# How to construct a nice homotopy?

Let $$(M,g)$$ be a closed simply-connected Riemannian manifold. Can we find a constant $$C$$, which depends on $$M$$, such that for any closed curve $$\alpha_0$$ with $$L(\alpha_0) \le 1$$, there exists a homotopy $$\{\alpha_s:0\le s \le 1\}$$ satisfying

(1) $$\alpha_1$$ is a point;

(2) For any $$0 \le s \le 1$$, $$L(\alpha_s) \le C$$?

Here $$L$$ denotes the length of a curve.

• This is a very nice and natural question. The result is a special case of a much more general result due to Lang and Schlichenmaier and I will write details later when I have time. – Piotr Hajlasz Jan 29 at 17:49
• If you want $C$ to depend only on $M$ and not on the metric $g$, then such $C$ can't exit. For example for $M=S^2$ take a double-ball movementfirst.sg/self-myofascial-release/… It has a shortish geodesic that can not be contracted without increasing its length significantly – Dmitri Panov Jan 29 at 23:45

The result is true and in fact we do not need the condition $$L(\alpha_0)\leq 1$$ since a stronger result is true:

Theorem 1. If $$(M,g)$$ is a closed simply-connected Riemannian manifold, then there is a constant $$C\geq 1$$ such that for every closed curve $$\alpha_0$$ of finite length $$L<\infty$$, there is a homotopy $$\alpha_s$$, $$0\leq s\leq 1$$ between $$\alpha_0$$ and a constant curve $$\alpha_1$$ such that $$L(\alpha_s)\leq CL$$ for all $$0\leq s\leq 1$$.

Since a reparametrization of a curve does not change its length, we can assume that the curve of length $$L$$ is parametrized by the constant speed which makes the curve $$L/(2\pi)$$-Lipschitz as a mapping from $$\mathbb{S}^1$$ (with arc-length distance) to $$(M,g)$$. Now the following result is a straightforward consequence of Theorem 5.1 in [1].

Theorem 2. If $$M$$ is a compact connected and simply-connected Riemannian manifold, then there is $$\gamma>0$$ such that every $$L$$-Lipschitz mapping $$\alpha:\mathbb{S}^1\to M$$ admits a $$\gamma L$$-Lipschitz extension $$A:\overline{\mathbb{B}}^2(0,1)\to M$$.

Indeed, using terminology from [1], $$M$$ is $$1$$-Lipschitz connected in the small, because it has a finite covering by balls that are bi-Lipschitz homeomorphic to Euclidean balls.

Now, as we pointed out at the beginning, we can assume that $$\alpha_0$$ is $$L/(2\pi)$$-Lipschitz. Therefore the extension $$A$$ is $$\gamma L/(2\pi)$$ Lipschitz and restrictions of $$A$$ to circles of radii $$0\leq t\leq 1$$ will give us a desired homotopy: the restriction to the circle of radius $$t$$ will have length at most $$\gamma Lt\leq \gamma L$$. This completes the proof of Theorem 1.

[1] Lang, U., Schlichenmaier, T., Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions. Int. Math. Res. Not. 2005, no. 58, 3625-3655. https://arxiv.org/abs/math/0410048