Time-like willmore surfaces in Lorentzian 3-space

Let R-1(3) be the Lorentzian 3-space with inner product (,). Let Q(3) be the conformal compactification of R-1(3), obtained by attaching a light-cone C-infinity to R-1(3) in infinity. Then Q(3) 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 Q(3) and dual theorem for Willmore surfaces in Q(3). Let M subset of R-1(3) be a time-like surface. Let n be the unit normal and H the mean curvature of the surface M. For any p is an element of M we define S-1(2)(p) = {X is an element of R(1)(3)vertical bar (X-c(p), X-c(p)) = (1)/(H(p)2)} with c(p) = p + (1)/(H(p))n(p) is an element of R-1(3). Then S-1(2)(p) is a one-sheet-hyperboloid in R-1(3), which has the same tangent plane and mean S2 (p), P 3 curvature as M at the point p. We show that the family {S-1(2)(p), p is an element of M} of hyperboloid in R-1(3) defines in general two different enveloping surfaces, one is M itself, another is denoted by (M) over cap (may be degenerate), and called the associated surface of M. We show that (i) if M is a time-like Willmore surface in Q(3) with non-degenerate associated surface (M) over cap, then A is also a time-like Willmore surface in Q(3) satisfying (M) over cap = M; (ii) if (M) over cap is a single point, then M is conformally equivalent to a minimal surface in R-1(3).