pseudo-Hermitian Geometry Research Articles

On the role of embeddability in conformal pseudo-hermitian geometry

Abstract In this article, we review some recent results about the role of embeddability in conformal CR (Cauchy-Riemann) geometry. We will show how this condition enters in the second variation of the pseudo-hermitian counterpart of the Einstein-Hilbert action, in the positivity of the pseudohermitian mass and in the extremality properties of the CR (Cauchy-Riemann) Yamabe quotient. Given also some explicit examples at hand, this aspect shows sharp differences compared to the Riemannian case.

On the variation of the Einstein–Hilbert action in pseudohermitian geometry

Abstract In this paper, we compute the first and second variation of the normalized Einstein–Hilbert functional on CR manifolds. We characterize critical points as pseudo-Einstein structures. We then turn to the second variation on standard spheres. While the situation is quite similar to the Riemannian case in dimension greater than or equal to five, in three dimensions, we observe a crucial difference, which mainly depends on the embeddable character of the perturbed CR structure.

Pointwise Slant Curves in Pseudo-Hermitian Geometry

Pointwise Slant Curves in Pseudo-Hermitian Geometry

Pseudo-harmonic Maps from Complete Noncompact Pseudo-Hermitian Manifolds to Regular Balls

In this paper, we give an estimate of sub-Laplacian of Riemannian distance functions in pseudo-Hermitian geometry which plays a similar role as Laplacian comparison theorem in Riemannian geometry, and deduce a prior horizontal gradient estimate of pseudo-harmonic maps from pseudo-Hermitian manifolds to regular balls of Riemannian manifolds. As an application, Liouville theorem is established under the conditions of nonnegative pseudo-Hermitian Ricci curvature and vanishing pseudo-Hermitian torsion. Moreover, we obtain the existence of pseudo-harmonic maps from complete noncompact pseudo-Hermitian manifolds to regular balls of Riemannian manifolds.

Dirichlet and Neumann eigenvalue problems on CR manifolds

We study the properties of Carnot–Caratheodory spaces attached to a strictly pseudoconvex CR manifold M, in a neighborhood of each point $$x \in M$$ , versus the pseudohermitian geometry of M arising from a fixed positively oriented contact form $$\theta $$ on M. The weak Dirichlet problem for the sublaplacian $$\Delta _b$$ on $$(M, \theta )$$ is solved on domains $$\Omega \subset M$$ supporting the Poincare inequality. The solution to Neumann problem for the sublaplacian $$\Delta _b$$ on a $$C^{1,1}$$ connected $$(\epsilon , \delta )$$ -domain $$\Omega \subset {{\mathbb {G}}}$$ in a Carnot group (due to Danielli et al. in: Memoirs of American Mathematical Society 2006) is revisited for domains in a CR manifold. As an application we prove discreetness of the Dirichlet and Neumann spectra of $$\Delta _b$$ on $$\Omega \subset M$$ in a Carnot–Cartheodory complete pseudohermitian manifold $$(M, \theta )$$ .

Invariant surface area functionals and singular Yamabe problem in 3-dimensional CR geometry

We express two CR invariant surface area elements in terms of quantities in pseudohermitian geometry. We deduce the Euler–Lagrange equations of the associated energy functionals. Many solutions are given and discussed. In relation to the singular CR Yamabe problem, we show that one of the energy functionals appears as the coefficient (up to a constant multiple) of the log term in the associated volume renormalization.

Parallel spinors and basic holonomy in pseudo-Hermitian geometry

We introduce in this paper an adjustment for the Webster–Tanaka connection of pseudo-Hermitian geometry on CR manifolds, which behaves naturally with respect to issues concerning parallel sections and holonomy. We call this connection and its holonomy group basic. The basic holonomy group is suitable to describe pseudo-Einstein spaces. Moreover, for pseudo-Hermitian spin manifolds, we discuss the existence of (transversally) parallel spinors. Such spinors exist on pseudo-Einstein spaces, no matter of the sign of the Webster scalar curvature.

Laplacians and Legendre Surfaces in Pseudo-Hermitian Geometry

In this paper, we prove that for a Legendre surface N of 5-dimensional Sasakian space forms \(M^5\), if N satisfies \({\hat{\triangle }} H=\lambda H\) and \({\mathrm {tr}}{\hat{\nabla \ }}{{\hat{T}}}(H)=0\) for a constant \(\lambda \), then \(\Vert H\Vert \) is a constant if and only if H is \({\hat{D}}\)-parallel, N is a Chen surface, and \(\mathrm{tr} S^2_{H}=\lambda \parallel H \parallel \). From this, for a Legendre surface N of \(M^5\) such that \(\Vert H\Vert \) is a constant, if N satisfies \({\hat{\triangle }} H=\lambda H\) and \({\mathrm {tr}}{\hat{\nabla \ }}{\hat{T}}(H)=0\) for a constant \(\lambda \), then N is a \({\hat{D}}\)-parallel Legendre–Chen surface. Moreover, we show that it is minimal, or a local product of a geodesic and a pseudo-Hermitian circle or two pseudo-Hermitian circles.

Comparison theorems in pseudo-Hermitian geometry and applications

In this paper, we study the theory of geodesics with respect to the Tanaka-Webster connection in a pseudo-Hermitian manifold, aiming to generalize some comparison results in Riemannian geometry to the case of pseudo-Hermitian geometry. Some Hopf-Rinow type, Cartan-Hadamard type and Bonnet-Myers type results are established.

Worm domains and Fefferman space–time singularities

Let W be a smoothly bounded worm domain in C 2 and let A = Null ( L θ ) be the set of Levi-flat points on the boundary ∂ W of W . We study the relationship between pseudohermitian geometry of the strictly pseudoconvex locus M = ∂ W ∖ A and the theory of space–time singularities associated to the Fefferman metric F θ on the total space of the canonical circle bundle S 1 → C ( M ) ⟶ π M . Given any point ( 0 , w 0 ) ∈ A , we show that every lift Γ ( φ ) ∈ C ( M ) , 0 ≤ φ − log | w 0 | 2 < π ∕ 2 , of the circle Γ w 0 : r = 2 cos [ log | w 0 | 2 − φ ] in M , runs into a curvature singularity of Fefferman’s space–time ( C ( M ) , F θ ) . We show that Σ = π − 1 ( Γ w 0 ) is a Lorentzian real surface in ( C ( M ) , F θ ) such that the immersion ι : Σ ↪ C ( M ) has a flat normal connection. Consequently, there is a natural isometric immersion j : O ( Σ ) → O ( C ( M ) , Σ ) between the total spaces of the principal bundles of Lorentzian frames O ( 1 , 1 ) → O ( Σ ) → Σ and adapted Lorentzian frames O ( 1 , 1 ) × O ( 2 ) → O ( C ( M ) , Σ ) → Σ , endowed with Schmidt metrics, descending to a map of bundle completions which maps the b -boundary of Σ into the adapted bundle boundary of C ( M ) , i.e. j ( Σ ̇ ) ⊂ ∂ adt C ( M ) .

Renormalized Chern-Gauss-Bonnet formula for complete Kähler-Einstein metrics

We present a renormalized Gauss-Bonnet formula for a strictly pseudoconvex manifold with a complete K\{a}hler metric given by a globally defined potential function near the boundary. When the metric is asymptotically Einstein, the boundary contribution in the formula is explicitly written down in terms of the pseudo-hermitian geometry of the boundary and is shown to be a global CR invariant. The CR invariant generalizes the Burns-Epstein invariant on 3-dimensional CR manifolds.

PSEUDOHERMITIAN LEGENDRE SURFACES OF SASAKIAN SPACE FORMS

From the point of view of pseudohermitian geometry, we classify Legendre surfaces of Sasakian space forms with non-minimal <TEX>${\hat{C}}$</TEX>-parallel mean curvature vector field for the Tanaka-Webster connection.

Pseudo-Einstein manifolds

Among the studies on pseudo-Hermitian geometry of strictly pseudo-convex almost CR manifolds, we study especially the two kinds of pseudo-Einstein structures and related problems.

Submanifolds with harmonic mean curvature in pseudo-Hermitian geometry

We classify Hopf cylinders with proper mean curvature vector field in Sasakian 3-manifolds with respect to the Tanaka-Webster connection.

Pseudo-Hermitian Geometry in 3-D

Pseudo-hermitian geometry is the study of contact form in conjunction with an almost complex structure on the contact planes. In this two part survey, we first discuss the theory of surfaces in 3-D pseudo-hermitian manifolds. The local geometry of the surface is governed by the p-mean curvature equation. In analogy with the Gauss and Codazzi equation of a surface in Euclidean three space, we develop their analogue of these two equations. Due to the degeneracy of the p-mean curvature equation, the regularity theory is quite interesting. In the second part of this article, we describe a CR-invariant condition for the embeddability question. In CR geometry in 3-D, there is no local integrability condition for the almost-complex structure, so the condition to impose is global in nature. The condition involves two geometrically defined linear operators which have their counterparts in conformal geometry. We also formulate a notion of CR mass and discuss a positive mass theorem to characterize the Heisenberg space. The solvability of the \({\bar\partial}\) Neumann problem is also a crucial ingredient in the positivity of the CR mass.

Construction of equivalence maps in pseudo-Hermitian geometry via linear partial differential equations

We discuss an equivalence problem of pseudo-Hermitian structures on 3-dimensional manifolds, and develop a method of constructing equivalence maps by using systems of linear partial differential equations. It is proved that a pseudo-Hermitian structure is transformed to a standard model of pseudo-Hermitian structure constructed on the Heisenberg group if and only if it has the vanishing pseudo-Hermitian torsion and the pseudo-Hermitian curvature. A system of linear partial differential equations whose coefficients are associated with a given pseudo-Hermitian structure is introduced, and plays a central role in this paper. The system is integrable if and only if the pseudo-Hermitian structure has vanishing torsion and curvature. The equivalence map is constructed by using a normal basis of the solution space of the system.

A gauged Fefferman construction for partially integrable CR geometry

A non-degenerate almost CR structure T 10 ⊂ T M ⊗ C of hypersurface type on a smooth manifold M n of odd dimension n = 2 m + 1 ≥ 3 is called partially integrable iff [ Γ ( T 10 ) , Γ ( T 10 ) ] ⊂ Γ ( T 10 ⊕ T 10 ¯ ) . For any choice of pseudo-Hermitian form θ there exists a canonical linear connection ∇ W on a partially integrable CR manifold ( M , T 10 ) . This connection is a natural generalisation of the Tanaka–Webster connection of pseudo-Hermitian geometry. We use the generalised connection for the construction of a Fefferman metric on the total space of the canonical circle bundle of any (strictly pseudoconvex) partially integrable CR manifold. The construction is a CR invariant. In fact, we invent here a slightly more general version of this construction by the use of a gauge form ℓ . This is called the gauged Fefferman construction of partially integrable CR geometry. The scalar curvature of the gauged Fefferman metric and the Laplacian of the given pseudo-Hermitian structure θ are calculated in explicit form.

Minimal surfaces in pseudohermitian geometry

We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the p-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a characteristic curve, and as a quantity in terms of calibration geometry. As a differential equation, the p-minimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set (i.e., the set where the (p-)area integrand vanishes), we formulate some extension theorems, which describe how the characteristic curves meet the singular set. This allows us to classify the entire solutions to this equation and to solve a Bernstein-type problem (for graphs over the xy-plane) in the Heisenberg group H1. In H1, identified with the Euclidean space R3, the p-minimal surfaces are classical ruled surfaces with the rulings generated by Legendrian lines. We also prove a uniqueness theorem for the Dirichlet problem under a condition on the size of the singular set in two dimensions, and generalize to higher dimensions without any size control condition. We also show that there are no closed, connected, C2 smoothly immersed constant p-mean curvature or p-minimal surfaces of genus greater than one in the standard S3. This fact continues to hold when S3 is replaced by a general pseudohermitian 3-manifold. Mathematics Subject Classification (2000): 35L80 (primary); 35J70, 32V20, 53A10, 49Q10 (secondary).

Affine biharmonic submanifolds in 3-dimensional pseudo-Hermitian geometry

The notion of biharmonic map between Riemannian manifolds is generalized to maps from Riemannian manifolds into affine manifolds. Hopf cylinders in 3-dimensional Sasakian space forms which are biharmonic with respect to Tanaka-Webster connection are classified.

Area-stationary surfaces in the Heisenberg group [formula omitted

We consider area-stationary surfaces, perhaps with a volume constraint, in the Heisenberg group H 1 endowed with its Carnot–Carathéodory distance. By analyzing the first variation of area, we characterize C 2 area-stationary surfaces as those with mean curvature zero (or constant if a volume-preserving condition is assumed) and such that the characteristic curves meet orthogonally the singular curves. Moreover, a Minkowski-type formula relating the area, the mean curvature, and the volume is obtained for volume-preserving area-stationary surfaces enclosing a given region. As a consequence of the characterization of area-stationary surfaces, we refine the Bernstein type theorem given in [Jih-Hsin Cheng, Jenn-Fang Hwang, Andrea Malchiodi, Paul Yang, Minimal surfaces in pseudohermitian geometry, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (1) (2005), 129–177. MR 2165405] and [Nicola Garofalo, Scott Pauls, The Bernstein problem in the Heisenberg group, arXiv: math.DG/0209065] to describe all C 2 entire area-stationary graphs over the xy-plane in H 1 . A calibration argument shows that these graphs are globally area-minimizing. Finally, by using the description of the singular set in [Jih-Hsin Cheng, Jenn-Fang Hwang, Andrea Malchiodi, Paul Yang, Minimal surfaces in pseudohermitian geometry, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (1) (2005), 129–177. MR 2165405], the characterization of area-stationary surfaces, and the ruling property of constant mean curvature surfaces, we prove our main results where we classify area-stationary surfaces in H 1 , with or without a volume constraint, and non-empty singular set. In particular, we deduce the following counterpart to Alexandrov uniqueness theorem in Euclidean space: any compact, connected, C 2 surface in H 1 , area-stationary under a volume constraint, must be congruent to a rotationally symmetric sphere obtained as the union of all the geodesics of the same curvature joining two points. As a consequence, we solve the isoperimetric problem in H 1 assuming C 2 smoothness of the solutions.