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

For a smooth scalar function $u$ on a compact Riemannian manifold $M$, its Dirichlet energy isThe Riemannian gradient measures local variation and the Riemannian volume weights its contribution. With source function $f$, consider $J(u)=E(u)-{\int }_{M}fud{\mathrm{vol}}_g$. On a manifold without boundary, or for variations vanishing at the boundary, the first variation in direction $v$ isHere ${\Delta }_g={\mathrm{div}}_g{\nabla }_g$ is the Laplace-Beltrami operator. A critical function therefore solves the Poisson equation $-{\Delta }_{g}u=f$.

On a closed manifold, adding a constant leaves $E$ unchanged. If ${\int }_{M}fd{\mathrm{vol}}_g\ne 0$, shifting that constant makes $J$ unbounded below. Thus zero integral of the source is a necessary compatibility condition for this unconstrained closed-manifold problem.

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.