- Open Access
© Ezhov and Schmalz; licensee Springer. 2015
- Received: 6 June 2014
- Accepted: 10 June 2014
- Published: 3 June 2015
- Spherical CR manifolds
- Rigid CR manifolds
- Normal form
- Zero curvature
The class of real hypersurfaces in complex space that are invariant with respect to an infinitesimal translation transversal to the complex tangent space is known as rigid hypersurfaces (see ). In this article, we consider rigid hypersurfaces in with coordinates z,w=u+iv. In this case, a rigid hypersurface can be locally described by an equation of the form v=h(z).
Stanton has raised the question whether this list was complete. Her method was based on the analysis of the holomorphic mappings that transform the infinitesimal translation ∂/∂ w into a vector field of a seven-parameter family of infinitesimal sphere automorphisms. She acknowledged that the listed examples correspond to a four-parameter subfamily.
where p is a homogeneous polynomial (see ). Stanton has solved this problem for polynomials of degree bigger than two. Our result completes the case of degree 2.
which for θ=0 coincides with (1) but was absent in Stanton’s list for θ≠0.
We show that Stanton’s mappings of a rigid sphere to the Heisenberg sphere can be modified by combining them with suitable sphere automorphisms so that all parameters can be covered.
where . Loboda has derived and studied PDE (4) in . In particular he found the solution (3). The PDE (4) appears in relation to the study of shear-free congruences of null geodesics in a Lorentzian geometry by Robinson and Wilson in . In particular, they establish a correspondence between the Taub-NUT congruence and a CR manifold that is equivalent to our CR manifold sinhv=e−v |z|2.
Condition (5) is imposed for convenience in computation and can be fixed at the next step.
with initial conditions p(0)=0, h(0)=0, h ′(0)>0. Using combinations with automorphisms of the Heisenberg sphere we may assume that α(0)=0, h ′(0)=1, p ′(0)=0, h ′′(0)=0. It follows that, for given c 22,c 23,c 33, system (6)-(8) has unique solutions α,p,h.
Stanton’s family realises all those c 22,c 23,c 33 for which the system of algebraic equations (12)-(14) has a solution , . A method to solve this system is to first express θ using Equation (12) then express r 2 through c 33 and |b|2 using Equation (14) and plug it into the absolute square of Equation (13). Solve the resulting cubic equation on |b|2 and finally find r,θ. However, the resulting values for |b|2 or r 2 could turn out to be negative.
- 2.For c 22=0 and c 23=2, only c 33≥−2 is feasible. Indeed, from Equation (12), we get θ=3|b|2. Then, (13) is equivalent to:
Notice that the initial conditions are p ′(0)=b and h ′′(0)=−2r.
On the other hand, the proposition below shows that the solutions of (9) with different vector fields X yield the missing rigid spheres.
and initial conditions Z(z,0)=z, W(z,0)=0.
For any choice of parameters c 22,c 23,c 33, we find the corresponding parameters θ,a,ρ and hence the infinitesimal sphere automorphism X. According to the Picard-Lindelöf theorem, the system (9) has a unique solution with initial conditions Z(z,0)=z, W(z,0)=0.
where the dots indicate terms of bidegree (2,4) and (4,2) and higher order.
It is an immediate consequence that the modified Chern-Moser normalisation of the Heisenberg sphere from Section 2 yields indeed rigid hypersurfaces.
where a=−b(r−i θ+2 i|b|2) and ϕ=|b|2.
for n≥1 and therefore is divisible by r 2+θ 2. Clearly, sinhr w is divisible by r. Therefore, P 1,P 2,Q are entire functions with respect to z,w,r,θ,a,ϕ. Moreover, they are even functions with respect to r.
We make formula (15) universal by allowing imaginary r and negative ϕ.
Notice that the cubic equation has real coefficients and therefore has at least one real solution. Let ϕ be any real solution.
Here, we take the principal branch of the cubic root on the right half-plane and the real cubic root on the real axis. Notice that ϕ is a continuous real-valued function, though not necessarily non-negative, and that ϕ=0 for a=0.
where P 1,P 2,Q are as in (15) and θ and r 2 are expressed as functions of τ,ρ,ϕ by (17).
Then, Z,W satisfy the system (9) on the real algebraic set given by (18).
factorise with a factor 4ϕ 3+4τ ϕ 2+(τ 2−ρ)ϕ−|a|2=4ϕ 3−4θ ϕ 2+(θ 2+r 2)ϕ−|a|2.
Therefore, Z,W satisfy the system (9) on the real algebraic set (18).
Now the rigid sphere formula can be obtained either from Stanton’s formula (2) by replacing r,θ,b by their expressions in τ,a,ρ,θ or by inserting Z,W into the standard Heisenberg sphere equation.
for n≥1 and therefore is divisible by r 2+θ 2. Clearly, sin2r v is divisible by r. It follows that the rigid sphere formula is an entire function with respect to all variables and parameters.
together with the condition that f does not contain antiholomorphic terms.
Though we did not succeed in solving the zero-curvature equation directly, we can demonstrate the special cases when M is circular, i.e. v=f(|z|2), or M is a tube, i.e. v=h(x).
which are affinely equivalent to the tubes (20).
The authors are grateful to Martin Kolář for numerous useful discussions from which some of the ideas used in this article arose. The research was supported by the Max-Planck-Institut für Mathematik Bonn and the ARC Discovery grant DP130103485.
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