continuously on $c$, with respect to the $C^0$ topology.

-   $c$ and $\Psi(c)$ are homotopic in $M$ relative to $\Sigma$, i.e. there exists a continuous family $c_t:[0,1] \to M$, $t \in [0,1]$, with end points on $\Sigma$ such that $c_0=c$ and $c_1=\Psi(c)$. Moreover, the family $c_t$ depends continuously on $c$.

The curve shortening process $\Psi$ involves subdividing the curves and connecting points on the curve by minimizing geodesic segments (additional care has to be taken at the end points). The construction depends on some fixed parameter (which depends on the geometry of $M$, $\Sigma$ and ${\operatorname{Length}}(c)$). However, it can be shown that for curves with uniformly bounded length, the parameters can be chosen uniformly to make (1) - (3) above hold. In fact (1) and (3) follows easily from the constructions, but (2) requires some convexity estimates (see [@Zhou16 Lemma 3.2]). Using (1) - (3), it is not difficult to see that the sequence $\{\Psi^i(c)\}_{i=1}^\infty$ either converges to a point on $\Sigma$ or has a subsequence converging to an orthogonal geodesic chord. Theorem \[T:LS\], \[T:Bos\] and \[T:double-normals\] then follows from the abstract Lusternik-Schnirelmann theory applied to families of curves with end points on $\Sigma$ which represent a non-trivial homology class 