Rotational Hypersurfaces In The Spheres

The study of minimal surfaces and surfaces with constant mean curvature in thespace form is of great importance in geometry of submanifolds. In1970, Lawson con-jectured that the Cliford torus is the only embedded miniamal torus in the three-sphere.In2012, S. Brendle proved Lawson conjecture using”non-collapsing” argument. In thesame year, Ben Andrews and Haizhong Li applied this “non-collapsing” argument toembedded torus with constant mean curvature in the three-sphere and proved that anembedded CMC torus in S³must be a surface of rotation, which solved the famousPinkall-Sterling conjecture. Moreover, they further studied the rotational surface in S³,and gave a complete classification of embedded CMC tori in S³. The study of mini-mal hypersurface and CMC hypersurfaces in the high dimensional is also interesting.In1970s, T. Otsuki proved that the only embedded minimal hypersurfaces with twodistinct principal curvature in sphere must be Clliford torus. In this thesis, we studyrotational hypersurfaces in Sⁿ⁺¹. Our main results contain the following two parts:Firstly, we study a kind of special rotational hypersurfaces in Sⁿ⁺¹with Prof.Ben Andrews and get a uniqueness reslut. Specifically, we prove that a rotationalhypersurfaces Σ embedded in Sⁿ⁺¹with two distinct principal curvaturesλ, μ. If thereexists a constantα>0such thatλ+αμ=0, then Σ must be product of Sn1and S1.Secondly, for the rotaional CMC hypersurfaces in Sⁿ⁺¹, we study the period func-tion K（H,n,C）. We prove that for n=3,4, the function K（H, n, C） is monotone de-creasing in C. Therefore, we prove that when n=3,4, for any H≥0, there exists aunique rotational hypersurface Σ with constant mean curvature H and m-fold symmetryif and only if the positive integer m satisfiescot（π/m）＜H＜(（m²-2）（n-1）^1/2)/n(（m²-1）^1/2)...