I am new to variational analysis and I am currently working in the following setup:

We denote the $2$-sphere with radius $r>0$ by $S_r^2$ and $S^2:=S_1^2$.
In coordinates $(\theta,\varphi)$, we have a metric on $S_r^2$ given by $$g_r=r^2 d\varphi^2+r^2\sin^2\varphi d\theta^2$$ For $r>0,\, r\neq 1$ I want to find a minimizer of the following functional: \begin{align} E:C^\infty(S_r^2,S^2)&\longrightarrow\mathbb R_{\geq 0} \\ f&\longmapsto\frac{1}{\text{Vol}(S_r)}\int_{S_r^2}\operatorname{dist}^2(df(x),SO_x(g_r,g))dx \end{align} where $SO_x(g_r,g)$ is the set of orientation preserving isometries $T_x S_r^2\rightarrow T_{f(x)} S^2$.

An obvious candidate would be $f:x\mapsto\frac{1}{r}x$. Then $df=\frac{1}{r}\operatorname{id}$ and therefore $$\operatorname{dist}^2(df(x),SO_x(g_r,g))=\left|\frac{1}{r}\operatorname{id}-\operatorname{id}\right|^2=\frac{(1-r)^2}{r^2}$$ for all $x$. Hence $\inf E\leq\frac{(1-r)^2}{r^2}$.

I am suspecting that $\inf E=E(f)=\frac{(1-r)^2}{r^2}$, so that $f$ is indeed a minimizer, but I am having trouble how I would go about proving this.



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