nn-dimensional area of minimal rotational hypersurfaces in spheres

Let MR(n)MR(n) denote the set of compact minimal rotational hypersurfaces in the (n+1)(n+1)-dimensional sphere Sn+1Sn+1. For any compact hypersurface M⊂Sn+1M⊂Sn+1, let us denote by |M||M| its nn-dimensional area. In this paper we show that for any integer n≥2n≥2, the set of numbers An={|M|:M∈MR(n)}An={|M|:M∈MR(n)} is a discrete set. We also show that if M∈MR(3)M∈MR(3), then either |M|=|S3||M|=|S3|, or |M|=|M0(3)||M|=|M0(3)|, or |M|>|M0(3)||M|>|M0(3)|, where M0(n)=Sn−1(n−1n)×S1(1n). Finally, for 2≤n≤1002≤n≤100 we show numerically evidence that if M∈MR(n)M∈MR(n), then either |M|=|Sn||M|=|Sn|, or |M|=|M0(n)||M|=|M0(n)|, or |M|>2|M0(n)||M|>2|M0(n)|. We strongly believe that for any M∈MR(n)M∈MR(n), either |M|=|Sn||M|=|Sn|, or |M|=|M0(n)||M|=|M0(n)|, or |M|>2|M0(n)||M|>2|M0(n)| for all nn.