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Integral Inequalities for Compact Hypersurfaces with Constant Scalar Curvature in the Euclidean Sphere
Mediterranean Journal of Mathematics ( IF 1.1 ) Pub Date : 2020-02-22 , DOI: 10.1007/s00009-020-1482-z
Luis J. Alías , Josué Meléndez

We study the rigidity of compact-oriented hypersurfaces with constant scalar curvature isometrically immersed into the unit Euclidean sphere \({\mathbb {S}}^{n+1}\). In particular, we establish a sharp integral inequality for the behavior of the norm of the total umbilicity tensor, equality characterizing the totally umbilical hypersurfaces, and a certain family of standard tori of the form \({\mathbb {S}}^1(\sqrt{1-r^2})\times {\mathbb {S}}^{n-1}(r)\). Moreover, under an appropriate constraint on the total umbilicity tensor, we are able to extend this result for any integer k, with \(2\le k\le n-1\), equality characterizing the totally umbilical hypersurfaces and a certain family of standard product of spheres of the form \({\mathbb {S}}^k(\sqrt{1-r^2})\times {\mathbb {S}}^{n-k}(r)\).

中文翻译:

欧氏球面中具有恒定标量曲率的紧致超曲面的积分不等式

我们研究了等量浸入单位欧几里得球体\({\ mathbb {S}} ^ {n + 1} \)中的具有恒定标量曲率的紧致定向超曲面的刚度。特别是,我们为总脐带张量的范数的行为建立了尖锐的积分不等式,其特征在于完全脐带超曲面的相等性,以及形式为\({\ mathbb {S}} ^ 1( \ sqrt {1-r ^ 2})\次{\ mathbb {S}} ^ {n-1}(r)\)。而且,在适当的总脐度张量约束下,我们可以将这个结果扩展到任何整数k,其中\(2 \ le k \ le n-1 \),等式表征了全部脐带超曲面和的某些族形式球的标准产品\({\ mathbb {S}} ^ k(\ sqrt {1-r ^ 2})\次{\ mathbb {S}} ^ {nk}(r)\)
更新日期:2020-02-22
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