Time-like Willmore surfaces in Lorentzian 3-space

Let R13 be the Lorentzian 3-space with inner product (,). Let Q3 be the conformal compactification of R13, obtained by attaching a light-cone C∞ to R13 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 ( ) R13 be a time-like surface.Let n be the unit normal andHthe mean curvature of the surface M. For any p ∈ M we define S12(p) = {X ∈ R13 | (X - c(p),X - c(p)) = 1/H(p)2} with c(p) = p + 1/H(p)n(p) ∈ R13.Then S12(p) is a one-sheet-hyperboloid in R13, which has the same tangent plane and mean curvature as M at the point p. We show that the family {S12(p),p ∈ M} of hyperboloid in R13defines in general two different enveloping surfaces, one is M itself, another is denoted by (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 (M), then (M) is also a time-like Willmore surface in Q3 satisfying (M)= M; (ii) if M is a single point, then M is conformally equivalent to a minimal surface in R13.