A characterization of homogeneous three-dimensional CR manifolds

We characterize homogeneous three-dimensional CR manifolds, in particular Rossi spheres, as critical points of a certain energy functional that depends on the Webster curvature and torsion of the pseudohermitian structure.

We recall that Rossi spheres (s = 0) are the simplest examples of nonembeddable CR 3-manifolds which (two to one) cover embeddable ones in C 3 [5, pp. 324-325].Apart from the non-embeddability property, Rossi spheres provide counterexamples in conformal pseudohermitian geometry.In relation to the problem of existence of minimizers for the CR Yamabe problem, each Rossi sphere (J (s) , θ), s = 0, has negative pseudohermitian mass, as defined in [7], for s close to 0 while the infimum of the CR Sobolev quotient coincides with the one for the standard 3-sphere (s = 0), but is not attained [7].The notion of pseudo-Einstein contact form plays an important role in CR geometry.Geometrically it is characterized by a volume-normalization condition while analytically in dimension 3 it relates R to A 11 in their first covariant derivatives as follows: (1) R ,1 = iA 11, (cf.( 27)).Equation ( 1) is useful in simplifying the expressions involving R and A 11 .Among others, one can equate the Burns-Epstein invariant to the total Q ′ -curvature (up to a negative constant) [4, Theorem 1.2 on pp.290-291].
In this paper we exhibit a functional whose critical points (in both J and θ) characterize homogeneous pseudohermitian manifolds among pseudo-Einstein ones.Precisely define the following energy functional Our main result is as follows.
Theorem 1.On a closed (i.e.compact with no boundary) contact 3manifold (M, ξ), suppose that (J, θ) is a pseudo-Einstein critical point of (2).Then the universal cover M of M with the structure naturally inherited from (ξ, J, θ), still denoted by the same notation, is homogeneous as a pseudohermitian 3-manifold.
In the Riemannian case, a variational characterization of space forms was given in [10].It is well known that homogeneous CR 3-manifolds have been classified by Cartan ([3]; see also [2]).In particular, when (M, ξ) = (S 3 , ξ), we have the following characterization for Rossi spheres.
Corollary 2. On the standard contact 3-sphere (S 3 , ξ), it holds that (J, θ) is a pseudo-Einstein critical point of (2) if and only if J is isomorphic to a Rossi sphere J (s) for some unique s ≤ 0 and θ is a constant multiple of θ.

Theorem 3. With the notation above, we obtain the second variation formulas
We remark that there is no characterization in general on sign of the second variation.In fact, using for examples the Fourier decompositions for the variations of J and θ as in [1], it is possible to find deformations along which the second differential is either positive or Science and Technology of Taiwan for the support: grant no.MOST 110-2115-M-001-015 and the National Center for Theoretical Sciences for the constant support.A.M. is supported by the project Geometric Problems with Loss of Compactness from Scuola Normale Superiore.He is also a member of GNAMPA as part of INdAM.P. Y. acknowledges support from the NSF for the grant DMS 1509505.
2. Rossi spheres, variations and energy functional 2.1.Rossi spheres.In this subsection we give a short introduction to Rossi spheres.See [8, Section 2.2] for more details.The standard contact form θ on Note that θ is SU(2)-invariant, where SU(2) acts on C 2 in the canonical way.Dual to θ 1 = z 2 dz 1 − z 1 dz 2 , we have Consider the deformation of CR structures described by giving type (0, 1) vectors as follows: where E11 is a deformation tensor associated to the CR structure J (cf.Subsection 2.2).Note that we use the notation E11 instead of E1 1 for convenience/simplicity; for a unitary frame/coframe they are equal.The derivative of Z1 (s) in s reads as We express Z 1 and Z1 in terms of Z 1(s) and Z1 (s) as follows: Substituting the second equality of (5) into (4) gives Substituting ( 6) into ( 7) and writing J(s) = 2E (s) Therefore we have Ė(s) For Rossi spheres, we take where dθ = iθ 1 ∧ θ 1, i.e., h 1 1 = 1.So from (10) it follows that (11) Suppose that the Webster curvature R of (J, θ) is a positive constant R 0 (see [13] and [14] for basic pseudohermitian geometry).Then we should take ω 1 1 = −iR 0 θ in the structure equation, so that dω ), where we have used that z 1dz 1 + z 2dz 2 + conjugate = 0 on S 3 .We can then determine, from the structure equation for (J (s) , θ), that It follows that (J (s) , θ) is pseudo-Einstein (see ( 27)) since R (s),1(s) = 0 = A 11(s), 1(s) .For a pseudohermitian structure (J, θ) we recall that the sublaplacian ∆ b acting on a function u reads as for a unitary frame (so h 1 1 = 1).
A direct computation shows that each Rossi sphere (S 3 , J (s) , θ) is a solution to (17) by (13) and noting that A 11,0 = T A 11 − 2ω 1 1 (T )A 11 .We notice that the coefficient of |A| 2 J,θ in the integrand of E(J, θ) is different from that in the integrand of the following energy functional which is known to be the total Q ′ -curvature (for pseudo-Einstein (J, θ)), whose critical points are spherical, see [4].

Substituting (27) into the second equation of (17) gives 3∆
. It follows that R = const.since M is closed.We now multiply the second equation of (17) by R and integrate over M with respect to the volume form θ ∧ dθ.
After integrating by parts we obtain (28) where From the first equation of ( 17) and the commutation relation We have completed the proof.
Let τ denote the torsion tensor of the pseudohermitian connection for any tangent vector fields U, V .Recall that the Reeb vector field T is the unique vector field such that θ(T ) = 1 and dθ(T, •) = 0.It is not hard to see from the formulas for Lie brackets in [13, page 418] that for Y, W ∈ ξ where g J,θ is the Levi metric defined by g J,θ (Y, W ) := dθ(Y, JW ) and U is any tangent vector.

Define the tensor A
Observe that (extending the defining domain of τ to complex tangent vectors by complex linearity) by [13, page 418].In fact we also have τ (f Z 1 , T ) = −A(f Z 1 ) for any complex function f.So it holds that It follows from ( 35) and (36) that by the condition A 11,1 = 0, A 11, 1 = 0 due to Proposition 5.By taking complex conjugation, (37) also holds for Z1 replacing Z 1 .So the RHS of (34) vanishes.We have shown ( We have completed the proof. We call a pseudohermitian automorphism φ of (M, ξ, J, θ) a subsymmetry at a point x if φ(x) = x and φ * | ξ x = −1 (φ * T x = T x hence the Reeb orbit through x is fixed by φ).A local sub-symmetry at a point x means that φ is only defined in a neighborhood of x.
By Lemma 6 we obtain that the curvature R and the torsion (tensor) τ are parallel along the (horizontal) direction of any contact vector.From the proof of [9, Theorem 2.1] (noting that the Levi metric g J,θ plays the role of the metric in the setting of [9]), for each point x we can find a local sub-symmetry φ x such that φ x (x) = x, φ 2 x = Id.Lift φ x to a local sub-symmetry φx on M , the universal cover of M, where x ∈ M is a lift of x, .i.e.π(x) = x, π : M → M is the natural projection.Since M is simply connected, we can extend φ x uniquely to a global pseudohermitian automorphism using the parabolic exponential map in [11, page 309] by a similar argument in [12, pp. 252-255] for extending an affine map.Observe that the fixed point set of φx is a Reeb orbit F x in M , {F x} x∈ M foliate M and { φx } x∈ M permutes the Reeb orbits {F x} x∈ M , since all φx 's are pseudohermitian automorphisms.The subsymmetry φx has the following properties: x = Id, so φ−1 x = φx .Let Aut ψ.h. ( M, ξ, J, θ) denote the group of all pseudohermitian automorphisms.We claim that (38) Aut ψ.h. ( M, ξ, J, θ) acts on ( M , ξ, J, θ) transitively.
Observe that M is complete (meaning that it is complete as a metric space) by, for instance, [15, Theorem 7.1 (b)].Next, given p, q ∈ M , we can find a Legendrian (horizontal) geodesic (with respect to the Levi-metric g J,θ ) γ connecting p and q, parametrized by the arc length of g J,θ by [15, Theorem 7.1 (a)].Let m ∈ γ be the middle point of the curve γ.It follows that φm maps p (resp.q) to q (resp.p).We have shown (38).That is, ( M, ξ, J, θ) is homogeneous as a pseudohermitian manifold.
Remark 7. We notice that in [9] the authors make the assumption on homogeneity to classify all possible sub-symmetric spaces through a Lie-theoretic argument.
Proof of Corollary 2. By (13) (together with (11)) we verify that a Rossi sphere (J (s) , θ) is pseudo-Einstein and satisfies (17) by noting that (θ = θ) So, (J (s) , θ) is a pseudo-Einstein critical point of (2) in view of Proposition 4. Conversely, by Theorem 1 (S 3 , ξ, J, θ) is homogeneous.In particular, it is a homogeneous CR 3-manifold.According to Cartan [3, page 69], the CR structure J must be left-invariant on SU(2) (=S 3 ).By [2, Proposition 5.1 (c)] we conclude that J is isomorphic to a Rossi sphere J (s) for a unique s ≤ 0 (by comparing (9) with the coframe taken in the proof of [2, Proposition 5.1], we get the parameter relation: so t ≥ 1 corresponds to s ≤ 0, where t is strictly decreasing as a function of s).Moreover, that θ is SU(2)-invariant implies that θ is a constant multiple of θ.We have thus completed the proof.

Appendix
We give some examples for second variations in θ and J of E at the standard pseudohermitian 3-sphere (S 3 , J (0) , θ).Substituting R = 1 and Â11 = 0 into (46) gives Let H p,q,1 denote the restriction to S 3 of the space of the homogeneous complex harmonic polynomials of degree p + q, where p is the holomorphic homogeneity and q the antiholomorphic one.Then for f ∈ H p,q,1 one has (see [1, Proposition 2.2 on p.10]; note that θ is twice the contact form θ 0 in [1, p.8], so the sublaplacian there is twice the sublaplacian here while T there is exactly the Reeb vector field here, i.e.T = T ).By (50) one reduces (47) to (52) Taking h = f + f in (51) and writing λ = 1 2 (pq + 1 2 (p + q)), µ = p−q 2 , we have -△ b h = λ(f + f ) and h ,0 = T h = iµ(f − f ).Substituting