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Let $\Sigma$ be a 2-sphere with a (smooth) Riemannian metric $g$ of nonnegative curvature. Let $\mathcal{C}$ be a simple closed smooth curve on $\Sigma$. Then $\mathcal{C}$ splits $\Sigma$ into two components $\Sigma^+$, $\Sigma^-$. Set $S(t):=\{p\in\Sigma^+\mid d(\mathcal{C},p)=t\}$ where $d$ is the distance function induced by $g$. If I rememeber correctly, the length $L(t)$ of $S(t)$ is well-defined for almost all $t\geq 0$.

Is it true that $$|L(t_1)-L(t_2)|\leq 2\pi|t_1-t_2|$$ whenever defined? (At least when $S(t_1),S(t_2)$ are non-empty.)

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Cut out a small arc $AB$ of the equatorial circle of the round $S^2$ and replace it by minimizing geodesics connecting $A$ and $B$ to the north pole. Denote the resulting curve $\mathcal C$. Then the curves $S_t$ south of $\mathcal C$ vary in length wildly for small $t$, violating your bound.

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  • $\begingroup$ Does it still have nonnegative curvature ? $\endgroup$
    – Thomas Richard
    Commented Jul 3, 2017 at 8:51
  • $\begingroup$ @Thomas: Only the surface was assumed to have nonnegative curvature. $\endgroup$
    – Pete
    Commented Jul 3, 2017 at 10:50
  • $\begingroup$ @Mikhail: Thank you, of course! Wondering if it might hold true with more assumptions on $\mathcal{C}$. I'm interested in the case where $\mathcal{C}$ itself is the distance sphere of a point on the other hemisphere. $\endgroup$
    – Pete
    Commented Jul 3, 2017 at 10:53

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