Normalized null hypersurfaces in the Lorentz-Minkowski space satisfying $L_r x =U x +b$

IF 0.7 Q2 MATHEMATICS Tamkang Journal of Mathematics Pub Date : 2022-08-21 DOI:10.5556/j.tkjm.54.2023.4851
Hans FOTSING TETSING, C. Atindogbe, Ferdinand Nkageu
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引用次数: 1

Abstract

In the present paper, we classify all normalized null hypersurfaces $x: (M,g,N)\to\R^{n+2}_1$ endowed with UCC-normalization with vanishing $1-$form $\tau$, satisfying $L_r x =U x +b$ for some (field of) screen constant matrix $U\in \R^{(n+2)\times(n+2)}$ and vector$b\in\R^{n+2}_{1}$, where $L_r$ is the linearized operator of the$(r+1)th-$mean curvature of the normalized null hypersurface for$r=0,...,n$. For $r=0$, $L_0=\Delta^\eta$ is nothing but the (pseudo-)Laplacian operator on $(M, g, N)$. We prove that the lightcone $\Lambda_0^{n+1}$, lightcone cylinders $\Lambda_0^{m+1}\times\R^{n-m}$, $1\leq m\leq n-1$ and $(r+1)-$maximal Monge null hypersurfaces are the only UCC-normalized Monge null hypersurface with vanishing normalization $1-$form $\tau$ satisfying the above equation. In case $U$ is the (field of) scalar matrix $ \lambda I$, $\lambda\in\R$ and hence is constant on the whole $M$, we show that the only normalized Monge null hypersurfaces $x: (M,g,N)\to\R^{n+2}_1$ satisfying $\Delta^\eta x =\lambda x +b$, are open pieces of hyperplanes.
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Lorentz-Minkowski空间中满足L_r x =U x +b的归一化零超曲面
本文对具有ucc归一化的所有归一化空超曲面$x: (M,g,N)\to\R^{n+2}_1$进行了分类,其中$L_r$是$r=0,...,n$归一化空超曲面$(r+1)th-$的平均曲率的线性化算子,并将$1-$从$\tau$中消失,满足$L_r x =U x +b$的屏幕常数矩阵$U\in \R^{(n+2)\times(n+2)}$和向量$b\in\R^{n+2}_{1}$。对于$r=0$, $L_0=\Delta^\eta$只是$(M, g, N)$上的(伪)拉普拉斯运算符。我们证明了光锥$\Lambda_0^{n+1}$、光锥柱面$\Lambda_0^{m+1}\times\R^{n-m}$、$1\leq m\leq n-1$和$(r+1)-$是满足上述方程的唯一具有消失归一化形式$1-$$\tau$的ucc归一化的最大Monge零超曲面。如果$U$是标量矩阵$ \lambda I$, $\lambda\in\R$的(域),因此在整个$M$上是常数,我们证明了满足$\Delta^\eta x =\lambda x +b$的唯一归一化的Monge零超曲面$x: (M,g,N)\to\R^{n+2}_1$是超平面的开放块。
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来源期刊
CiteScore
1.50
自引率
0.00%
发文量
11
期刊介绍: To promote research interactions between local and overseas researchers, the Department has been publishing an international mathematics journal, the Tamkang Journal of Mathematics. The journal started as a biannual journal in 1970 and is devoted to high-quality original research papers in pure and applied mathematics. In 1985 it has become a quarterly journal. The four issues are out for distribution at the end of March, June, September and December. The articles published in Tamkang Journal of Mathematics cover diverse mathematical disciplines. Submission of papers comes from all over the world. All articles are subjected to peer review from an international pool of referees.
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