the Dirichlet energy
/ DEER-ish-lay /
Imagine a rubber sheet stretched over a frame, or a map that drapes one surface onto another. The more it has to stretch, the more elastic energy it stores. The Dirichlet energy is the precise bookkeeping of that total stretching: add up, point by point, how much the map is stretching in all directions, and integrate. Minimizing it is the mathematical version of letting the sheet relax, and that single functional is the source of harmonic functions, harmonic maps, geodesics, and the modern solution of Plateau's problem.
For a function u: M -> R on a Riemannian manifold the Dirichlet energy is E(u) = (1/2) integral over M of |grad u|^2 dV. For a map f: (M, g) -> (N, h) it is E(f) = (1/2) integral over M of |df|^2 dV, where |df|^2 = g^{ij} h_{ab}(f) (partial_i f^a)(partial_j f^b) is the squared norm of the differential — the trace of the pulled-back target metric. The first variation gives the Euler-Lagrange equation: for functions it is Laplace's equation Delta u = 0 (so minimizers are harmonic functions), and for maps it is the harmonic-map equation tau(f) = 0. The key structural fact, special to a two-dimensional domain, is conformal invariance: E(f) is unchanged if you rescale the domain metric conformally, so on a surface the Dirichlet energy and the area functional share the same critical maps after a conformal change — this is exactly the bridge Douglas and Rado used to solve Plateau's problem and the reason energy is analytically nicer than area.
The Dirichlet energy is the workhorse functional of geometric analysis because it is quadratic in derivatives, hence convex enough to give existence by the direct method (minimize over a Sobolev space, extract a weakly convergent minimizing sequence, pass to the limit by lower semicontinuity), and because its critical points organize a vast swath of geometry under one roof. The honest cautions: the energy controls only first derivatives, so a finite-energy map can still be discontinuous in high dimensions (a Sobolev W^{1,2} map into a sphere can have point singularities for domain dimension at least three). And conformal invariance, the magic that makes the surface theory work, is precisely what causes energy to concentrate and bubble in the borderline dimension two — a sequence of maps can have bounded energy yet lose energy into shrinking spheres in the limit.
For the identity map of a flat torus to itself, |df|^2 equals the dimension at every point and the Dirichlet energy is just (dimension/2) times the total volume; any homotopic map that stretches more has strictly larger energy, so the identity is the harmonic (and minimizing) representative of its class.
The identity map minimizes Dirichlet energy in its homotopy class on a flat torus.
Dirichlet energy and area agree as functionals only on a two-dimensional conformally parametrized domain; in higher domain dimensions they are genuinely different, and 'minimize energy to get a minimal surface' is a surface-specific trick, not a general identity.