# Isometric immersions and metrics in the same conformal class

Let $$\phi:\Sigma^2\to M^3$$ an conformal isometric immersion into a Riemannian 3-manifold $$(M,g)$$.

I would like to know what kind of informations is preserved (about the immersion) when we change $$g$$ by $$e^f g$$.

For example, since $$g$$ is in the same conformal class of $$e^fg$$ then $$\phi:\Sigma^2\to M^3$$ is also an conformal isometric immersion into $$(M,e^fg)$$.

If we suppose that $$\Sigma$$ is an oriented Riemannian surface, is it possible to relate the normal vector field of $$\phi(\Sigma)$$ into $$(M,g)$$ and of $$(M,e^f g)$$?

If $$:\Sigma^2\to (M^3,g)$$ has constant mean curvature then $$\phi:\Sigma^2\to (M^3,e^fg)$$ has constant mean curvature too?

I appreciate any help and book recommendations.

The functional which is invariant under conformal change of the ambient metric is the Willmore functional (and not the area functional): $$\mathcal W(\phi)=\int_\Sigma (H^2-K+\bar K )dA,$$ where $$H$$ is the mean curvature, $$K$$ is the curvature of the induced metric, $$\bar K$$ is the sectional curvature of the tangent plane of $$\phi$$ and $$dA$$ is the area form, all w.r.t. to the metric $$g.$$ But in fact, the integrand $$(H^2-K+\bar K )dA$$ is invariant under conformal changes of the metric. This was known by Sophie Germain, Blaschke and others for Moebius transformations, but first shown in the general case by "B.Y. Chen, Some conforaml invariants of submanifolds and their applications, Bol. Un. Mat. Ital, (1974)", see also Joel Weiner: On a probelm of Chen, Willmore et.al., Indiana Math Journal (78).
• big thanks for your comments. Do you have any information about the normal vector field? Because, when $\Sigma$ is oriented, then we can suppose that the normal of the two immersion are the same, I guess. Feb 14, 2020 at 8:27