Global analysis, analysis on manifolds Definition

Energy of a map between Riemannian manifolds

Builds on Riemannian manifold Smooth manifold

Definition

Let $(M, g)$ be a compact Riemannian manifold of dimension $m$ , possibly with boundary, and let $(N, h)$ be a Riemannian manifold of dimension $n$ . Let $u \colon M \to N$ be a smooth map. The energy density of $u$ is the function $e(u) = \frac{1}{2} \abs{du}^2$ on $M$ . Here $\abs{du}^2$ is the squared norm of $du_x \colon T_x M \to T_{u(x)} N$ for the metrics $g$ and $h$ . In local coordinates $x^i$ on $M$ and $y^\alpha$ on $N$ it is $\abs{du}^2 = \sum_{i,j,\alpha,\beta} g^{ij}(x) \, h_{\alpha\beta}(u(x)) \, \frac{\partial u^\alpha}{\partial x^i} \frac{\partial u^\beta}{\partial x^j}.$ The energy of $u$ is $E(u) = \int_M e(u) \, dV_g = \frac{1}{2} \int_M \abs{du}^2 \, dV_g,$ where $dV_g$ is the Riemannian volume measure of $g$ .

Remarks

Equivalently, $\abs{du}^2 = \sum_i \abs{du(e_i)}_h^2$ for an orthonormal basis $(e_i)$ of $T_x M$ , which is the trace with respect to $g$ of the pull-back metric $u^* h$ . The same formula defines $E(u)$ for maps of Sobolev class $W^{1,2}$ , and for non-compact $M$ when the integral converges. For $N = \R$ it is the Dirichlet energy $\frac12 \int_M \abs{\nabla u}^2 \, dV_g$ , and for $M$ an interval it is the energy of a path. When $\dim M = 2$ , the energy is invariant under conformal changes of the metric $g$ . Its critical points are the harmonic maps.

Example: for the identity map of a compact $(M, g)$ of dimension $m$ , $du(e_i) = e_i$ , so $\abs{du}^2 = m$ and $E(\id) = \frac{m}{2} \operatorname{Vol}(M, g)$ .

Non-example: the formula gives no finite energy for the identity map of $\R$ with the Euclidean metric, since $\R$ is not compact. There $e = \frac{1}{2}$ everywhere, and the integral over $\R$ diverges.

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Riemannian manifold concept
Smooth manifold concept
Energy of a map between Riemannian manifolds definition

Tag	`5N1825CM10`
URI	`m2://5N1825CM10`
Kind	definition
Writing	W0 stub
MSC	58E20, 53C43
Defines	Energy of a map between Riemannian manifolds
Uses	Riemannian manifold, Smooth manifold

Energy of a map between Riemannian manifolds

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