2021/01/01 by Lynn Heller, Heller, Lynn, Sebastian Heller +3
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.2108.10214
For every g ≫ 1, we show the existence of a complete and smooth family of closed constant mean curvature surfaces fφg, φ∈ [0, \tfracπ2], in the round 3-sphere deforming the Lawson surface ξ1, 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= \tfrac12(g+1). This allows us to give an iterative algorithm to compute the power series expansion of the DPW potential and area of fφg at t= 0 explicitly. In particular, we obtain for large genus Lawson surfaces ξ1,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 \tfrac94ζ(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 \citeHHT).