Complete families of embedded high genus CMC surfaces in the 3-sphere (with an appendix by Steven Charlton)

TL;DR

This paper constructs a continuous family of high genus constant mean curvature surfaces in the 3-sphere, connecting Lawson surfaces to geodesic spheres, and provides explicit power series expansions for their geometric properties.

Contribution

It introduces an implicit function theorem approach to generate and analyze high genus CMC surfaces, including explicit power series computations of their area and DPW potential.

Findings

Existence of a smooth family of CMC surfaces deforming Lawson surfaces to geodesic spheres.

Explicit power series expansion of area and DPW potential in terms of multiple polylogarithms.

Identification of the third order coefficient of the area expansion with 9/4 a(3).

Abstract

For every $g \gg 1$, we show the existence of a complete and smooth family of closed constant mean curvature surfaces $f_\varphi^g,$ $ \varphi \in [0, \tfrac{\pi}{2}],$ in the round $3$-sphere deforming the Lawson surface $\xi_{1, g}$ to a doubly covered geodesic 2-sphere with monotonically increasing Willmore energy. To construct these we use an implicit function theorem argument in the parameter $t= \tfrac{1}{2(g+1)}$. This allows us to give an iterative algorithm to compute the power series expansion of the DPW potential and area of $f_\varphi^g$ at $t= 0$ explicitly. In particular, we obtain for large genus Lawson surfaces $\xi_{1,g}$ % due to the real analytic dependence of its area and DPW potential on $t,$ a scheme to explicitly compute the coefficients of the power series in $t$ in terms of multiple polylogarithms. Remarkably, the third order coefficient of the area expansion is identified with $\tfrac{9}{4}\zeta(3),$ where $\zeta$ is the Riemann $\zeta$ function (while the first and second order term were shown to be $\log(2)$ and $0$ respectively in \cite{HHT}).