2026/07/07 by Dipesh Bhandari
#math.DG
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=λ2H. 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/L2. 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 rank-two generator in \mathfrakso(2,2). For the generators arising from the reduction, the cubic identity distinguishes elliptic, hyperbolic, and index-three 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. For the real-principal timelike parabolic branch with spacelike profile and timelike orbit, we prove an analogous local analytic existence theorem and give a second explicit null-coordinate reconstruction.