We know that for the unit sphere in $\mathbb{R}^3$, the standard metric on such a sphere can be written as $$g=\frac{4}{(1+x_1^2+x_2^2)^2}(dx_1^2+dx_2^2)$$by sterographic projection map from the sphere to the complex plane.

I'm curious about the following question:

Given a closed Riemannian surface in $\mathbb{R}^3$, if the metric is globally conformal to $\mathbb{R}^2$, that is, $g=e^{2u}(dx_1^2+dx_2^2)$. Moreover, if we further assume that $u$ is radially symmetric, then does the surface have to be a sphere? Are there any other possible shapes except spheres? Can anyone give some specifice examples?

Any comments or ideas would be really appreciated.

We know that for the unit sphere in $\mathbb{R}^3$, the standard metric on such a sphere can be written as $$g=\frac{4}{(1+x_1^2+x_2^2)^2}(dx_1^2+dx_2^2)$$by a stereographic projection map from the sphere to the complex plane.

I'm curious about the following question:

Given a closed Riemannian surface in $\mathbb{R}^3$, if the metric is globally conformal to $\mathbb{R}^2$, that is, $g=e^{2u}(dx_1^2+dx_2^2)$, and if we further assume that $u$ is radially symmetric, then does the surface have to be a sphere? Are there any other possible shapes except spheres? Can anyone give some specific examples?

Any comments or ideas would be really appreciated.