Area and Boundary Length of Surfaces Diffeomorphic to Annuli

AREA AND BOUNDARY LENGTH OF SURFACES DIFFEOMORPHIC TO ANNULI TSZ-KIU AARON CHOW Abstract. In this paper, we give a proof to a statement in Perelman’s paper for finite extinction time of Ricci flow [5]. Our proof draws on different techniques from the one given in Morgan-Tian’s exposition [3] and is extrinsic in nature, which relies on the co-area formula instead of the Gauss-Bonnet theorem, and is potentially generalizable to higher dimensions. 1. Introduction The qualitative nature of singularities in Ricci flow of dimension 3 was signifi- cantly studied in Perelman’s three renowned papers [4–6]. The arguments in [4–6] were detailedly addressed in the expository articles published by Cao-Zhu [1], Kleiner-Lott [2] and Morgan-Tian [3]. In the third paper of Perelman [5] where he proved the extinction time for Ricci flow is finite using the technique of curve shortening flow, Perelman stated an estimate which concerns the lengths of two curves which are close in the sense that they are the boundary of an annulus with small area. More specifically speaking, given two non-intersecting closed connected curves c0 and c1 such that the disjoint union of them forms the boundary of an annulus Σ, we quote the statement given by Perelman in [5] here:“ If ε> 0 is small then, given any r> 0, one can findµ ¯, depending only on r and on upper bound for sectional curvatures of the ambient space, such that if the length of c0 is at least r, each arc of c0 with length r has total curvature at most ε, and Area(Σ) ≤ µ¯, then Length(c1) ≥ (1 − 100ε)Length(c0).” Perelman’s statement was addressed by Morgan and Tian in Chapter 18.6 of their exposition [3]. To our knowledge, this is the only place in the literature where this statement is addressed. In this paper, we will present a proof of Perelman’s statement which draws on different techniques. Our proof is extrinsic in nature, and relies on the co-area formula instead of the Gauss-Bonnet theorem. In particular, our proof is potentially generalizable to higher dimensions. To that end, we will arXiv:2107.00335v1 [math.DG] 1 Jul 2021 first give a slight variant of Perelman’s statement. We will first present the full argument to the case where R3 is the ambient space, as the main argument is easier to follow when it is presented in the R3 case. After that we will present how the argument can be carried over to the Riemannian case where the ambient space is a fixed Riemannian manifold. The statement corresponding to the R3 case is given as follows: 1 Theorem 1.1. Let 0 <ε< 10000 be small. If Γ0 and Γ1 are two connected closed embedded curves in R3 such that 1 (i) Γ0 is a C curve satisfying |T (x) − T (y)| ≤ ε for all x, y ∈ Γ0 such that distΓ0 (x, y) ≤ 1. Here, T denotes the unit tangent vector field along Γ0; (ii) Length(Γ0) ≥ 1; 2 (iii) Γ0 ⊔ Γ1 = ∂Σ for a smooth annulus Σ with Area(Σ) ≤ ε 1 2 TSZ-KIU AARON CHOW Then we have Length(Γ1) ≥ (1 − Cε)Length(Γ0), where C is a positive constant. Moreover, the statement corresponding to the Riemannian case is given as follows: 1 Theorem 1.2. Let 0 <ε< 10000 be small. Let (M,g) be a compact Riemannian manifold of dimension 3 with inj(M,g) ≥ 1000. If Γ0 and Γ1 are two connected closed embedded curves in M such that 1 (i) Γ0 is a C curve satisfying |Py,x(T (y)) − T (x)| ≤ ε for all x, y ∈ Γ0 such that distΓ0 (x, y) ≤ 1. Here, T denotes the unit tangent vector field along the curve Γ0 and Py,x : TyM → TxM is the parallel transport along Γ0; (ii) Length(Γ0) ≥ 1; 2 (iii) Γ0 ⊔ Γ1 = ∂Σ for a smooth annulus Σ ⊂ M with Area(Σ) ≤ ε . Then we have Length(Γ1) ≥ (1 − Cε)Length(Γ0), where C is a positive constant depending only on the ambient space. Let us point out here that Theorem 1.2 implies Perelman’s statement in [5], which we summarize here formally: 1 Corollary 1.3. Let 0 <ε< 10000 be small. Let (M,g) be a compact Riemannian manifold of dimension 3 and let K > 0 be an upper bound of the sectional curvatures of M. Then given any r> 0, the following holds: If Γ0 and Γ1 are two connected closed embedded curves in M such that (i) The length of Γ0 is at least r. And |k|ds ≤ ε ZI for any sub-segment I of Γ0 of length r. Here k denotes the curvature of the curve Γ0. 1 2 2 (ii) Γ0 ⊔ Γ1 = ∂Σ for a smooth annulus Σ ⊂ M with Area(Σ) ≤ 1000K r ε . Then we have Length(Γ1) ≥ (1 − Cε)Length(Γ0), where C is a positive constant depending only on the ambient space. Corollary 1.3 will be proved in Appendix B. Now we illustrate the arguments we used to prove Theorem 1.1 and Theorem 1.2. The idea is to first construct a one- parameter family of disks around Γ0 such that when the parameter is restricted to a suitable segment, the corresponding disks form a foliation. Using that, we consider the intersection curve of each disk with the annulus Σ. Since the annulus has small area and the tangent vector field along Γ0 cannot change too much along any sub- segment of length 1, one could then use the co-area formula to deduce that most of the intersection curves intersect with Γ1 and do not cross with each others. From this, we can deduce the desired length inequality. The main difficulty with this argument is the construction of the desired family of disks, which is due to the fact that we do not have a point-wise control on the curvature of Γ0. It could happen that some points on Γ0 have arbitrarily high curvature, so that near these points AREAANDBOUNDARYLENGTHOFANNULI 3 the disks perpendicular to Γ0 may have intersection out to any fixed distance. Nev- ertheless, we could get around this difficulty by first constructing a smooth curve 1 m that is C -close to Γ0 and has uniform control on the C -bound, where m ≥ 2. After that a canonical one-parameter family of disks could be constructed using the smoothed curve in a way that that each disk is perpendicular to the smoothed curve. In Section 2, we will construct a canonical one-parameter family of disks around 3 Γ0 when the ambient space is R . This will be incorporated into Section 3 to prove Theorem 1.1. After that we will demonstrate in Appendix A how to adapt the arguments in Section 2 and 3 to the Riemannian case, and Theorem 1.2 will follows accordingly. Lastly, we will give a proof of Corollary 1.3 from Theorem 1.2 in Appendix B. Throughout the paper, a curve is understood to mean a connected curve. And the symbol C is served to denote a positive constant depending only on the ambient space. Acknowledgement: The author would like to express his gratitude to his advi- sor Professor Simon Brendle for his continuing encouragement and many inspiring ideas. 2. A one-parameter family of disks in R3 By the assumptions in Theorem 1.1, we immediately have the following lemma: Lemma 2.1. Let γ0 : R → Γ0 be a parametrization of Γ0 by arc-length. Then for |s − s0|≤ 100 we have ′ (i) |γ0(s) − γ0(s0) − (s − s0)γ0(s0)|≤ Cε; ′ ′ (ii) |γ0(s) − γ0(s0)|≤ Cε. Corollary 2.2. Under the assumptions of Theorem 1.1, Length(Γ0) ≥ 100. Proof. Let’s denote by L the length of Γ0. Let γ0 : R → Γ0 be a parametrization of Γ0 by arc-length, then γ0 is periodic with period L. In particular, we have γ0(L)= γ0(0). Now if L< 100, then Lemma 2.1 implies 1 ≤ L ≤ Cε, which is a contradiction. Corollary 2.3. Suppose that Γˆ is a sub-segment in Γ0 with end-points x,ˆ yˆ. If |xˆ − yˆ|≤ 10ε and Length(Γ)ˆ ≥ 1, then Length(Γ)ˆ ≥ 50. Proof. Let’s assume in contrary that Length(Γ)ˆ < 50. Let γ0 : R → Γ0 be a parametrization of Γ0 by arc-length and denote by L the length of Γ.ˆ Without loss of generality we may assume γ0(0) =x ˆ and γ0(L)=ˆy. Then Lemma 2.1 implies 4 TSZ-KIU AARON CHOW that ′ 2 Cε ≥ |γ0(L) − γ0(0) − Lγ0(0)| 2 ′ ≥ L − 2hγ0(L) − γ0(0), Lγ0(0)i 2 ≥ L − 2L|γ0(L) − γ0(0)| ≥ 1 − 1000ε, which is a contradiction. Next, we would like to construct a one-parameter family of disks which form a smooth foliation when they are restricted to short sub-segments of Γ0. The idea is to replace Γ0 by a piecewise linear path, and then smooth out the corners using a cut-off function. We first fix a smooth cut-off function χ : R → R such that: 1 • χ =0on(−∞, − 4 ]; 1 • χ = 1 on [ 4 , ∞). We also fix a parametrization γ0 : R → Γ0 by arc-length, so that γ0 is periodic with period L = Length(Γ0). By Lemma 2.1, we have L > 100. Now we fix a number L 1 k ∈ [L, 2L] and divide [0,L] into sub-intervals of length k ∈ [ 2 , 1]. Associated with the curve γ0, we define a curveγ ˜0 by jL ks ks (j + 1)L jL γ˜ (s) := γ + − j χ − j γ − γ 0 0 k L L 0 k 0 k ks ks (j − 1)L jL + j − 1 − χ − j γ − γ , (2.1) L L 0 k 0 k (j−1/2)L (j+1/2)L 1 3 1 for s ∈ k , k , j = 2 , 2 , .., k − 2 .

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