The Primitive and Diamond surfaces locally minimize the variance of Gauss curvature

Abstract

We prove that the Schwarz' Primitive (P) and Diamond (D) surfaces are local minimizers of the variance of Gaussian curvature among local deformations within the moduli space of triply periodic minimal surfaces of genus 3 (TPMSg3s). Our approach interprets the branch values of the Gauss map as a configuration of eight points on the sphere and expresses the variance of Gaussian curvature as the product of two integrals involving exponentials of Green's functions. We then show that the Hessian of this functional is positive definite at the cubic configuration when restricted to antipodal deformations, thereby establishing the local minimality of the P and D surfaces. Along the other deformation directions, the Hessian is positive definite except for a two-dimensional eigenspace with slightly negative eigenvalues, leaving promising hope that the Gyroid is also a local minimizer.