There exist the notion of the mean curvature sphere for surfaces $f\colon M\to R^3$ in 3-space: for $p\in M$ the sphere $S(p)$ is defined to be the unique sphere which goes through $f(p),$ which has at $f(p)$ the same tangent space, and which has the same mean curvature. Then one can show that this notion is Moebius invariant. Moreover, the energy of the sphere in the space of spheres is the exactly the Willmore functional. Of course, one has to do some computations for this, but the mean curvature sphere is clearly an important object in the field of Moebius invariant surface geometry.

Some Literature: Bryant: A duality theorem for Willmore surfaces. Journal of Dif- ferential Geometry 20, Burstal, Ferus, Leschke, Pedit, Pinkall: Conformal geometry of surfaces in S4. Lec- ture Notes in Mathematics 1772

There exist the notion of the mean curvature sphere for surfaces $f\colon M\to R^3$ in 3-space: for $p\in M$ the sphere $S(p)$ is defined to be the unique sphere which goes through $f(p),$ which has at $f(p)$ the same tangent space, and which has the same mean curvature. Then one can show that this notion is Moebius invariant. Moreover, the energy of the mean curvature sphere in the space of spheres is the exactly the Willmore functional. Of course, one has to do some computations for this, but the mean curvature sphere is clearly an important object in the field of Moebius invariant surface geometry.

Some Literature: Bryant: A duality theorem for Willmore surfaces. Journal of Dif- ferential Geometry 20, Burstal, Ferus, Leschke, Pedit, Pinkall: Conformal geometry of surfaces in S4. Lec- ture Notes in Mathematics 1772