Dirichlet's Principle (Harmonic Functions)/Riemannian Manifold

Theorem
Let $\struct {M, g}$ be a compact connected $n$-dimensional Riemannian manifold with nonempty boundary.

Let $\map {C^{\infty}} M$ be the smooth function space.

Let $u \in \map {C^\infty} M$ be a smooth real function.

Let $\rd V_g$ be the Riemannian volume form.

Let $\grad$ be the gradient operator.

Let $\size {\, \cdot \,}$ be the Riemannian inner product norm.

Then $u$ is harmonic $u$ minimizes


 * $\ds \int_M \size {\grad u}^2 \rd V_g$

among all smooth real function with the same boundary values.