SHARP RESULTS FOR SPHERICAL METRIC ON FLAT TORI WITH CONICAL ANGLE 6π AT TWO SYMMETRIC POINTS

In this paper, we investigate the following curvature equation: (0.1) [formula presented]. Here E_τ represents a flat torus and _ωk/2 is one of the half periods of E_ωk/2. Our primary objective is to establish a necessary and sufficient criterion for the existence of a non-even family of solutions (see the definition in Section 1). Remarkably, this is equivalent to determining the presence of solutions for the equation with a single conical singularity: [formula presented]. This study marks the first exploration of the structure of non-even families of solutions to the curvature equation with multiple singular sources in the literature. Building on our findings, we provide a comprehensive analysis of the solution structure for equation (0.1) for all τ. This analysis is facilitated by Theorem 1.3, which will play a central role in our exploration of cases involving general parameters in the future, such as: [formula presented]. As an application, we offer explicit descriptions for solutions to equation (0.1) in the context of both rectangle tori and rhombus tori. See Corollary 1.4 as well as Corollary 1.5.