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 1 2 (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 1 2 (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.