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The Curve Shortening Flow in the Metric-Affine Plane

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The Curve Shortening Flow in the Metric-Affine Plane mathematics Article The Curve Shortening Flow in the Metric-Affine Plane Vladimir Rovenski Department of Mathematics, University of Haifa, Mount Carmel, Haifa 31905, Israel; [email protected] Received: 31 March 2020; Accepted: 21 April 2020; Published: 2 May 2020 Abstract: We investigated, for the first time, the curve shortening flow in the metric-affine plane and prove that under simple geometric condition (when the curvature of initial curve dominates the torsion term) 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 a Euclidean plane. Keywords: curve shortening flow; affine connection; curvature; convex MSC: 53C44; 53B05 1. Introduction The one-dimensional mean curvature flow is called the curve shortening flow (CSF), because it is the negative L2-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 g in the plane R2 with a Euclidean metric g = h · , · i and the Levi-Civita connection r, satisfying the initial value problem (with parabolic partial differential equation) ¶g/¶t = kN, gj t=0 = g0. (1) Here, k is the curvature of g with respect the unit inner normal vector N and g0 is an embedded plane curve, see survey in [1,2]. 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 [3] describes this flow of convex curves. Theorem 1. (a) Under the CSF (1), a convex closed curve in the Euclidean plane smoothly shrinks to a point in finite time. (b) Rescaling in order to keep the length constant, the flow converges exponentially fast to a circle in C¥. This theorem and further result by M.A. Grayson, [4] (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 in a Euclidean plane, see ([1] Section 3.4), ¶g/¶t = (F(q)k + l Y(q)) N, gj t=0 = g0, (2) where F > 0 and Y are 2p-periodic functions of the normal to g(· , t) angle q and l 2 R, generalizes the CSF. Anisotropy of the flow (2), studied when Y is also positive, is indispensable in dealing with phase transition, crystal growth, frame propagation, chemical reaction, and mathematical biology. The particular case of ACEF, when F and Y are positive constants, serves as a model for essential biological processes, see [5]. On the other hand, (2) is a particular case of the flow in a Euclidean plane 2 R (x1, x2), ¶g/¶t = F(g, q, k)N, gj t=0 = g0, Mathematics 2020, 8, 701; doi:10.3390/math8050701 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 701 2 of 11 1 2 4 see ([1] Chapter 1), where g = (g , g ) and F = F(x1, x2, x3, x4) is a given function in R , 2p-periodic in q = x3. In recent decades, many results have appeared in the differential geometry of a manifold with an affine connection r¯ (which is a method for transporting tangent vectors along curves), e.g., collective monographs [6,7]. The difference T = r¯ − r (of r¯ and the Levi-Civita connection r of g), is a (1,2)-tensor, called contorsion tensor. Two interesting particular cases of r¯ (and T) are as follows. (1) Metric compatible connection: r¯ g = 0, i.e., hT(X, Y), Zi = −hT(X, Z), Yi. Such manifolds appear in almost Hermitian and Finsler geometries and are central in Einstein-Cartan theory of gravity, where the contorsion tensor is related to the spin tensor of matter. (2) Statistical connection: r¯ is torsionless and the rank 3 tensor r¯ g is symmetric in all its entries, i.e., hT(X, Y), Zi is fully symmetric. Statistical manifold structure, which is related to geometry of a pair of dual affine connections, is central in Information Geometry, see [8]; affine hypersurfaces in + Rn 1 are a natural source of statistical manifolds. There are no results about the CSF in metric-affine geometry. In this paper, the metric-affine plane is R2 (a two-dimensional real vector space) endowed with a Euclidean metric g and an affine connection r¯ . Our objective is to study the CSF in the metric-affine plane and to generalize Theorem1 for convex curves in (R2, g, r¯ ). Thus, we replace (1) by the following initial value problem: ¶g/¶t = k¯ N, gj t=0 = g0, (3) where k¯ is the curvature of a curve g with respect to r¯ and g0 is a closed convex curve. Such flow can be interesting as a geometrically natural analogue of the CSF, which could perhaps show some different behavior and it is not clear that it is a gradient flow for length and is variational in nature. Note that (3) is the particular case (when F = 1 = l) of the ACEF, where Y may change its sign. Put k0 := minfk(x) : x 2 g0g > 0. 2 Let fe1, e2g be the orthonormal frame in (R , g, r¯ ). In the paper we assume the following k Condition 1: T has constant components Tij = hT(ei, ej), eki (that is the contorsion tensor T is r-parallel) and constant norm kTk = c ≥ 0. Let g : S1 ! R2 be a closed curve in the metric-affine plane with the arclength parameter s. Then T = ¶g/¶s is the unit vector tangent to g. In this case, k = hrT T, Ni and the curvature of g with respect to an affine connection r¯ is k¯ = hr¯ T T, Ni, we obtain k¯ = k + Y, (4) where Y is the following function on g: Y = hT(T, T), Ni. (5) kTk = jT( )j = k k = k k = By assumptions : supX,Y2R2nf0g X, Y c, see Condition 1, and T N 1, we have j Yj ≤ c. (6) As far as we are aware such variations to the CSF have not been considered before. In this novel case, the new term defining the difference from the classical case is of a different order of homogeneity with respect to the curvature itself, so the extra term becomes weaker and weaker where the curvature is large, unlike the Finsler/anisotropic setting, where the anisotropic effects remain at all scales. The convergence of the ACEF (2) when F and Y are positive has been studied in ([1] Chapter 3). However, our function Y in (4) takes both positive and negative values, and ([1] Theorem 3.23) is not Mathematics 2020, 8, 701 3 of 11 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”. The main result of the paper states that a “sufficiently convex” closed curve (i.e., with curvature bounded below by a positive constant chosen to bound the size of the extra term), the curve shrinks to a point under our new flow, and is asymptotic to a shrinking circle solution of the classical CSF. The following theorem generalizes Theorem1(a). Theorem 2. Let g0 be a closed convex curve in the metric-affine plane with condition k0 > 2 c. Then (3) has a unique solution g(·, t), and it exists at a finite time interval [ 0, w), and as t " w, the solution g(·, t) converges ( ) to a point. Moreover, if k > 3 c then w ≤ A g0 · k0−2c , where A(g ) is the area enclosed by g . 0 2p k0−3c 0 0 Nonetheless, the approach of [1] to the normalized flow of (2) in the contracting case still works without the positivity of Y, see ([1] Remark 3.14). Based on this result and Theorem2, we obtain the following result, generalizing Theorem1(b). Theorem 3. Consider the normalized curves g˜(·, t) = (2(w − t))1/2g(·, t), see Theorem2, and introduce a new time variable t = −(1/2) log(1 − w−1 t) 2 [ 0, ¥). Then the curves g˜(·, t) converge to the unit circle smoothly as t ! ¥. In Section2, we prove Theorem2 in several steps, some of them generalize the steps in the proof of ([2] Theorem 1.3). In Section3, we prove Theorem3 about the normalized flow (3), following the proof of convergence of the normalized flow (2) in the contracting case. Theorem2 can be easily extended to the case of non-constant contorsion tensor T of small norm, but we can not now reject the Condition 1 for Theorem3, since its proof is based on the result for the normalized ACEF, see [1], where Y depends only on q. 2. Proof of Theorem 2 Recall the axioms of affine connections r¯ : XM × XM ! XM on a manifold M, e.g., [7]: r¯ = r¯ + r¯ r¯ ( + ) = r¯ + ( ) · + r¯ f X1+X2 Y f X1 Y X2 Y, X f Y1 Y2 f XY1 X f Y1 XY2 for any vector fields X, Y, X1, X2, Y1, Y2 and smooth function f on M. 1 2 Let q be the normal angle for a convex closed curve g : S ! R , i.e., cos q = −hN, e1i and sin q = −hN, e2i. Hence, N = −[cos q, sin q], T = [− sin q, cos q]. (7) Lemma 1. The function Y given in (5) has the following view in the coordinates: 3 3 Y = a30 sin q + a03 cos q + a12 sin q + a21 cos q, (8) where aij are given by 2 2 1 1 1 2 a12 = T12 + T21 − T11, a21 = T12 + T21 − T22, 2 1 1 1 1 2 2 2 a03 = T22 − T22 − T12 − T21, a30 = T11 − T11 − T12 − T21.
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