- Vladimir Ezhov
^{2}Email author and - Gerd Schmalz
^{3}Email author

**1**:2

https://doi.org/10.1186/2197-120X-1-2

© Ezhov and Schmalz; licensee Springer. 2015

**Received: **6 June 2014

**Accepted: **10 June 2014

**Published: **3 June 2015

### Keywords

Spherical CR manifolds Rigid CR manifolds Normal form Zero curvature## 1 Introduction

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 [1]). In this article, we consider rigid hypersurfaces in
with coordinates *z*,*w*=*u*+i*v*. In this case, a rigid hypersurface can be locally described by an equation of the form *v*=*h*(*z*).

In 1991, Stanton [8] developed a normal form for rigid hypersurfaces. Other normal forms that reflect the presence of symmetries have been constructed by Kolář [5] and Ezhov et al. [2] more recently.

*v*=|

*z*|

^{2}, via its normal form. As an application of the normal form, Stanton derived a list of examples of normal forms equivalent to the sphere:

*b*and

*c*=

*r*+i

*θ*are complex parameters. The remaining examples from Stanton’s list can be obtained by letting

*r*and

*θ*converge to 0. In particular, for

*b*=0 and

*θ*=0, one finds the pendant to (1)

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 [8]). Stanton has solved this problem for polynomials of degree bigger than two. Our result completes the case of degree 2.

*ϕ*of bidegrees (2,2), (2,3), (3,2) and (3,3) with respect to being allowed to be any constants. These four real constants completely control the rigid hypersurface. Thus, the normal form equation becomes:

*c*

_{22}, and

*c*

_{33}depend in an algebraic way on Stanton’s parameters

*b*and

*c*. It can be shown that not all coefficients can be attained in this way, which indicates that Stanton’s list is incomplete. In the case when

*c*

_{23}=0, Stanton’s example reduces to:

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.

*M*in normal form:

where
. Loboda has derived and studied PDE (4) in [6]. 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 [7]. In particular, they establish a correspondence between the Taub-NUT congruence and a CR manifold that is equivalent to our CR manifold sinh*v*=e^{−v
}|*z*|^{2}.

## 2 Modified normalisation

*w*

_{2}=|

*z*

_{2}|

^{2}to obtain a hypersurface of the form:

*c*

_{22},

*c*

_{23},

*c*

_{33}, such normalisation mapping is uniquely determined up to automorphisms of the Heisenberg sphere. All rigid spheres can be found by this procedure, though the resulting hypersurface does not have to be rigid in the higher-order terms

*a priori*. We construct the inverse mapping as a composition of two:

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*.

## 3 Stanton’s mapping

*X*of the Heisenberg sphere back to . The relevant infinitesimal automorphisms of the Heisenberg sphere are well-known and form a seven-parametric family consisting of:

*W*(

*z*,0)≡0. Stanton solved this system in the particular case

*a*=

*ρ*=0, that is when (9) is linear. The solutions are:

*v*=|

*z*|

^{2}yields (2). We show that not all values of the parameters

*c*

_{22},

*c*

_{23},

*c*

_{33}can be realised by these mappings. Direct computation shows that the explicit sixth-order expansion of (2) is:

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)=−2*r*.

On the other hand, the proposition below shows that the solutions of (9) with different vector fields *X* yield the missing rigid spheres.

###
**Proposition**
**1**

*c*

_{22},

*c*

_{23},

*c*

_{33}, there exists a rigid sphere. It can be realised by the solutions

*Z*(

*z*,

*w*),

*W*(

*z*,

*w*) of (9) with parameters

*c*=i

*θ*,

*a*,

*ρ*and Re

*c*=

*b*=0, where:

and initial conditions *Z*(*z*,0)=*z*, *W*(*z*,0)=0.

### Proof

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.

## 4 Twisting Stanton’s mapping

*a*,

*ρ*is to apply the adjoint action of SU(2,1) on the coefficient matrix of the linear system in a suitable way. Geometrically, this amounts to compose Stanton’s mapping with a sphere automorphism. The composition of Stanton’s mapping

*z*

_{1}(

*z*,

*w*),

*w*

_{1}(

*z*,

*w*) with parameters

*c*=

*r*+i

*θ*,

*b*and the sphere automorphism:

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, sinh*r*
*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.

### Theorem 1

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).

###
*Proof*.

factorise with a factor 4*ϕ*
^{3}+4*τ*
*ϕ*
^{2}+(*τ*
^{2}−*ρ*)*ϕ*−|*a*|^{2}=4*ϕ*
^{3}−4*θ*
*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, sin2*r*
*v* is divisible by *r*. It follows that the rigid sphere formula is an entire function with respect to all variables and parameters.

## 5 The zero-curvature equation

*M*in to a sphere can be characterised by vanishing of its Cartan curvature. In [3], an explicit expression of the Cartan curvature has been computed. In the case of rigid hypersurfaces, this expression considerably simplifies. If

*M*is given by the equation:

with
. The condition that *h* is a real function translates into
.

*h*are encoded in the choice of the solution of the auxiliary equation:

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).

## Declarations

### Acknowledgements

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.

## Authors’ Affiliations

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