原始曲面与钻石曲面局部最小化高斯曲率的方差
The Primitive and Diamond surfaces locally minimize the variance of Gauss curvature
AI总结：
该研究证明施瓦茨原始与钻石曲面是亏格3三重周期极小曲面模空间内高斯曲率方差的局部极小值，其海森矩阵在特定形变方向正定，为螺旋曲面的局部极小性提供了可能。
AI中文摘要：
我们证明，在亏格为3的三重周期极小曲面（TPMSg3）模空间内的局部形变中，施瓦茨原始（P）曲面与钻石（D）曲面是高斯曲率方差的局部极小值。我们的方法将高斯映射的分支值解释为球面上8个点的构型，并将高斯曲率的方差表示为涉及格林函数指数的两个积分的乘积。随后，我们证明该泛函的海森矩阵在三次构型上，在限制于对跖形变时为正定，从而确立了P和D曲面的局部极小性。在其他形变方向上，海森矩阵除了具有特征值略为负的二维本征空间外均为正定，这为螺旋（Gyroid）曲面也为局部极小值留下了良好的希望。
英文摘要：
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.
