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 $u=λ<sup>2H$. 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 $-2/L<sup>2$. 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.

• The paper establishes that proper biharmonic conformal immersions in AdS3 show rigidity in the CMC case with constant dilation.
• It develops a local analytic existence theory via a cohomogeneity-one reduction, yielding explicit ODE systems and reconstruction formulae.
• It classifies rotational families through a conserved invariant that distinguishes elliptic, hyperbolic, and parabolic types in AdS3.

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

Introduction and Context

This paper provides a comprehensive analytic investigation of biharmonic conformal immersions of nondegenerate surfaces into the three-dimensional anti-de Sitter space ($AdS_3$), systematically addressing the existence, rigidity, and classification of such immersions. The approach unites the spacelike and timelike surface cases under a unified pseudo-Riemannian formalism, offering new existence results beyond previous classifications, which predominantly focused on isometric or linearly modified bitension-field settings.

The key innovation is a detailed local existence theory for proper biharmonic conformal immersions into $AdS_3$ with nonconstant mean curvature and nonconstant conformal dilation, as well as a complete analytic reduction for the rotationally invariant (cohomogeneity-one) parabolic families in both causal types. The analysis is grounded in a rigorous ODE approach, with explicit reconstruction formulae in null coordinates.

Rigidity for Spacelike Constant Mean Curvature (CMC) Surfaces

The paper establishes a strong rigidity theorem for biharmonic conformal immersions of spacelike surfaces with nonzero CMC into $AdS_3$. Specifically, the only such immersions are those with constant conformal dilation, which are locally totally umbilical hyperbolic planes with intrinsic curvature $-2/L^2$ for radius $L$ of the ambient $AdS_3$.

This result is obtained by leveraging the structure of the biharmonic conformal surface equation (see Theorem 1):

• If $g = \phi^* h = \lambda^2 \bar g$, and $u = \lambda^2 H$, then the immersion is biharmonic if and only if

$$\Delta_g u + (-\varepsilon\, \mathrm{tr}_g A^2 + 2c)u = 0, \qquad A(\mathrm{grad}_g u) + \varepsilon u\, \mathrm{grad}_g H = 0$$

where $\varepsilon$ is the normal signature and $AdS_3$0 the ambient curvature.

Any nonminimal CMC conformal-biharmonic immersion forces the conformal factor $AdS_3$1 to be constant, so the only nontrivial examples are homothetic images of the standard proper biharmonic isometric branch. The structure is strictly local; no topological or completeness assumption is required.

This behavior contrasts sharply with the corresponding Riemannian cases, where totally umbilical CMC surfaces in nonpositive sectional curvature do not admit proper biharmonic conformal immersions, highlighting a genuinely Lorentzian phenomenon.

Local Existence Theory and Cohomogeneity-One Reductions

For the non-CMC case, the paper constructs a general local analytic existence theory for properly biharmonic conformal immersions via a cohomogeneity-one reduction. The approach is based on the warped-metric ansatz with totally geodesic orbits and principal coordinate frames, yielding a closed analytic ODE system on the profile curve. The equations encode metric, second fundamental form, and the weighted mean curvature function.

On an open set of initial data (where both $AdS_3$2 and $AdS_3$3 are nonconstant), this ODE system admits unique local analytic solutions. The dynamical system’s profile is governed by four parameters modulo orbit rescaling, source metric homothety, and ambient isometries.

Orbit Classification and the Ambient Invariant

A significant result is the identification of an orbit-type invariant $AdS_3$4 defined by

$AdS_3$5

which is conserved along the immersion and distinguishes elliptic, hyperbolic, and parabolic rotational types via the sign of $AdS_3$6. The ambient generator $AdS_3$7 defined by this invariant satisfies the minimal polynomial equation $AdS_3$8.

Figure 1: Meridian generating curves $AdS_3$9 in the $AdS_3$0 null-coordinate section, showing both a non-CMC parabolic branch and the totally umbilical constant-dilation branch.

The sign of $AdS_3$1 directly determines the type of orbit in the $AdS_3$2 isometry group:

• $AdS_3$3: elliptic (rotation on definite two-planes)
• $AdS_3$4: hyperbolic (Lorentzian boosts)
• $AdS_3$5: parabolic (null rotations)

Analytic Reduction for Parabolic Rotational Families

The most explicit analysis occurs in the parabolic (degenerate) rotational class, for both spacelike and timelike cases. The paper provides a scalar third-order analytic ODE for the profile parameter $AdS_3$6, which encodes all the geometric quantities via elementary quadrature once solved. For spacelike parabolic immersions, the relevant equations for mean curvature, conformal dilation, metric, and principal curvatures are all derived directly from $AdS_3$7, with an explicit ambient reconstruction in null coordinates.

The construction yields open local families of biharmonic conformal immersions, parametrized by three essential profile parameters (after factoring out geometric symmetries and gauge redundancies), in both spacelike and timelike parabolic sectors.

The explicit parametric realization in null coordinates provides all geometric quantities, notably giving an explicit one-parameter subgroup action of the isometry group, whose generator connects back to the central invariant $AdS_3$8.

Implications and Future Directions

This work demonstrates analytically that non-CMC proper biharmonic conformal immersions into $AdS_3$9 with nonconstant mean curvature and nonconstant dilation do exist locally on an open set of initial data, in contrast to previous results for isometric or $-2/L^2$0-biharmonic immersions. The local theory is complete, with explicit formulas for rotationally symmetric families and clear classification of the possible orbit types.

The numerical audit, accompanying source code, and perturbation tests reported in the appendix rigorously support the analytic existence and correctness of the constructed examples.

The primary open questions are global in nature: whether such local nonconstant-dilation solutions can be extended globally, whether periodic or complete profiles exist in the elliptic or hyperbolic sectors, and under which conditions complete immersions can be achieved. Such global results require delicate control of the third-order analytic ODEs defining the profiles, and understanding global blow-up or continuation criteria remains an important direction.

Conclusion

The paper systematically develops the local analytic existence, rigidity, and classification theory for biharmonic conformal immersions into anti-de Sitter three-space. The results reveal both the rigidity in the CMC case—a unique local form up to scaling and isometry—and the flexibility of the generic non-CMC branches, with explicit local parabolic profiles in both spacelike and timelike causal classes. The central role of the orbit invariant integrates the local and extrinsic differential geometry, enabling a full analytic characterization of rotationally symmetric biharmonic conformal immersions. The work leaves important and compelling questions for global theory and further classification.