Biharmonic Conformal Immersions into Anti-de Sitter Three-Space: Rigidity, Local Existence, and Parabolic Rotational Families

Abstract

We study biharmonic conformal immersions of nondegenerate surfaces into three-dimensional anti-de Sitter space. Using a sign convention adapted simultaneously to spacelike and timelike surfaces, we express the biharmonic equation in terms of the induced metric, shape operator, scalar mean curvature, and the weighted mean curvature $u=λ^2H$. For spacelike surfaces, we prove that a nonminimal constant-mean-curvature biharmonic conformal immersion has constant dilation and is locally totally umbilical, with intrinsic curvature $-2/L^2$. We then derive a cohomogeneity-one analytic system and prove local existence for an open set of initial data for which both the mean curvature and the dilation are nonconstant. An ambient moving-frame calculation produces a conserved orbit invariant and a constant generator in $\mathfrak{so}(2,2)$ whose minimal polynomial distinguishes elliptic, hyperbolic, and parabolic rotational types. On the generic spacelike parabolic branch, the equations reduce to a scalar third-order analytic ODE. We give an explicit null-coordinate reconstruction by quadratures and concrete initial data defining a local proper biharmonic conformal immersion with nonconstant dilation. The corresponding timelike parabolic reduction is also recorded.