Continuous point map for spherical domains

Question

Consider the space $J$ of Jordan domains on the sphere $\textbf{S}^2$, i.e., continuous injective maps from the unit disk into $\textbf{S}^2$ modulo homeomorphisms of the disk. How can one construct a continuous map $f\colon J\to\textbf{S}^2$ which is equivariant under isometries of $\textbf{S}^2$, i.e., for every $\rho\in O(3)$ and $D\in J$, $f(\rho(D))=\rho(f(D))$?

For Jordan domains in Euclidean plane $\textbf{R}^2$, there are several examples for $f$ such as the center of mass, or circumcenter. But these notions do not seem to generalize readily to $\textbf{S}^2$.

Notes:

1. In a paper with Igor Belegradek, it was shown that $f$ exists for $C^1$ domains, using equivariant fiber bundle theory. We were also able to choose the points in the interior of the domains; however, the selection was not canonical.

2. $f$ does not exist if the topology on the domains is induced by Hausdorff distance and we stipulate that the point lies in the domain; see Remark 3.4 in the paper with Belegradek.

3. I first heard the question of continuous point selection from the interior of Jordan domains from Eugenio Calabi in 1995.

Answer 1

Here’s an attempt to define such a map which is not an answer but couldn’t fit in a comment.

Consider a Riemann mapping $\varphi:\mathbb{H}^2 \to \overset{\circ}{D}$ (which has a continuous extension to the closure by Carathéodory) and define $F(D) = \varphi( bar(\varphi^\ast\mu_{\mathbb{S}^2}))$. Here $\mu_{\mathbb{S}^2}$ is the spherical measure, $\varphi^\ast\mu_{\mathbb{S}^2}$ is the pulled back measure, and $bar$ is the barycenter *. This is well-defined (if the barycenter exists) since the barycenter is geometric and any two Riemann mappings will differ by composition with an isometry of $\mathbb{H}^2$.

• The catch is that the barycenter might not be well-defined if the pullback of the area measure has infinite second moment.

Let’s make another stab at defining such a barycenter that circumvents the issue of having infinite second moment. The Riemann mapping $\varphi$ is conformal, so each point $x\in \mathbb{H}^2$ has a conformal factor $c_x$, and this is well-defined independent of $\varphi$. As $x\to \infty$ in $\mathbb{H}^2$, $c_x \to 0$ (I think this can be deduced from Koebe’s theorem). There is a unique cutoff $C$ such that $\int_{c_x > C} \varphi^*(\mu_{\mathbb{S}^2}) =\frac12 \int_{\mathbb{H}^2} \varphi^*(\mu_{\mathbb{S}^2}) =\frac12 Area(D)$. Then $U=\{x\in\mathbb{H}^2 | c_x>C\}$ is bounded, and thus $bar(\varphi^*(\mu_{\mathbb{S}^2})_{|U})$ is well-defined.

I believe that this definition should be continuous on $J$, but I haven’t checked it carefully. One issue is that $U$ could change drastically with a small perturbation of $D$, with a little island where $c_x>C$ appearing far from the origin. However, I think the mass of this will be small, so shouldn’t affect the barycenter much. But I certainly haven’t thought this through carefully, and this may be where this proposal breaks down.

Answer 2

Igor Belegradek and I just finished another paper where we construct a continuous point selection from the interior of Jordan domains in Riemannian surfaces, which is equivariant under isometries of the surface, or even conformal transformations in some cases. This is a sequel to the earlier paper mentioned above.

We use equivariant fiber bundle theory, including a theorem of Palais on slices of proper group actions, and conformal mapping techniques based on Caratheodory's kernel theorem. Another approach, using the notion of reach in the sense of Federer, is also discussed for smooth planar domains.

An application is that the space of Jordan curves in a surface of constant curvature admits a strong deformation retraction to the space of round disks in an equivariantly way with respect to the isometries of the surface, which answers a question in a related post.