\leq \frac{2}{\ell} \|f\|^2_{L^2}$.

It follows directly from (\[E:norm\]) and (\[E:DN\]).

Let $C=C_{p,q}:I \to {\mathbb{R}}^n$ be a chord joining two distinct points $p,q$ on $\Sigma$. We define the tangential outward conormal $\eta^T:\partial I=\{0,1\} \to {\mathbb{R}}^n$ to be the tangential component (relative to $\Sigma$) of the outward unit conormal of $C$, i.e. (recall (\[E:1st-var-b\]) and Definition \[D:Cpq\]) $$\label{E:eta-T}
\eta^T(u)=\pi_{C(u)} \eta \qquad \text{for $u=0,1$}.$$

\[L:length-evolution\] Suppose $C(u,t):I \times [0,T) \to {\mathbb{R}}^n$ is a solution to the chord shortening flow relative to $\Sigma$ as in Definition \[D:CSF\]. If we denote the length of the chord $C(t)$ at time $t$ by $$\ell(t):=d(C(0,t),C(1,t)),$$ then $\ell$ is a non-increasing function of $t$ and (recall (\[E:norm\]) and (\[E:eta-T\])) $$\label{E:ell-evolution}
\frac{d \ell}{dt} = - \|\eta^T\|_{L^2}^2 \leq 0 .$$

It follows directly from the first variation formula (\[E:1st-var-b\]).

\[T:eta-T-evolution\] Suppose $C(u,t):I \times [0,T) \to {\mathbb{R}}^n$ is a solution to the chord shortening flow relative to $\Sigma$ as in Definition \[D:CSF\]. Then the tangential outward conormal $\eta^T$ of the chord $C(t)$ satisfies the following evolution equation: $$\begin{aligned}
\label{E:eta-T-evolution}
\frac{\partial}{\partial t} \eta^T &=& - {\Delta^{\frac{1}{2}}}\eta^T + \frac{1}{\ell} \|\eta^T\|^2_{L^2} \eta^T  - \sum_{i=1}^k {\langle}A(\eta^T,e_i),\eta^N {\rangle}e_i  \\
&& \phantom{aaaaaaaaa} - \frac{1}{\ell}  (\overline{\eta^T}-\eta^T)^N - A(\eta^T,\eta^T), \nonumber\end{aligned}$$ where $\{e_i\}_{i=1}^k$ is an orthonormal basis of $T\Sigma$ at the 