Definition of quasi-harmonic map

Question

What is the definition of quasi-harmonic maps between two Riemannian manifolds?

Any reference will be highly appreciated!

Answer 1

For a smooth map $\phi:(M,g)\to(N,h)$ between two Riemannian manifolds $(M,g)$ and $(N,h)$, its energy $E(\phi)$ is defined by $$E(\phi)={1\over 2}\int_M |d\phi(x)|^2\;dM,$$ where $|d\phi(x)|^2$ is the square of the norm of the differential $d\phi(x)$ at a point $x\in M$.

A map $\phi:(M,g)\to(N,h)$ is called harmonic if $\phi$ is a critical point of the energy functional $E$.

In the literature, one can also find the notion of quasi-harmonic map. Suppose that $N$ is a smooth compact Riemannian manifold. We call $\phi:(M,g)\to(N,h)$ a quasi-harmonic map, if $\phi$ is a non-constant smooth map which is a critical point of the quasi-energy functional $E_q$ with respect to any smooth compactly supported variation, where $$E_q(\phi)={1\over 2}\int_M |d\phi(x)|^2e^{-{|y|^2\over 4}}\;dy.$$