Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 16 / 6 / c

Choose a Riemannian metric $h$ on $\Sigma$ compatible with its complex structure $j$, and use $g_J=\omega (\cdot ,J\cdot )$ on the target. The Dirichlet energy of a map isIt is independent of the particular conformal representative $h$: rescaling $h$ multiplies the squared differential norm by the inverse factor and the area element by the same factor. For a J-holomorphic curve, the Energy identity for a J-holomorphic curve isTo prove it, take an oriented $h$-orthonormal frame $e_1,e_2=j{e}_1$ and put $a=du(e_1)$, $b=du(e_2)$. The J-holomorphic curve equation says $b=Ja$. The pointwise energy density is therefore $\frac{1}{2}(|a{|}_{g_J}^2+|Ja{|}_{g_J}^2)=|a{|}_{g_J}^2$, while the pulled-back area density is $\omega (a,Ja)=|a{|}_{g_J}^2$. Integrating proves the identity.

More generally, the same frame givesWith the full tensor norm of ${\bar{\partial }}_{J}u$, its two frame components are $(a+Jb)/2$ and $(b-Ja)/2$, so their squared norms sum to $\frac{1}{2}|b-Ja{|}^2$. Hence the full identity isThis also fixes the normalization of the error term. The energy is nonnegative and vanishes exactly when $du=0$. The formulas apply whenever the relevant integrals are defined, in particular on compact source surfaces.