Given any $P,P^{\prime}$ on $L$, we write the potential as a line integral of the Liouville 1-form

$$f_{L}(P)-f_{L}(P^{\prime})=\int_{P^{\prime}}^{P}\lambda\leq\left\lVert\lambda\right\rVert_{C^{0}}dist(P,P^{\prime}).$$

Thus the intrinsic ball volume lower bound implies that if $a$ lies in the range of $f_{L}$, then

$$\text{Vol}(\{|f_{L}-a|<\frac{1}{2}\})\geq C^{-1}.$$

The range of $f_{L}$ is by assumption a closed interval. If the interval has length $\geq N$, then we can find $N$ distinct values of $a$ with disjoint $\{|f_{L}-a|<\frac{1}{2}\}$, so

$$C^{-1}N\leq\text{Vol}(L)\leq\frac{1}{\sin\epsilon}\int_{L}\text{Re}(\Omega)\leq C,$$

This provides an a priori bound on $N$, hence on the potential oscillation. ∎