Time-like Willmore surfaces in Lorentzian 3-space |
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Authors: | DENG Yanjuan WANG Changping |
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Institution: | LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China |
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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
with
. 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
(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
, then
is also a time-like Willmore surface in ℚ3 satisfying
; (ii) if
is a single point, then M is conformally equivalent to a minimal surface in ℝ1
3. |
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Keywords: | Lorentzian 3-space Willmore surfaces conformal Gauss map dual theorem |
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