# Definition of quasi-harmonic map

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

Any reference will be highly appreciated!

## 1 Answer

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 $$E_q(\phi)={1\over 2}\int_M |d\phi(x)|^2e^{-{|y|^2\over 4}}\;dy.$$

• Thanks a lot for the answer. So, in a way, a quasi-harmonic map from M to N is an harmonic map from M endowed with that gaussian density to N. Why are quasi-harmonic maps studied? I can see that if M and N are Eucludean spaces, the Euler-Lagrange equation given by the quasi-energy is the drifted-Laplace equation. But what is the general motivation? – Onil90 Aug 4 '17 at 13:36
• @Onil90 I presume that the study of this energy has some kind of physical meaning, but unfortunately I can't give you more details. I study harmonic maps in geometry as part of my research, and I knew that there were some other harmonic-like maps derived from changing the Dirichlet energy, but I don't know the details about their motivation. – Edu Aug 4 '17 at 13:47
• @Onil90 : In mean curvature flow, the singularity model is more or less a critical points of such a drifted harmonic equations. Similar happens for Ricci flow and I believe also for other flows. – user99914 Aug 15 '17 at 14:08
• @JohnMa You are talking about the fact that shrinking solitons (which model the type I singularities) can be viewed as minimal surfaces w.r.t. that Gaussian density, right? Where can I find this phrased in terms of quasi harmonic maps? – Onil90 Aug 15 '17 at 14:19
• In Colding-Minicozzi's work here. Or more explicitly in my work on branched self-shrinkers in dimension 2 (see section 4 here) @Onil90 – user99914 Aug 15 '17 at 14:28