Sphere and boundary conformal-Killing modes for tree gauge fixing

Write a global holomorphic vector field in the affine chart as $f(z){\mathrm{\partial }}_z$, with $f$ entire. Since ${\mathrm{\partial }}_z=-w^2{\mathrm{\partial }}_w$, regularity at $w=0$ means $w^{2}f(1\mathrm{/}w)$ is holomorphic near zero, hence $f(z)=O(\mathrm{\mid }z{\mathrm{\mid }}^2)$ at infinity. Thus for sufficiently large $R$, ${\sup }_{\mathrm{\mid }z\mathrm{\mid }=R}\mathrm{\mid }f(z)\mathrm{\mid }\le C{R}^2$ for a constant $C$. By [F1], for every $n\ge 3$, $\mathrm{\mid }{f}^{(n)}(0)\mathrm{\mid }\le n!C{R}^{2-n}$; letting $R\to \mathrm{\infty }$ gives $f^{(n)}(0)=0$. The entire Taylor series therefore has degree at most two. Conversely, if $f(z)=a+bz+c{z}^2$, then $-w^{2}f(1\mathrm{/}w)=-(a{w}^2+bw+c)$ is holomorphic at $w=0$, so each such polynomial defines a global vector field. The three monomials are linearly independent, proving dimension three.

An infinitesimal real matrix $M(t)=I+t\left(\begin{array}{cc}\alpha & \beta \\ \gamma & -\alpha \end{array}\right)+O(t^2)$ acts by $z\mapsto ((1+t\alpha )z+t\beta )\mathrm{/}(t\gamma z+1-t\alpha )+O(t^2)$. Differentiating at $t=0$ gives $\beta +2\alpha z-\gamma z^2$. The real parameters $\alpha ,\beta ,\gamma$ are independent, so the adopted boundary group has real Lie algebra spanned by the same three polynomial vector fields over $\mathbb{R}$ and dimension three.