Does there exist a harmonic map from S^2 to 3d hyperbolic space

Question

My question is, does there exist a harmonic map from $S^2$ to $\mathbb{H}^3$ , $\mathbb{H}^3$ means the 3d hyperbolic space. In addition, if it exist, could we directly construct the map?

Thank you very much!

Answer 1

We can prove that the only such maps are the constants using the Bochner technique.

Consider the Bochner Identity for harmonic maps:

If $f:(M,g)\to (N,h)$ is harmonic then $$-\frac12\Delta|Df|^2 = |\nabla Df|^2 + \sum_i h(Df(\text{Rc}^M(e_i)), Df(e_i)) - \sum_{i,j}\text{Rm}^N(Df(e_i),Df(e_j),Df(e_i),Df(e_j))$$ for $e_i$ an orthonormal frame for $g$.

If we had a harmonic map $f:\Bbb S^2 \to \Bbb H^3$, then the second term on the right becomes $|Df|^2 \ge 0$ and the third is a (negative multiple of) a sectional curvature of $\Bbb H^3$. Integrating gives zero on the LHS (Stokes Theorem) and the sum of three non-negative integrals on the RHS; so the three quantities on the RHS must be identically zero. In particular $|Df|^2 = 0$ and thus $f$ is constant.