Preliminaries on the curve shortening flow

In this section we rather closely follow [A-G].

2.1 Let $M$ be a closed $n$-dimensional manifold, $n\ge 3,$ and let $g^t$ be a smooth family of riemannian metrics on $M$ evolving by the Ricci flow on a finite time interval $[t_0,t_1].$ It is known [B] that $g^t$ for $t>t_0$ are real analytic. Let $c^t$ be a solution to the curve shortening flow in $(M,g^t),$ that is $c^t$ satisfies the equation $\frac{d}{dt}c^t(x)=H^t(x),$ where $x$ is the parameter on $\mathbb{S}^1,$ and $H^t$ is the curvature vector field of $c^t$ with respect to $g^t.$ It is known [G-H] that for any smoothly immersed initial curve $c$ the solution $c^t$ exists on some time interval $[t_0,t_1'),$ each $c^t$ for $t>t_0$ is an analytic immersed curve, and either $t_1'=t_1,$ or the curvature $k^t=g^t(H^t,H^t)^{\frac{1}{2}}$ is unbounded when $t\to t_1'.$

Denote by $X^t$ the tangent vector field to $c^t,$ and let $S^t=g^t(X^t,X^t)^{-\frac{1}{2}}X^t$ be the unit tangent vector field; then $H=\nabla_S S$ (from now on we drop the superscript $t$ except where this omission can cause confusion). We compute

$$\frac{d}{dt}g(X,X)=-2\mathrm{Ric}(X,X)-2g(X,X)k^2,$$ (1)

which implies

$$[H,S]=(k^2+\mathrm{Ric}(S,S))S$$ (2)

Now we can compute

$$\frac{d}{dt}k^2=(k^2)''-2g((\nabla_S H)^{\perp},(\nabla_S H)^{\perp})+2k^4 + ...,$$ (3)

where primes denote differentiation with respect to the arclength parameter $s,$ and where dots stand for the terms containing the curvature tensor of $g,$ which can be estimated in absolute value by $\mathrm{const}\cdot(k^2+k).$ Thus the curvature $k$ satisfies

$$\frac{d}{dt}k\le k''+k^3+\mathrm{const}\cdot(k+1)$$ (4)

Now it follows from (1) and (4) that the length $L$ and the total curvature $\Theta=\int kds$ satisfy

$$\frac{d}{dt}L\le \int (\mathrm{const}\ -k^2)ds,$$ (5)

$$\frac{d}{dt}\Theta\le \int \mathrm{const}\cdot (k+1)ds$$ (6)

In particular, both quantities can grow at most exponentially in $t$ (they would be non-increasing in a flat manifold).

2.2 In general the curvature of $c^t$ may concentrate near certain points, creating singularities. However, if we know that this does not happen at some time $t^{\ast},$ then we can estimate the curvature and higher derivatives at times shortly thereafter. More precisely, there exist constants $\epsilon , C_1, C_2,...$ (which may depend on the curvatures of the ambient space and their derivatives, but are independent of $c^t$), such that if at time $t^{\ast}$ for some $r>0$ the length of $c^t$ is at least $r$ and the total curvature of each arc of length $r$ does not exceed $\epsilon ,$ then for every $t\in (t^{\ast},t^{\ast}+\epsilon r^2)$ the curvature $k$ and higher derivatives satisfy the estimates $k^2=g(H,H)\le C_0 (t-t^{\ast})^{-1},\ \ g(\nabla_S H,\nabla_S H)\le C_1 (t-t^{\ast})^{-2},...\ \$ This can be proved by adapting the arguments of Ecker and Huisken [E-Hu]; see also [A-G,§4].

2.3 Now suppose that our manifold $(M,g^t)$ is a metric product $(\bar{M},\bar{g}^t)\times \mathbb{S}^1_{\lambda },$ where the second factor is the circle of constant length $\lambda ;$ let $U$ denote the unit tangent vector field to this factor. Then $u=g(S,U)$ satisfies the evolution equation

$$\frac{d}{dt}u=u''+(k^2+\mathrm{Ric}(S,S))u$$ (7)

Assume that $u$ was strictly positive everywhere at time $t_0$ (in this case the curve is called a ramp). Then it will remain positive and bounded away from zero as long as the solution exists. Now combining (4) and (7) we can estimate the right hand side of the evolution equation for the ratio $\frac{k}{u}$ and conclude that this ratio, and hence the curvature $k,$ stays bounded (see [A-G,§2]). It follows that $c^t$ is defined on the whole interval $[t_0,t_1].$

2.4 Assume now that we have two ramp solutions $c_1^t, c_2^t,$ each winding once around the $\mathbb{S}^1_{\lambda }$ factor. Let $\mu^t$ be the infimum of the areas of the annuli with boundary $c_1^t\cup c_2^t.$ Then

$$\frac{d}{dt}\mu^t\le (2n-1)\vert\mathrm{Rm}^t\vert\mu^t,$$ (8)

where $\vert\mathrm{Rm}^t\vert$ denotes a bound on the absolute value of sectional curvatures of $g^t.$ Indeed, the curves $c^t_1$ and $c^t_2,$ being ramps, are embedded and without substantial loss of generality we may assume them to be disjoint. In this case the results of Morrey [M] and Hildebrandt [Hi] yield an analytic minimal annulus $A,$ immersed, except at most finitely many branch points, with prescribed boundary and with area $\mu.$ The rate of change of the area of $A$ can be computed as

$$\int_A (-\mathrm{Tr}(\mathrm{Ric}^T)) +\int_{\partial A} (-k_g) \le \int_A (-\mathrm{Tr}(\mathrm{Ric}^T)+K)$$

$$\le \int_A (-\mathrm{Tr}(\mathrm{Ric}^T)+\mathrm{Rm}^T) \le (2n-1)\vert\mathrm{Rm}\vert\mu ,$$

where the first inequality comes from the Gauss-Bonnet theorem, with possible contribution of the branch points, and the second one is due to the fact that a minimal surface has nonpositive extrinsic curvature with respect to any normal vector.

2.5 The estimate (8) implies that $\mu^t$ can grow at most exponentially; in particular, if $c^t_1$ and $c^t_2$ were very close at time $t_0,$ then they would be close for all $t\in [t_0,t_1]$ in the sense of minimal annulus area. In general this does not imply that the lengths of the curves are also close. However, an elementary argument shows that if $\epsilon >0$ is small then, given any $r>0,$ one can find $\bar{\mu},$ depending only on $r$ and on upper bound for sectional curvatures of the ambient space, such that if the length of $c^t_1$ is at least $r,$ each arc of $c^t_1$ with length $r$ has total curvature at most $\epsilon ,$ and $\mu^t\le\bar{\mu},$ then $L(c^t_2)\ge (1-100\epsilon )L(c^t_1).$