Dirichlet energy on a Riemannian manifold ($E(u)$)

For a function $u$ on a compact Riemannian manifold, its Dirichlet energy is $\frac{1}{2}{\int }_M|\nabla u{|}_g^{2}d{\mathrm{vol}}_g$. Adding a linear source term $-{\int }_{M}fud{\mathrm{vol}}_g$ gives Euler-Lagrange equation $-{\Delta }_{g}u=f$.

For a scalar function $u$ on a Riemannian manifold, the Dirichlet energy is $E(u)=\frac{1}{2}{\int }_M|\nabla u{|}_g^{2}d{\mathrm{vol}}_g$.

For a smooth map between Riemannian manifolds, its Dirichlet energy is half the integral of the squared norm of its differential, using the source and target metrics. When the source is two-dimensional, a conformal change of its metric leaves this energy unchanged. For a J-holomorphic curve into a symplectic manifold with a compatible almost complex structure, the Energy identity for a J-holomorphic curve identifies it with the pulled-back symplectic area. The norm of a differential here is the full tensor norm, summing its squared values on an orthonormal source frame.