2024/07/09 by Steven Charlton, Charlton, Steven, Lynn Heller +5 · 1 citation
Computer Science · Mathematics · #11G55 #11M32 #53A10 #53C42 #53C43 #Advanced Mathematical Identities #Analytic Number Theory Research #Differential Geometry (math.DG) #FOS: Mathematics #Graph Labeling and Dimension Problems #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2407.07130
openalex publication_date 2024/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we show for every sufficiently large integer g the existence of a complete family of closed and embedded constant mean curvature (CMC) surfaces deforming the Lawson surfaces ξ1,g parametrized by their conformal type. When specializing to the minimal case, we discover a pattern resulting in the coefficients of the involved expansions being alternating multiple zeta values (MZVs), which generalizes the notion of Riemann's zeta values to multiple integer variables. This allows us to extend a new existence proof of the Lawson surfaces ξ1,g to all g≥ 3 using complex analytic methods and to give closed form expressions of their area expansion up to order 7. For example, the third order coefficient is \tfrac94ζ(3) (the first and second order term were shown to be log(2) and 0 respectively in \citeHHT). As a corollary, we obtain that the area of ξ1,g is monotonically increasing in their genus g for all g≥ 0.