Time-like Willmore surfaces in Lorentzian 3-space

Abstract

Let ℝ₁ ³ be the Lorentzian 3-space with inner product (,). Let ℚ³ be the conformal compactification of ℝ₁ ³, obtained by attaching a light-cone C _∞ to ℝ₁ ³ in infinity. Then ℚ³ has a standard conformal Lorentzian structure with the conformal transformation group O(3,2)/{±1}. In this paper, we study local conformal invariants of time-like surfaces in ℚ³ and dual theorem for Willmore surfaces in ℚ³. Let M ⊂ ℝ₁ ³ be a time-like surface. Let n be the unit normal and H the mean curvature of the surface M. For any p ∈ M we define $S_1^2 (p) = \{ X \in \mathbb{R}_1^3 |(X - c(p),X - c(p)) = \tfrac{1}{{H(p)^2 }}\}$ with $c(p) = p + \tfrac{1}{{H(p)^2 }}n(p) \in \mathbb{R}_1^3$. Then S ²₁ (p) is a one-sheet-hyperboloid in ℝ₁ ³, which has the same tangent plane and mean curvature as M at the point p. We show that the family {S ²₁ (p),p∈M} of hyperboloid in ℝ₁ ³ defines in general two different enveloping surfaces, one is M itself, another is denoted by $\hat M$ (may be degenerate), and called the associated surface of M. We show that (i) if M is a time-like Willmore surface in ℚ³ with non-degenerate associated surface $\hat M$, then $\hat M$ is also a time-like Willmore surface in ℚ³ satisfying $\hat \hat M$; (ii) if $\hat M$ is a single point, then M is conformally equivalent to a minimal surface in ℝ₁ ³.