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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 $1 \le s \le 1$, $L(\alpha_s) \le C$?

Here $L$ denotes the length of a curve.

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.