The curve shortening flow in the metric-affine plane

We investigate for the first time the curve shortening flow in the metric-affine plane and prove that under simple geometric condition it shrinks a closed convex curve to a"round point"in finite time. This generalizes the classical result by M. Gage and R.S. Hamilton about convex curves in Euclidean plane.


Introduction
The one-dimensional mean curvature flow is called the curve shortening flow (CSF), because it is the negative L 2 -gradient flow of the length of the interface, and it is used in modeling the dynamics of melting solids. The CSF deals with a family of closed curves γ in the plane R 2 with a Euclidean metric g = · , · and the Levi-Civita connection ∇, satisfying the initial value problem (with parabolic partial differential equation) ∂γ/∂t = kN, γ| t=0 = γ 0 . (1) Here, k is the curvature of γ with respect the unit inner normal vector N and γ 0 is an embedded plane curve, see survey in [3,10]. The flow defined by (1) is invariant under translations and rotations Recall that the curvature of a convex plane curve is positive. The next theorem by M. Gage and R.S. Hamilton [4] describes this flow of convex curves. This theorem and further result by M.A. Grayson, [5] (that the flow moves any closed embedded in the Euclidean plane curve in a finite time to a convex curve) have many generalizations and applications in natural and computer sciences. For example, the anisotropic curvature-eikonal flow (ACEF) for closed convex curves, see [3,Section 3.4], ∂γ/∂t = (Φ(θ)k + λ Ψ(θ)) N, γ| t=0 = γ 0 , where Φ > 0 and Ψ are 2π-periodic functions of the normal to γ(· , t) angle θ and λ ∈ R, generalizes the CSF. Anisotropy of (2) is indispensable in dealing with phase transition, crystal growth, frame propagation, chemical reaction, and mathematical biology. The particular case of ACEF, when Φ and Ψ are positive constants, serves as a model for essential biological processes, see [6]. On the other hand, (2) is a particular case of the flow During last decades, many results have appeared in the differential geometry of a manifold with an affine connection∇ (which is a method for transporting tangent vectors along curves), e.g., collective monographs [2,8]. The difference T =∇ − ∇ (of∇ and the Levi-Civita connection ∇ of g), is a (1,2)-tensor, called contorsion tensor. Two interesting particular cases of∇ (and T) are as follows.
1) Metric compatible connection:∇g = 0, i.e., T(X, Y ), Z = − T(X, Z), Y . Such manifolds appear in almost Hermitian and Finsler geometries and are central in Einstein-Cartan theory of gravity, where the torsion is represented by the spin tensor of matter.
2) Statistical connection:∇ is torsionless and the rank 3 tensor∇g is symmetric in all its entries, i.e., T(X, Y ), Z is fully symmetric. Statistical manifold structure, which is related to geometry of a pair of dual affine connections, is central in Information Geometry, see [7]; affine hypersurfaces in R n+1 are a natural source of statistical manifolds.
There are no results about the CSF in metric-affine geometry. The metric-affine plane is R 2 endowed with a Euclidean metric g and an affine connection∇. Our objective is to study the CSF in the metric-affine plane and to generalize Theorem 1 for convex curves in (R 2 , g,∇). Thus, we replace (1) by the following initial value problem: wherek is the curvature of a curve γ with respect to∇ and γ 0 is a closed convex curve. Note that (3) is the particular case (when Φ = 1 and λ Ψ(θ) = Ψ(θ)) of the ACEF. Put Let {e 1 , e 2 } be the orthonormal frame in (R 2 , g,∇). In the paper we assume that the contorsion tensor T is ∇-parallel, i.e., T has constant components T k ij = T(e i , e j ), e k and constant norm T = c ≥ 0.
Let γ : S 1 → R 2 be a closed curve in the metric-affine plane with the arclength parameter s. Then T = ∂γ/∂s is the unit vector tangent to γ. In this case, k = ∇ T T, N and the curvature of γ with respect to an affine connection∇ isk = ∇ T T, N , we obtain where Ψ is the following function on γ: By the assumptions T = c, see (4), and T = N = 1, we have The convergence of the ACEF (2) when Φ and Ψ are positive has been studied in [3,Chapter 3]. However, our function Ψ in (5) takes both positive and negative values, and [3, Theorem 3.23] is not applicable to our flow of (3). By this reason, we independently develop the geometrical approach to prove the convergence of (3) to a "round point". Our main goal is the following theorem, generalizing Theorem 1(a).
Theorem 2. Let γ 0 be a closed convex curve in the metric-affine plane with condition k 0 > 2 c. Then (3) has a unique solution γ(·, t), and it exists at a finite time interval [ 0, ω), and as t ↑ ω, the solution γ(·, t) converges to a point.
Nonetheless, the approach of [3] to the normalized flow of (2) in the contracting case still works without the positivity of Ψ, see [3,Remark 3.14]. Based on this result and Theorem 2, we obtain the following result, generalizing Theorem 1(b).
In Section 2, we prove Theorem 2 in several steps, some of them generalize the steps in the proof of [10, Theorem 1.3]. In Section 3, we prove Theorem 3 about the normalized flow (3), following the proof of convergence of the normalized flow (2) in the contracting case.
Theorem 2 can be easily extended to the case of non-constant contorsion tensor T of small norm, but we can not now reject the assumption (4) for Theorem 3, since its proof is based on the result for the normalized ACEF, see [3], where Ψ depends only on θ.

Proof of Theorem 2
Recall the axioms of affine connections∇ : X M × X M → X M on a manifold M, e.g., [8]: for any vector fields X, Y, X 1 , X 2 , Y 1 , Y 2 and smooth function f on M.
The support function S of a convex curve γ is given by, e.g. [10], For example, a circle of radius ρ has S(θ) ≡ ρ. Since ∂γ/∂θ, N = 0, the derivative S θ is and γ can be represented by the support function and parameterized by θ, see [10], This yields the following known formula for the curvature of γ(θ): Then, according to (5) and (15),k be a family of closed curves satisfying (3). We will use the normal angle θ to parameterize each curve: γ(θ, t) = γ(u(θ, t), t).
By the theory of parabolic equations we have the following.
From Lemma 2, see also (21), we conclude the following.
Proposition 5. If a 30 = a 03 = 0, see (9) and (10), then the problem (3) in the metricaffine plane reduces to the classical problem (1) in the Euclidean plane for modified by parallel translation of γ t curvesγ t = γ t + t[a 21 , a 12 ].
Example 2. One may show that is the support function of a special solution of (3) with a 30 = a 03 = 0. We claim that the solution is a family of round circles of radius ρ(t) (t ≥ 0) shrinking to a point at the time t 0 = 1 2 ρ 2 (0). Indeed, by (14), S(θ, t) corresponds to a family of circles with centers ( −ǫ 2 (t), −ǫ 1 (t) ) and the curvature k = −1/ρ(t). We then calculate Thus, S θθ + S = ρ(t) holds, and (17) reduces to where θ is arbitrary. We get the system of three ODEs: Its solution with initial conditions ǫ i (0) = 0 is (22).

Example 3. (a)
The projective connections∇ = ∇ + T are defined by the condition where U is a given vector field, e.g., [8]. Then Ψ = T T T, N = 0, see (6). Thus, (3) in the metric-affine plane with a projective connection is equal to (1) in the Euclidean plane.
(b) The semi-symmetric connections∇ = ∇ + T are defined by the condition where U is a given vector field, e.g. [9]. Such connections are metric compatible, and for them the formulas (12)  Then, see (10), a 30 = U, e 2 − e 1 = −a 03 . Let U be a constant vector field on R 2 , then we can take the orthonormal frame {e 1 , e 2 } in (R 2 , g,∇) such that U is orthogonal to e 1 − e 2 . Thus, see Proposition 5, the problem (3) in the metric-affine plane with a semi-symmetric connection and constant U reduces to the problem (1) in the Euclidean plane.
Proposition 6 (Finite time existence). Let a convex closed curve γ 0 in the metric-affine plane with condition k 0 > 2 c be evolved by (3). Then, the solution γ t must be singular at some time ω > 0.
Note that a point or a line segment are the only compact convex sets of zero area in R 2 .
Lemma 3 (Enclosed area). Let a convex closed curve γ 0 in the metric-affine plane with condition k 0 > 2 c be evolved by (3). Then γ(·, ω) is either a point or a line segment.
Proof. Suppose the lemma is not true. We may assume the origin is contained in the interior of the region enclosed by γ(·, ω). We can draw a small circle, with radius 2ρ and centered at the origin, in the interior of the region enclosed by γ(·, ω).
Since the solution γ(·, ω) becomes singular at the time ω, we know from the evolution equation (15) that the curvature k(·, t) becomes unbounded as t → ω. To derive a contradiction, we only need to get a uniform bound for the curvature. Consider For anyω < ω, we can choose (θ 0 , t 0 ) such that Without loss of generality, we may assume t 0 > 0. Then at (θ 0 , t 0 ), On the other hand, By the above, Sincek = k + Ψ(θ) withk > 0, see the proof of Proposition 4, and using (7), we obtain From quadratic inequality (24) we conclude that Thus, k is bounded as t ↑ ω, -a contradiction. Thus, the area enclosed by γ(·, t) tends to zero as t ↑ ω.
To complete the proof of Theorem 2, observe that if the flow (3) does not converge to a point as the enclosed by γ(·, t) area tends to zero, then min θ∈S 1 k(θ, t) tends to zero as t ↑ ω, -a contradiction to Proposition 4.
The following steps for the ACEF, see [3], are applicable to the normalized flow (3): 1) The entropy for the normalized flow, E(γ(·, τ )) = 1 2π 2π 0 logk dθ, is uniformly bounded for τ ∈ [ 0, ∞), see [3, pages 63-68]. The bound on the entropy yields upper bounds for the diameter and length of the normalized flow, and also thatk and its gradient are uniformly bounded. 4) With two-sided bounds fork, the convergence of the normalized flow (3), as τ → ∞, follows. Namely, noting that e −τ Ψ → 0 when τ → ∞, for any sequence τ j → ∞, we can find a subsequence j k such that solution S(·, τ j k ) of (27) converges in C ∞ topology (as k → ∞) to a solution of the corresponding stationary equation S =k. (28) Based on the fact [1] that the only embedded solution of (28) is the unit circle, we conclude (similarly as in [3, p. 73] for ACEF with Φ = 1) thatγ(·, τ ) converges, as τ → ∞, to the unit circle in C ∞ , that completes the proof of Theorem 3.

Conclusion
The main contribution of this paper is a geometrical proof of convergence of CSF for convex closed curves in a metric-affine plane. In the future, we will study several related problems on convergence of flows in metric-affine geometry, for example: 1) CSF (3) for non-constant contorsion tensor T and for not just convex γ 0 , 2) Anisotropic CSF in metric-affine geometry, 3) The mean curvature flow for convex hypersurfaces in the metric-affine R n . 4) Numerical experiments (as in [1]) for solutions of (3) when γ 0 is not embedded.