Complete families of embedded high genus CMC surfaces in the 3-sphere (with an appendix by Steven Charlton)
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| Format: | Preprint |
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2021
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| author | Heller, Lynn Heller, Sebastian Traizet, Martin |
| author_facet | Heller, Lynn Heller, Sebastian Traizet, Martin |
| contents | For every $g \gg 1$, we show the existence of a complete and smooth family of closed constant mean curvature surfaces $f_φ^g,$ $ φ\in [0, \tfracπ{2}],$ in the round $3$-sphere deforming the Lawson surface $ξ_{1, g}$ 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 $f_φ^g$ at $t= 0$ explicitly. In particular, we obtain for large genus Lawson surfaces $ξ_{1,g}$ % 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 $\tfrac{9}{4}ζ(3),$ where $ζ$ is the Riemann $ζ$ function (while the first and second order term were shown to be $\log(2)$ and $0$ respectively in \cite{HHT}). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2108_10214 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Complete families of embedded high genus CMC surfaces in the 3-sphere (with an appendix by Steven Charlton) Heller, Lynn Heller, Sebastian Traizet, Martin Differential Geometry 53A10, 53C42, 53C43 For every $g \gg 1$, we show the existence of a complete and smooth family of closed constant mean curvature surfaces $f_φ^g,$ $ φ\in [0, \tfracπ{2}],$ in the round $3$-sphere deforming the Lawson surface $ξ_{1, g}$ 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 $f_φ^g$ at $t= 0$ explicitly. In particular, we obtain for large genus Lawson surfaces $ξ_{1,g}$ % 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 $\tfrac{9}{4}ζ(3),$ where $ζ$ is the Riemann $ζ$ function (while the first and second order term were shown to be $\log(2)$ and $0$ respectively in \cite{HHT}). |
| title | Complete families of embedded high genus CMC surfaces in the 3-sphere (with an appendix by Steven Charlton) |
| topic | Differential Geometry 53A10, 53C42, 53C43 |
| url | https://arxiv.org/abs/2108.10214 |