We have a map $f \in \mathcal{C}^{\infty}(M, N)$ with two manifolds $M$ and $N$ (with dimensions $m:\dim(M)$ and $n:=\dim(N)$). We define the graph $F: M \to M \times N$ by $F(p)=(p, f(p))$. I wish to prove:
1.) $e(F)=\frac{m}{2}+e(f)$, where $e$ is the energydensity.
2.) $f$ is harmonic iff $F$ is harmonic.
Thank you and best regards.
It is intended that $M\times N$ is endowed with the direct sum Riemann structure. In this case, for any smooth map $f:L\to M\times N$ it is true that $\frac{1}{2}|Df|^2=\frac{1}{2}|Df_1|^2+\frac{1}{2}|Df_2|^2$, whence (1) since the energy density of id is m/2. Also, $f$ is a local minimizer of the local energy integral $E(f)=E(f_1)+E(f_2)$ if and only of both $f_1$ and $f_2$ are, that implies (2).
rmk. Here of course "local minimizer" is a map f such that for any point p of the domain there exists a nbd U of p such that the integral of the density energy of f over U is minimum with respect to variations with compact support in U. This property is equivalent to harmonicity (and of course does not require that the total energy be finite).
