Vanishing dλ-energy on all discs implies vanishing dλ-energy on the plane

Let $S=H^{-1}(1)$ be a level set with the induced contact form $\lambda$, and let $w:\mathbb{C}\to\mathbb{R}^{2n+2}$ be any map. Write $E_{d\lambda}(w;A)=\int_A w^*d\lambda$ for the $d\lambda$-energy of $w$ on a set $A\subseteq\mathbb{C}$, that is, the lower Lebesgue integral of the positive part of the $d\lambda$-density. If $E_{d\lambda}(w;B_R(0))=0$ for every $R\in\mathbb{R}$, then $E_{d\lambda}(w;\mathbb{C})=0$.

This lets a statement about the $d\lambda$-energy on every disc be read as a statement about the whole plane, for example when comparing with results on finite-energy planes.

Formalization Note The $d\lambda$-energy is dlambdaEnergy H A w, the lintegral over A of ENNReal.ofReal (dlambdaDensity H w z). No regularity of $w$ is assumed. For $R\le0$ the disc ball 0 R is empty.