Making Surfaces Biharmonic by Conformal Reparametrization in Anti-de Sitter Three-Space
Harmonic immersions of surfaces are minimal, while biharmonic maps form a fourth-order extension of harmonic-map theory. Because every harmonic map is automatically biharmonic, the basic existence problem is to find \emph{proper} biharmonic maps, namely biharmonic maps that are not harmonic. This paper asks a more geometric question: when can a fixed nondegenerate surface in three-dimensional anti-de Sitter space be made proper biharmonic by changing only the conformal metric on its domain? Equivalently, how much of biharmonicity is determined by the immersed surface, and how much can be created by conformal reparametrization? Writing the induced metric as $g=\lambda^2\bar g$ and introducing the weighted mean curvature $u=\lambda^2H$, we first reduce the map equation to a normal scalar equation coupled to a tangential first-order constraint. The resulting system reveals a sharp rigidity--existence dichotomy. A nonminimal spacelike constant-mean-curvature solution must have constant dilation and is locally the totally umbilical hyperbolic plane of curvature $-2/L^2$. Once the constant-mean-curvature assumption is removed, however, there is an open set of local analytic solutions for which both $H$ and $\lambda$ vary. A moving-frame invariant then identifies the ambient one-parameter symmetry and separates elliptic, hyperbolic, and index-three parabolic orbit types. In the parabolic class the geometric system reduces to a scalar third-order analytic equation, from which we reconstruct explicit local spacelike and real-principal timelike families in null coordinates. The paper therefore locates the rigid branch, proves that the rigidity can be escaped, and gives an explicit mechanism for producing the resulting non-CMC surfaces.