L1-2-type surfaces in 3-dimensional De Sitter and anti De Sitter spaces

TL;DR

This work studies orientable surfaces in $\mathbb{S}^3_1$ or $\mathbb{H}^3_1$ that are of $L_1$-2-type, i.e., admit a spectral decomposition under the operator $L_1$ with two distinct eigenvalues. It develops the Lorentzian $L_1$-spectral theory, derives key identities for $L_1$ acting on the immersion and normal, and proves that for such surfaces the conditions of constant principal curvature, constant mean curvature $H$, and constant second mean curvature $H_2$ are equivalent. Consequently, $L_1$-2-type surfaces are classified as open portions of standard pseudo-Riemannian products or $B$-scrolls over null curves, or as surfaces with all curvatures non-constant, depending on the ambient space. The results extend finite-type spectral geometry to Lorentzian space forms and provide a Lorentzian analogue of the isoparametric classification in $\mathbb{S}^3_1$ and $\mathbb{H}^3_1$.

Abstract

Let $M$ be an orientable surface immersed in the De Sitter space $S_1^3$ in $R^4_1$ or anti de Sitter space $H_1^3$ in $R^4_2$. In the case that $M$ is of $L_1$-2-type we prove that the following conditions are equivalent to each other: $M$ has a constant principal curvature; $M$ has constant mean curvature; $M$ has constant second mean curvature. As a consequence, we also show that an $L_1$-2-type surface is either an open portion of a standard pseudo-Riemannian product, or a $B$-scroll over a null curve, or else its mean curvature, its Gaussian curvature and its principal curvatures are all non-constant.

Theorems & Definitions (11)

• Lemma 1
• Example 1
• Example 2
• Example 3
• Example 4
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4