Global analysis, analysis on manifolds Definition

Harmonic map heat flow

Builds on Riemannian manifold Tension field Energy of a map between Riemannian manifolds Heat equation

Definition

Let $(M, g)$ be a compact Riemannian manifold without boundary, let $(N, h)$ be a Riemannian manifold, and let $u_0 \colon M \to N$ be a smooth map. Let $0 < T \le \infty$ . A solution of the harmonic map heat flow with initial map $u_0$ on $[0, T)$ is a map $u \colon M \times [0, T) \to N$ with these properties. It is smooth on $M \times (0, T)$ , continuous up to $t = 0$ , and $\frac{\partial u}{\partial t}(x, t) = \tau\big(u(\cdot, t)\big)(x) \quad \text{on } M \times (0, T), \qquad u(x, 0) = u_0(x),$ where $\tau$ is the tension field.

Remarks

The energy decreases along a solution: $\frac{d}{dt} E\big(u(\cdot, t)\big) = -\int_M \abs{\tau(u(\cdot, t))}^2 \, dV_g \le 0.$ If $N$ is compact, a solution exists on some interval $[0, T)$ with $T > 0$ , since the equation is a semilinear parabolic system. A map $u_0$ is a fixed point of the flow exactly when it is a harmonic map.

Example: for $N = \R$ the Christoffel symbols vanish, and the flow is the heat equation $\partial_t u = \Delta_g u$ on $M$ . Its solutions exist for all time and converge to the constant function equal to the mean of $u_0$ .

Non-example: a family with $\partial_t u = -\tau(u)$ is not a solution of the harmonic map heat flow but of the backward flow, along which the energy increases. For $N = \R$ this is the backward heat equation, which is ill-posed.

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Energy of a map between Riemannian manifolds concept
Heat equation concept
Riemannian manifold concept
Tension field concept
Harmonic map heat flow definition

Tag	`X6QC23AAB9`
URI	`m2://X6QC23AAB9`
Kind	definition
Writing	W0 stub
MSC	58E20, 53E99, 35K55
Defines	Harmonic map heat flow
Uses	Riemannian manifold, Tension field, Energy of a map between Riemannian manifolds, Heat equation

Harmonic map heat flow

Definition

Remarks

Reading path

Further reading

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