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A short note on a class of Weingarten hypersurfaces in $$\mathbb {R}^{n + 1}$$ R n + 1
Geometriae Dedicata ( IF 0.5 ) Pub Date : 2020-10-30 , DOI: 10.1007/s10711-020-00580-0
Eudes L. de Lima

We provide sharp bounds for the squared norm of the second fundamental form of a wide class of Weingarten hypersurfaces in Euclidean space satisfying \(H_r = aH + b\), for constants \(a, b \in \mathbb {R}\), where \(H_r\) stands for the rth mean curvature and H the mean curvature of the hypersurface. Besides we are able to characterize those hypersurfaces for which these bounds are attained by showing that they must be a cylinder of the type \(\mathbb {R}\times \mathbb {S}^{n - 1}(\rho )\). Moreover, with the additional assumption that the Gauss–Kronnecker curvature of the hypersurface does not change sign, we prove that it is the cylinder \(\mathbb {R}\times \mathbb {S}^{n - 1}(\rho )\). In order to proof our results, we will use the method introduced in Alías and Garca-Martínez (Geom Dedic 156:31–47, 2012), Alías and Meléndez (Geom Dedic 182:117–131, 2016), which is based on the so called principal curvature theorem due to Smyth and Xavier (Invent Math 90:443–450, 1987).



中文翻译:

关于$$ \ mathbb {R} ^ {n + 1} $$ R n + 1中的一类Weingarten超曲面的简短说明

对于常数\ {a,b \ in \ mathbb {R} \}中的欧几里得空间中满足\(H_r = aH + b \)的宽泛类Weingarten超曲面的第二基本形式的平方范数,我们提供了清晰的界,其中\(H_r \)代表第r个平均曲率,H代表超曲面的平均曲率。除此之外,我们还可以通过显示出它们必须是\(\ mathbb {R} \ times \ mathbb {S} ^ {n-1}(\ rho)\ )。此外,还假设超曲面的高斯-克朗内克曲率不改变符号,我们证明它是圆柱体\(\ mathbb {R} \ times \ mathbb {S} ^ {n-1}(\ rho)\)。为了证明我们的结果,我们将基于在Alías和Garca-Martínez(Geom Dedic 156:31–47,2012),Alías和Meléndez(Geom Dedic 182:11​​7–131,2016)中引入的方法,该方法基于由于Smyth和Xavier(Invent Math 90:443-450,1987)而产生的所谓主曲率定理。

更新日期:2020-11-02
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