On the role of embeddability in conformal pseudo-hermitian geometry

1 Introduction

The Yamabe problem consists in finding on a Riemannian manifold a metric conformal to a background one such that its scalar curvature is constant. This represents a higher-dimensional counterpart of the classical uniformization problem in two dimensions, and it has been introduced in [45].

Let ( M , g ) be a Riemannian manifold of dimension n ≥ 3 , and that for simplicity, we assume closed, i.e. compact and without boundary. If one denotes the scalar curvature by R g and performs the conformal change of metric g ˜ = w 4 n − 2 g with w positive on M , then the scalar curvature R g ˜ of g ˜ is given by

where L g stands for the conformal Laplacian: L g w = − 4 ( n − 1 ) n − 2 Δ g w + R g w . This operator enjoys the covariance law

By this formula, the Yamabe problem amounts to finding a positive solution of

for some R ¯ ∈ R , which can be interpreted as a Lagrange multiplier: noting that for u smooth one has

one can search for critical points of the Yamabe quotient on W 1 , 2 ( M ) ≔ { u : M → R ∣ u , ∇ u ∈ L 2 ( M ) } .

where 2 * = 2 n n − 2 = n + 2 n − 2 + 1 . Such critical points, that one could also look for constrained to the unit sphere of L 2 * ( M ) and taken non-negative, are solutions of (Y). Given the conformal covariance of L g , Q g satisfies the invariance property Q g ˜ ( u ) = Q g ( w u ) if g ˜ = w 4 n − 2 g .

The quantity Q g has a clear geometric interpretation: define the normalized Einstein-Hilbert action as the Riemannian functional given by

The factor Vol g ( M ) − n − 2 n makes ℛ ( g ) scaling invariant. Its critical points are Einstein metrics, but its second variation at those is strongly indefinite, which makes its variational analysis difficult in general. On the round sphere ( S n , g S n ) , in particular, it is known that conformal deformations of the metric give rise to non-negative second variations, while tt-variations of the metric, for which one takes out the conformal part and the invariance by diffeomorphisms, have negative second variation [42]. The study by Koiso [28] is one of the first in which a more systematic study of the second variation properties of ℛ ( g ) at Einstein metrics was initiated.

The relation of Q g to ℛ ( g ) is that if g ˜ = u 4 n − 2 g , then Q g ( u ) = ℛ ( g ˜ ) , so while Einstein metrics are free critical points of ℛ ( g ) , those constrained to a fixed conformal class are Yamabe metrics. Even extremizing within a given conformal class is not easy, since the lack of compactness of the embedding W 1 , 2 ( M ) ↪ L 2 * ( M ) makes the extremization of Q g challenging, because minimizing sequences for Q g might a priori not converge. This is due to the presence of bubbling, which means concentration of the H 1 - or of the L 2 * -norms at arbitrarily small scales.

Problem (Y) was solved by considering the Yamabe quotient, which is defined as follows:

The aforementioned covariance of Q g implies that indeed Y ( M , g ) depends only on the conformal class [ g ] of g , and will be therefore denoted by Y ( M , [ g ] ) . Conformal classes [ g ] for which Y ( M , [ g ] ) > 0 (respectively, = 0 or < 0 ) are said to be of positive Yamabe class (respectively, of null or negative class). In the study by Trudinger [41], it was proven that for every n ≥ 3 , there exists ε n > 0 such that Y ( M , [ g ] ) is attained when Y ( M , [ g ] ) ≤ ε n , which holds in particular when the Yamabe class is negative or null.

This result was sharpened by Aubin [5], where it was proven that the infimum of Q g is attained whenever

where g S n denotes the round metric of the sphere. As ( S n , g S n ) is conformally equivalent to R n via stereographic projection, Y ( S n , [ g S n ] ) coincides up to a dimensional constant with the Sobolev constant of R n , whose extremals (classified also in [39]) decay at infinity like the fundamental solution of the Laplacian. Such profiles, after a proper dilation, can be glued using normal coordinates to any point p of M : the geometry of ( M , g ) affects the expansion of the Yamabe quotient depending on the scaling parameter. Being the decay of the Green’s function of R n faster in higher dimensions, the role of the geometry is more localized, and it is ruled by the Weyl tensor W g at p when n ≥ 6 . Instead, for n ≤ 5 (or when g is locally conformally flat), it is determined by the mass of ( M , g ) once it is conformally deformed by the Green’s function of L g with pole at p . Using the positive mass theorem from [36,37] and [34], the aforementioned inequality (1.5) was proven also for the latter cases in the study by Schoen [35].

The purpose of this article is to explore the counterparts of the aforementioned facts in Cauchy-Riemann (CR) and pseudohermitian geometry, which sometimes are surprisingly in sharp contrast to the Riemannian case. All the basic definitions will be recalled in Section 2, while here we will only list some main concepts. Pseudohermitian manifolds are ( 2 n + 1 ) -dimensional manifolds with an n -dimensional complex sub-bundle H of the complexified tangent bundle T C M of M , verifying H ∩ H ¯ = { 0 } , [ H , H ] ⊆ H , and on which a complex rotation J acts. Letting H ( M ) = Re ( H ⊕ H ¯ ) , there exists a one-form θ , the contact form, such that H ( M ) = ker θ . Classical examples are hypersurfaces of C n or the Heisenberg group H n . If the Levi form L θ ( W , Z ¯ ) ≔ − i d θ ( W , Z ¯ ) is non-degenerate, pseudohermitian manifolds carry a natural connection along H ( M ) , the Tanaka-Webster connection [18,40,43]. The trace W J , θ of the corresponding curvature tensor is known as the Webster curvature, and represents a pseudohermitian counterpart of the scalar curvature. Furthermore, θ ∧ ( d θ ) n acts naturally as a volume form.

In the study of properties of CR geometry, we will stress a crucial difference between the cases n = 1 and higher due to the embeddable or non-embeddable character of the structures. By a well-known theorem by Boutet de Monvel [9], if a closed CR manifold M with n ≥ 2 is strictly pseudoconvex, i.e., when the Levi form is positive-definite), then it can be CR embedded in C N for some natural integer N . This fact is not always true when n = 1 [13], and in fact generically false for perturbations of the standard S 3 [10], causing differences in all counterparts of the above-mentioned results concerning the Riemannian case.

We begin considering the second variation of the counterpart of the normalized pseudohermitian Einstein-Hilbert action, which we define by

Here, Q = 2 n + 2 stands for the homogeneous dimension of the manifold M , which is equal to the topological dimension plus 1. We also note that since a local deformation of the contact structure can be achieved via contactomorphisms to the original one by a result in [21], it is sufficient for us to vary only J and θ .

Embedding S 2 n + 1 into C n , the sphere naturally inherits a pseudohermitian structure ( J 0 , θ 0 ) from the ambient, see Section 2 for the explicit definitions. It can be shown that first variations of W ˜ vanishes at ( S 2 n + 1 , J 0 , θ 0 ) , while for second-order ones (see Section 2 for explicit formulas), we have then the following result.

For reasons of brevity, in the aforementioned statement, we did not characterize the deformations that infinitesimally preserve embeddability. These were described first in [8] and depend on a Fourier expansion of a suitable complex-valued function with respect to the standard coordinates ( z 1 , z 2 ) of C 2 , see Section 2 for more details.

We consider next the Yamabe problem on closed CR manifolds, whose study was initiated in previous studies [25,27]. Similarly as mentioned earlier, fixing the CR structure J , one wants to find a conformal change of contact form so that the Webster curvature becomes constant. As we will see, also in this case, a Yamabe-type quotient is fundamental to attack this problem.

Referring to (2.4) for the notation of the sub-Laplacian Δ b , it is well-known that the Webster curvature trasforms under a change of contact form: by the law

in complete analogy to (1.1). The operator on the left-hand side is the CR sublaplacian and and has the following covariance property under a conformal change of contact form

see (1.2), where L ˜ b stands for the conformal sublaplacian relative to ( J , θ ˜ ) . The CR Yamabe equation then becomes

Similarly to the Riemannian case, solutions can be found by considering the CR Yamabe quotient, or Webster quotient, defined by

where θ ˜ is any contact form which annihilates ξ = H ( M ) (the contact bundle) and where ∣ ∇ b u ∣ 2 stands for the squared sub-gradient of u , see Section 2.

Jerison and Lee [25] proved that Y ( M , J ) ≤ Y ( S 2 n + 1 , J 0 ) , and that if strict inequality holds, then the infimum is attained. They also showed that if n ≥ 2 and ( M , J ) is not locally spherical, which means that the Chern tensor does not vanish identically, then indeed one has strict inequality. This was done using an argument in the spirit of [5] and a classification result from [26] about extremal functions for the Sobolev quotient in the Heisenberg group, the flat model of CR manifold.

In lower dimension, it is again expected that a positive mass theorem might play a role since the extremal functions still behave like the fundamental solution of Δ b , and have a slower decay when n = 1 . A concept of pseudohermitian mass can be defined for a suitable class of aymptotically flat pseudohermitian manifolds manifolds. Such manifolds can be obtained in particular from conformal blow-ups of compact ones in positive Webster class. This means that the infimum in (1.7) is positive, which coincides with the positivity of the operator L b , which for n = 1 has the expression

Under the assumption Y ( J ) > 0 , we have that L b is invertible, so for any p ∈ M , there exists a Green’s function G p for which

One can show that in CR normal coordinates ( z , t ) (which are pseudo-hermitian coordinates for a proper conformal change θ ↦ θ ˆ , see Section 3) G p admits the following expansion:

where A is some real constant and where we have set ρ 4 ( z , t ) = ∣ z ∣ 4 + t 2 , z ∈ C , t ∈ R , with ρ being the homogeneous norm on H 1 . We consider then the new pseudohermitian manifold with a blow-up of contact form

where θ ˆ is a contact form used in the definition of the above coordinates. The asymptotics of G p near p determines the geomerty of N at infinity, which becomes the Heisengberg group H 1 in the limit. Using a divergence structure appearing in the variation of the integral of W J , θ on N (similarly to the Riemannian case), one can define a notion of mass for such manifolds, see Definition 3.4. It also turns out that this mass is proportional to the constant A , see Lemma 3.5. One then has the following positive mass theorem for the three-dimensional case, under the assumption of embeddability of the CR structure.

The aforementioned theorem was proved by Cheng et al. [16] for manifolds with non-negative CR Paneitz operator, see Proposition 3.6, using crucially a result from the study by Hsiao and Yung [24]. The fact that embeddability implies non-negativity of the Paneitz operator was shown by Takeuchi [38]. For more relations between these two conditions, see the studies by Case et al. [11] and Chanillo et al. [12]. In higher dimension, we are only aware of the results in Cheng et al. [15] and Cheng and Chiu [14], and it would be desirable to have more general ones.

The positivity of the mass allows to prove the attainment of Y ( M , J ) .

Even though the resolution of the CR Yamabe problem was known before from [19] using the theory of critical points at infinity from [6] (see also [20] for the locally spherical case), there was no previous result about minimal solutions in three dimensions. More recently, a compactness result for solutions of the CR Yamabe equation was derived by Afeltra [1] for n = 1 concerning embeddable manifolds, and it is an open problem to have higher-dimensional counterparts.

We discuss next the sharpness of the above assumptions by exhibiting examples in striking contrast to the Riemannian case. One is given by Rossi spheres [33]: these are a one-parameter-family of CR structures on the 3-sphere of the form S s 3 ≔ ( S 3 , J ( s ) , θ 0 ) , where θ 0 and J ( 0 ) = J 0 are the standard contact form and CR structure of S 3 and J ( s ) is characterized by

with Z 1 being the vector field Z 1 = z ¯ 2 ∂ z 1 − z ¯ 1 ∂ z 2 restricted to S 3 . Rossi spheres are interesting prototype example because they are homogeneous and non-embeddable CR structures on the three-sphere. One has indeed the following theorem.

We saw before (in both low-dimensional Riemannian and CR cases) that positivity of the mass implies attainment of the Sobolev quotient. We also strengthen the relation by means of the following result, again in sharp contrast with the Riemannian case.

It is an open problem to prove counterparts of the last two theorems for more general perturbations of the standard S 3 or to provide other classes of such examples.

The plan of the article is the following. In Section 2, we discuss the variation of Webster curvature with respect to variations of pseudo-hermitian structure and give an idea for the proof of Theorem 1.1. In Section 3, we introduce the notion of pseudo-hermitian mass and sketch the proof of Theorem 1.4. In Section 4, we discuss instead the proofs of Theorems 1.4 and 1.5.

2 Second variation of the normalized Einstein-Hilbert action

In this section, after recalling some useful basic material on pseudohermitian manifolds, as well as some calculation concerning the variation of the contact form or of the CR structure, we display first- and second-variation formulas for the Webster curvature. About the forthcoming review material, we refer the reader to [18] and [30], and for the rest of the calculations to [2].

A CR manifold is a real smooth manifold M endowed with a complex sub-bundle H = T 1 , 0 M of the complexified tangent bundle of M , T C M , such that H ∩ H ¯ = { 0 } and [ H , H ] ⊆ H . We take M to be of hypersurface type, that is, dim M = 2 n + 1 and dim C H = n . Let H ( M ) denote Re ( H ⊕ H ¯ ) (we use sometimes the notation ξ ). Then there exists a natural complex structure on H ( M ) given by

The CR structure is uniquely determined by H ( M ) and J . For H ( M ) and J to generate a CR structure, it is necessary that H is closed under the Lie bracket operation. In three dimension, H is one dimensional, so the condition [ H , H ] ⊆ H automatically holds.

There also exists a non-zero real differential form θ whose kernel at every point coincides with H ( M ) ; this is unique up to scalar multiplication by a non-zero function. A triple ( M , J , θ ) as above is called a pseudohermitian structure. On a pseudohermitian manifold, the Levi form is defined as follows:

A CR manifold is said to be strictly pseudoconvex (respectively, non-degenerate) if it admits a positive definite (respectively, non-degenerate) Levi form. Non-degeneracy is equivalent to the fact that θ is a contact form (see Proposition 1.9 and formula (1.66) in [18]). In this case, we have a unique vector field T such that i T d θ = 0 and θ ( T ) = 1 . For example, if z 1 , … , z n + 1 are complex coordinates on C n + 1 , then on the unit sphere S 2 n + 1 standard choices are given by the formulas:

with J = J 0 being the restriction of the ambient complex rotation to H ( S 2 n + 1 ) . As mentioned in Section 1, classical examples of pseudohermitian manifolds are the Heisenberg group or boundaries of pseudoconvex domains in complex spaces.

On a non-degenerate pseudohermitian manifold, there is a connection, known as the Tanaka-Webster connection. To define it, we recall some useful facts, mostly from [30]. If { T , Z α , Z α ¯ } is a frame dual to { θ , θ α , θ α ¯ } , we express the Levi form as follows:

The matrix h α β ¯ = δ α β will be used in a standard way to raise and lower indices. The Webster connection forms ω α β and the torsion forms τ β = A β α θ α are defined by the following equations:

Also, the curvature forms

satisfy the structure equations

The Ricci tensor and the pseudohermitian scalar (or Webster) curvature are defined by the contractions:

The covariant differentiation is characterized by the following equations:

For a tensor S with components S ⋅ ⋅ , we will use the notation

We also have the following commutation rules for second-order covariant derivatives of functions u and ( 1 , 0 ) -forms σ = σ α θ α , see Lemma 2.3 in [30]:

As a consequence of Bianchi’s identities (see Lemma 2.2 in [30]), we have in particular that

The sub-Laplacian of a scalar function u ∈ C ∞ ( M ) is defined as (see formula (4.10) in [29])

while the squared subgradient is given by

Given the contact bundle ξ , consider a smooth family t ↦ J ( t ) of CR structures on ξ . Then, for all values of t , J ( t ) : ξ → ξ satisfies J ( t ) 2 = − I d . Take a basis of eigenvectors ( Z α ( t ) ) α such that J ( t ) Z α ( t ) = i Z α ( t ) , which implies for the conjugate vector fields J ( t ) Z ¯ α ( t ) = − i Z ¯ α ( t ) . In this way, J ( t ) writes as

Differentiating with respect to t the relation J ( t ) 2 = − I d and using the integrability conditions

which hold along all the deformation, one finds the following result.

Differentiating the structure equations, one also finds:

Taking one more derivative in t of the latter formula, we also obtain also the following result.

With this formula at hand, one is able to compute the second variation of W ˜ with respect to J . We refer to Proposition 3.3 in [18] for a general pointwise second variation formula regarding Webster’s curvature.

We are now in position to discuss the proof of Theorem 1.1.

3 Pseudo-hermitian mass

We consider here the case of three-dimensional pseudo-hermitian manifolds. We recall the notion of pseudohermitian normal coordinates, see [27] (Theorem 2.1), which are defined as follows. Given a vector V + c T ∈ R 2 × R ≃ H 1 ≃ T p M , consider the curve γ V , c solving the ordinary differential equation