双曲空间中悬链面的重整化面积
Renormalized Area of Catenoids in the Hyperbolic Space
AI总结：
研究双曲空间\(\mathbb{H}^{2n + 1}\)中悬链面，证明其生成曲线是特定\(p\)-弹性曲线，利用变分特征与公式给出重整化面积超椭圆积分表达式，分析得出该面积无下界且\(n\)为偶数时无符号。
AI中文摘要：
我们证明了双曲空间\(\mathbb{H}^{2n + 1}\)中非完全测地的球面旋转极小超曲面（简称为悬链面）的生成曲线是\(p=(2n - 1)/(2n)\)的\(p\)-弹性曲线。我们利用这种变分特征，结合局部共形平坦流形的陈 - 高斯 - 博内公式，给出了悬链面重整化面积的超椭圆积分显式表达式。进一步分析这些特殊积分，表明悬链面的重整化面积从负无穷连续变化到\(\mathbb{H}^{2n}\subset\mathbb{H}^{2n + 1}\)中完全测地超曲面重整化面积的两倍。因此，我们得出重整化面积无下界，且当\(n\)为偶数时，\(\mathbb{H}^{2n + 1}\)中极小超曲面的重整化面积无符号。
英文摘要：
We show that the generating curves of non-totally geodesic spherical rotational minimal hypersurfaces (catenoids, for simplicity) of the hyperbolic spaces $\mathbb{H}^{2n+1}$ are $p$-elastic curves for $p=(2n-1)/(2n)$. We employ this variational characterization, together with the Chern--Gauss--Bonnet formulas for locally conformally flat manifolds, to present an explicit expression for the renormalized area of catenoids in terms of hyperelliptic integrals. Further analyzing these special integrals, we show that the renormalized area of catenoids varies continuously from negative infinity to twice the renormalized area of the totally geodesic hypersurfaces $\mathbb{H}^{2n}\subset\mathbb{H}^{2n+1}$. Therefore, we conclude that the renormalized area is not bounded below and that, when $n$ is even, the renormalized area of minimal hypersurfaces in $\mathbb{H}^{2n+1}$ does not have a sign.
