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

单位3-球面中的平坦环面的直径猜想指出单位3-球面中等距浸入的平坦环面的外直径等于π。在本研究中，我们在浸入的平均曲率为非负或非正的假设下证明了该猜想。利用这个结果，我们证明了：如果f是Clifford环面到平均曲率为非负或非正的单位球面的等距浸入，则浸入f与Clifford环面的标准嵌入全等.

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.