Let $\{(M^2,g_t)\}_{t \in (a,b)}$ be a one-parameter family of Riemannian surfaces with boundary. Let $\Gamma$ be a piecewise $C^1$ curve on $M$ such that its endpoints lie on the boundary of $M$, and suppose it separates. Let $\Omega$ be one of the components.
If $\operatorname{Area}_{g_t}(\Omega)$ is constant with respect to $t$, is there a way to compute the first variation of the perimeter of $\Omega$? That is, is there a formula to compute
$$ \frac{d}{dt}L_{g_t}(\Gamma) $$ ?