Studies on some open problems concerning flat tori in the unit 3-sphere

单位 3 球体中平面环面的直径猜想表明，单位 3 球体中等距浸没的平面环面的外在直径等于 π。在本研究中，我们在浸没平均曲率非负或非正的假设下证明了该猜想。利用这个结果，我们证明，如果 f 是 Clifford 环面在平均曲率非负或非正的单位 3 球体中的等距浸入，则浸入 f 与 Clifford 环面的标准嵌入一致。

Diameter conjecture on flat tori in the unit 3-sphere states that the extrinsic diameter of isometrically immersed flat tori in the unit 3-sphere is equal to π. In this research, we proved the conjecture under the assumption that the mean curvature of the immersion is nonnegative or nonpositive. Using this result, we proved that if f is an isometric immersion of a Clifford torus into the unit 3-sphere whose mean curvature is nonnegative or nonpositive, then the immersion f is congruent to the standard embedding of the Clifford torus.