d/l) < 1.
Lemma 4.1. Let Σ = X(R) be an embedded curve. Let p, q ∈ R, p = q, such that Σ crosses the connecting line between X(p) and X(q) at X(r) with r / ∈ [p, q]. Then (d/l)(p, q) cannot be the infimum.
Proof. Let Σ 0 = X 0 (R) be an embedded curve satisfying (1.5) and (1.8). Then for all t ∈ (0, T ). Lemma 3.3 implies that θ 0 ∈ [−π, α + 2π]. From the maximum principle for θ, Theorem 3.4, it follows that for all p, q ∈ S 1 and t ∈ (0, T ). Since d/l is continuous and initially positive, Let p, q ∈ R, p = q, be points where a local spatial minimum of d/l at t 0 is attained and assume w.l.o.g. that s(p, t 0 ) < s(q, t 0 ). We have for all In the following, we always calculate at the point (p, q, t 0 ). The spatial derivatives of d and l are all given in [18] (for detailed calculations, see [5, Lems. 6.2 and 7.4]).
The first spatial derivative of d/l at (p, t 0 ) in direction of the vector ξ = τ p ⊕ 0 is given by At (q, t 0 ) and for the vector ξ = 0 ⊕ τ q , we have Since d/l ∈ (0, 1), and by (4.3) and (4.4), there exists β ∈ (0, π) with By Lemma 3.5, (4.2) and (4.5), either for k ∈ Z. We use the evolution equation (1.1) and Lemma 2.1 to differentiate the ratio in time, We are now considering four different cases.
(iv) Assume that θ ∈ (π, α + 2π). By (4.5), w, τ q = w, τ p = d/l ∈ (0, 1). Since Σ t0 is embedded with ends going to ∞ and X( · , t 0 ) is continuous, the curve has to cross the line segment between X(p, t 0 ) and X(q, t 0 ) at least once at X(r, t 0 ) with r / ∈ [p, q]. Lemma 4.1 implies that d/l cannot attain the infimum at (p, q, t 0 ) (it could still, however, attain a local minimum a this point). Hence, is either the infimum from (4.1) or a local minimum as discussed in cases (i), (ii) and (iii).
Assume that d/l falls below c := min{c 1 , c 2 }, where c 1 and c 2 are given in (4.1) and (4.16), and attains Λ ∈ (0, c) for the first time at time t 2 ∈ (0, T ) and points p, q ∈ S 1 , p = q, so that is the infimum and ∂ ∂t |t=t 2 d l (p, q, t) ≤ 0 .  Remark 4.4. Counterexample 5.5 shows that in order for embeddedness to be preserved it is crucial to assume that the initial local total curvature lies above −π.

Distance comparison principle for closed curves
We continue to adapt the methods from Huisken [18]. Let X(S 1 ) be a circle of radius R. Then d(p, q) 2 = L 2π sin πl(p, q) L for all p, q ∈ S 1 . This motivates the definition of the function ψ : Remark 5.1. Since sin(π − α) = sin(α), we have ψ(p, q, t) = ψ(q, p, t). Hence, we will later assume that l ≤ L/2. Embedded curves satisfy d/ψ > 0. If a closed curve Σ t is not a circle, then there exist p, q ∈ S 1 so that d(p, q, t) < ψ(p, q, t) and thus min S 1 ×S 1 (d/ψ) < 1.
Lemma 5.2. Let Σ = X(S 1 ) be an embedded, closed curve. Let p, q ∈ S 1 , p = q, such that Σ crosses the connecting line between X(p) and X(q). Then (d/ψ)(p, q) cannot be a global minimum.
Assume that d/ψ falls below c and attains Λ ∈ (0, c) for the first time at time t 2 ∈ (0, T ) and points p, q ∈ S 1 , p = q, so that is a global minimum and The next example shows, why the condition min θ 0 ≥ −π is sharp.
Pihan [25,Section 5.4] gave an incomplete proof for its validity which we will fix here. If we allow local total curvature smaller than −π, then there exist counterexamples for any given minimum min θ 0 < −π. For the curve in Figure 2, θ min = θ(p 1 , p 2 ) < −π. We will construct a solution of (1.1) with embedded initial curve Σ 0 that intersects itself in finite time. Fix K 0 > 0. Let S be the set of all smooth, embedded curves in R 2 that satisfy where L 0 is chosen big enough so that curves like in Figure 2 are in S. By the short time existence, see [16, p. 36] In particular, where X 1 := X, e 1 , and, by (1.1) and (5.25), for all p ∈ S 1 and for all t ∈ [0, T /2], where ν 1 := ν, e 1 . Assume h(0) > 0 and set Then (5.26) holds for t ∈ [0, t 1 ]. Let Σ ∈ S be a curve like in Figure 2, which is symmetric about the x 2 -axis. Let p, q ∈ S 1 be located as in the picture so that for t ∈ [0, t 1 ]. Since min θ 0 < −π, we can smoothly deform a curve like in Figure 2 to achieve arbitrarily small distance between X(p, 0) and X(q, 0) without exceeding the upper bound K 0 in (5.24) or changing the length or enclosed area. Hence, we can choose an embedded initial curve Σ 0 with Then, by (5.29) and (5.31) and by (5.30) and (5.31) so that the curve has crossed itself by the time t 1 .

Singularity analysis
Proposition 2.4 states that the curvature blows up if T < ∞. In this section, we assume T < ∞ and investigate curvature blow-ups for embedded flows (1.1) that satisfy (1.8), (1.9) and (1.10). We adapt techniques from the theory of CSF to show that the curvature does not blow up in finite time and conclude T = ∞. Proposition 2.4 motivates the following definition. We say that a solution X : Lemma 6.1. Let X : I × [0, T ) → R 2 be a solution of (1.1) satisfying (1.10) and with maximal time T < ∞. Then, for all t ∈ (0, T ), Proof. The proof is as in [ Like for CSF, we distinguish between two kinds of singularities according to the blow-up rate from Lemma 6.1. Let X : I × (0, T ) → R 2 be a solution of (1.1) with T < ∞. We say that a singularity is of type I, if there exists a constant C 0 > 0 so that max for all t ∈ (0, T ). A singularity is said to be of type II, if such a constant does not exist, that is, lim sup Type-I singularities have already been exploited in [21,Section 4]. We refer also to [26, Section 11] for a characterisation of singularities for almost Brakke flows with bounded global terms, using a monotonicity formula and a result of [20]. Since the global term is bounded, it will vanish in any limit flow of a type-I rescaling where we rescale by the maximal curvature. Also, since the lengths of the curves are bounded away from zero, the curves of any limit flow will be of infinite length. Like in the analysis in [17] of type-I singularities of mean curvature flow, a monotonicity formula, see [21,Proposition 4.9] or [5, Theorem 8.5], yields that any limit flow of a type-I rescaling is an embedded homothetically shrinking solution of CSF with non-vanishing curvature. By [1], this is an embedded shrinking circle. This contradicts the unbounded length.
Proof. Theorem 6.3 yields that the limit flow consists of strictly convex or concave curves Σ ∞ τ for τ ∈ R satisfying sup R×R |κ ∞ | = |κ ∞ (0, 0)| = 1. If κ ∞ < 0, we change the direction of parametrisation so that κ ∞ > 0. Since the curvature attains its maximum at the point (0, 0) ∈ R × R, [14,Main Theorem B] yields that X ∞ is a translating solution of CSF. [2,Thm. 8.16] implies that Σ ∞ τ is the grim reaper for every τ ∈ R. The grim reaper is asymptotic to two parallel lines of distance π from inside. Let τ ∈ R. We can find a sequence of points (p j , Proof. By Theorems 6.2 and 6.4 neither a type-I nor a type-II singularity can form at T so that curvature stays bounded on [0, T ] by a constant C(Σ 0 , T ). We can extend the flow beyond T and repeat the above argument. Hence, for every time T < ∞, there exists a constant C(Σ 0 , T ) < ∞ so that max p∈S 1 |κ(p, t)| ≤ C for all t ∈ [0, T ). Applying Proposition 2.4 yields that the short time solution can be extended to a smooth solution on (0, ∞).

Convexity in finite time
In this section, we show that a smooth, embedded solution X : S 1 × (0, ∞) → R 2 of (1.1) with a global term h satisfying (1.11) becomes convex in finite time.
Theorem 7.9. Let X : S 1 × [0, ∞) → R 2 be a smooth, embedded solution of (1.1) with initial curve Σ 0 and h satisfying (1.11). Then there exists a time T 0 ≥ 0 such that Σ t is strictly convex for t > T 0 .

Longtime behaviour
In this section we show that convex solutions of (1.1) that exist for all positive times converge exponentially and smoothly to a round circle. This was already shown in [8] for the APCSF and in [25] for the LPCF. We repeat and extend the arguments here for h satisfying (1.11) for the sake of completeness. We mostly follow the lines of [9, Section 5] for rescaled convex CSF, [8] for convex APCSF, and [25,Chapter 7] for convex LPCF. For further details, see [5,Chapter 11].
Lemma 8.1 (Isoperimetric inequality, Gage [7]). For a closed, convex C 2 -curve in the plane, with equality if and only if the curve is a circle.
We summarise our results in the following and two theorems.
Proof. By Theorem 7.9, there exists a time T 0 > 0 so that the curves are strictly convex on (T 0 , ∞). Like in [9], [25,Section 7.5] and [5,Section 11.4], we can show for convex curves with the help of Wirtinger's inequality and the smooth convergence of Theorem 8.11 exponential decay of the L 2 -norm of the derivative of the curvature. The proof is independent of the particular form of h, which is why we do not repeat it here. Interpolation inequalities then yield that for β ∈ (0, 1) and m, n ∈ N ∪ {0}, m + n > 0, there exist constants C n,m > 0 such that max ϑ∈S 1 ∂ m ∂τ m ∂ n κ ∂ϑ n (ϑ, τ ) ≤ C n,m exp − βτ (n + 2m + 1)R 2 (8.13) for τ large enough. To prove (i), we follow the lines of [25,Prop. 7.27]. For t ≥ 0, let p 1 , p 2 ∈ S 1 be the points where the curvature attains its maximum and minimum. By Lemma 7.2 and (8.13), there exists a time-independent constant C > 0 so that max p∈S 1 κ(p, t) − min p∈S 1 κ(p, t) = |κ(p 2 , t) − κ(p 1 , t)| ≤ Σt ∂κ ∂s ds t ≤ C exp − βt R 2 .
Then there exists a unique, smooth, embedded solution X : S 1 ×[0, ∞) → R 2 to (1.1) with initial curve Σ 0 and h satisfying (1.2). The evolving curves Σ t = X(S 1 , t) are contained in a uniformly bounded region and converge smoothly and exponentially to a circle of radius R.
Proof. By the short time existence, there exists a unique solution X ∈ C ∞ (S 1 × [0, T )) By Lemma 7.2, c ≤ L ≤ C so that h is uniformly bounded from above and below away from zero. By Corollary 5.4 the curves remain embedded on (0, T ). Corollary 6.5 yields that T = ∞. Hence, we can apply Theorem 8.12.