# Minimizing elastic energy of shrinking balls

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 prooving this.

Any help would be very much appreciated!