Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms

Abstract

Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in $S^4$, are studied in this paper. We define two kinds of transforms for such a surface, which produce the so-called left/right polar surfaces and the adjoint surfaces. These new surfaces are again conformal Willmore surfaces. For them holds interesting duality theorem. As an application spacelike Willmore 2-spheres are classified. Finally we construct a family of homogeneous spacelike Willmore tori.Comment: 19 page

Spacelike S-Willmore spheres in Lorentzian space forms

“…Later this study was extended to Willmore 2-spheres in Möbius geometry [Bryant 1984;Ejiri 1988b;Musso 1990;Montiel 2000] and Lorentzian conformal geometry [Alías and Palmer 1996;Ma and Wang 2008]. In this paper we generalize some of these results for spacelike S-Willmore spheres in Lorentzian space forms.…”

“…For further details, see [Alías and Palmer 1996;Ma and Wang 2008]. Our treatment here follows the surface theory in [Burstall et al 2002].…”

“…The three n-dimensional Lorentzian space forms with constant sectional curvature c = 0, +1, −1 can be conformally embedded as a proper subset of Q Definition 2.1 [Bryant 1984;Ejiri 1988b;Alías and Palmer 1996;Ma and Wang 2008]. For a conformally immersed surface y : M → Q n 1 with canonical lift Y (with respect to a local coordinate z), we define…”

“…In higher-dimensional Lorentzian space forms, the duality theorem does not hold in general either [Ma and Wang 2008]. While we can define the so-called polar transforms for surfaces in 4-dimensional Lorentzian space forms, such transforms keep the Willmore property.…”

“…Theorem A [Ma and Wang 2008]. Any spacelike Willmore 2-sphere in Q 4 1 is either congruent to a complete spacelike stationary surface (that is, H = 0) in R 4 1 , or a polar surface of such a surface.…”