We introduce a global Cauchy-Riemann($CR$)-invariant and discuss its behavior on the moduli space of $CR$-structures. We argue that this study is related to the Smale conjecture in 3-topology and the problem of counting complex structures. Furthermore, we propose a contact-analogue of Ray-Singer's analytic torsion. This ``contact torsion'' is expected to be able to distinguish among ``contact lens'' spaces. We also propose the study of a certain kind of monopole equation associated with a contact structure.
We give a self-contained proof of a formula computing the Fredholm index for asymptotically non-degenerate Cauchy-Riemann operators on surfaces with boundary punctures using the method of large antilinear deformations. This method for computing the index was introduced in the case of closed surfaces by Taubes and generalized to the case with interior punctures by Gerig. One novel feature arising from our proof is that the Euler characteristic term in the index formula involves a non-standard weighted count of boundary zeros. We hope that this formulation of the index formula will be useful to …
This is a sequel to the paper [Oh5] (or ArXiv:math.SG/0206092). The main purpose of the paper is to give the proof of an existence theorem, with energy bounds, of certain pseudo-holomorphic sections of the mapping cylinder that is needed for the proof of nondegeneracy of the homological invariant pseudo-norm which the author has constructed on general symplectic manifolds [Oh4,5]. The existence theorem is also the crux of the author's recent proof of an optimal energy-capacity inequality given in [Oh5]. In this paper, we prove a more general existence result than needed in that we study Floer'…
We study the action of the group of contact diffeomorphisms on CR deformations of compact three-dimensional CR manifolds. Using anisotropic function spaces and an anisotropic structure on the space of contact diffeomorphisms, we establish the existence of local transverse slices to the action of the contact diffeomorphism group in the neighbourhood of a fixed embeddable strongly pseudoconvex CR structure.
We compute a recently introduced geometric invariant of stricly pseudoconvex CR 3-manifolds for certain circle invariant spherical CR structures on Seifert manifolds. We give applications to the problem of filling the CR manifold by a complex hyperbolic manifold, and more generally by a Kaehler-Einstein or an Einstein metric.
We introduce a new class of lightlike submanifolds, namely, Screen Transversal Cauchy Riemann (STCR)-lightlike submanifolds, of indefinite Kaehler manifolds. We show that this new class is an umbrella of screen transversal lightlike, screen transversal totally real lightlike and CR-lightlike submanifolds. We give a few examples of a STCR lightlike submanifold, investigate the integrability of various distributions, obtain a characterization of such lightlike submanifolds in a complex space form and find new conditions for the induced connection to be a metric connection. Moreover, we investiga…
We show that one can define a spectral curve for the Cauchy-Riemann operator on a punctured elliptic curve if one imposes appropriate boundary conditions. Algebraic curves of the type thus obtained appear as irreducible components of spectral curves of minimal tori with planar ends in R^3. It appears that these curves coincide with the spectral curves of certain elliptic KP solitons as studied by Krichever.
Fiedler and Mallet-Paret prove a version of the classical Poincaré-Bendixson Theorem for scalar parabolic equations. We prove that a similar result holds for bounded solutions of the non-linear Cauchy-Riemann equations. The latter is an application of an abstract theorem for flows with a(n) (unbounded) discrete Lyapunov function.
We consider a connection $\nabla^X$ on a complex line bundle over a Riemann surface with boundary $M_0$, with connection 1-form $X$. We show that the Cauchy data space of the connection Laplacian (also called magnetic Laplacian) $L:={\nabla^X}^*\nabla^X + q$, with $q$ a complex valued potential, uniquely determines the connection up to gauge isomorphism, and the potential $q$.
We relate a recently introduced non-local geometric invariant of compact strictly pseudoconvex Cauchy-Riemann (CR) manifolds of dimension 3 to various eta-invariants in CR geometry: on the one hand a renormalized eta-invariant appearing when considering a sequence of metrics converging to the CR structure by expanding the size of the Reeb field; on the other hand the eta-invariant of the middle degree operator of the contact complex. We then provide explicit computations for a class of examples: transverse circle invariant CR structures on Seifert manifolds. Applications are given to the probl…
The equivalence problem for second order ODEs given modulo point transformations is solved in full analogy with the equivalence problem of nondegenerate 3-dimensional CR structures. This approach enables an analog of the Feffereman metrics to be defined. The conformal class of these (split signature) metrics is well defined by each point equivalence class of second order ODEs. Its conformal curvature is interpreted in terms of the basic point invariants of the corresponding class of ODEs.
Let $M$ be a compact abstract $CR$ manifold of arbitrary $CR$ codimension. Under certain conditions on the Levi form we prove the infinite dimensionality of some global cohomology groups of $M$.
For certain real hypersurfaces in the projective space, of signature (1,n), we study the filling problem for small deformations of the CR structure (the other signatures being well understood). We characterize the deformations which are fillable, and prove that they have infinite codimension in the set of all CR structures. This result generalizes the cases of the 3-sphere and of signature (1,1) to higher dimension.
The lower-order cr-invariant variational problem for Legendrian curves in the 3-sphere is studied and its Euler-Lagrange equations are deduced. Closed critical curves are investigated. Closed critical curves with non-constant cr-curvature are characterized. We prove that their cr-equivalence classes are in one-to-one correspondence with the rational points of a connected planar domain. A procedure to explicitly build all such curves is described. In addition, a geometrical interpretation of the rational parameters in terms of three phenomenological invariants is given.
Let $\mathrm S^3$ be the unit sphere of $\mathbb C^2$ with its standard Cauchy-Riemann (CR) structure. This paper investigates the CR geometry of curves in $\mathrm S^3$ which are transversal to the contact distribution, using the local CR invariants of $\mathrm S^3$. More specifically, the focus is on the CR geometry of transversal knots. Four global invariants of transversal knots are considered: the phase anomaly, the CR spin, the Maslov index, and the CR self-linking number. The interplay between these invariants and the Bennequin number of a knot are discussed. Next, the simplest CR invar…
We prove that an integral Cauchy-Riemann inequality holds for any pair of smooth functions $(f,h)$ on the 2-sphere $\mathbb{S}^2$, and equality holds iff $f$ and $h$ are related $λ_1$-eigenfunctions. We extend such inequality to 4-tuples of functions, only valid on the $L^2$-orthogonal complement of a suitable nonzero finite dimensional space of functions. As a consequence we prove that 2-spheres are not $Ω$-stable surfaces with parallel mean curvature in $\mathbb{R}^7$ for the associative calibration $Ω$.
In the paper we develop a framework for the alternative way of the study of a local geometry of almost cosymplectic manifolds with Kahlerian leaves. The main idea is to apply the concept of a geometry and analysis of CR manifolds. Locally the almost cosymplectic manifold is modeled on the 'mixed' space RxCn. There is given a complete local description of the underlying almost contact metric structure in the system of local, mixed - real, complex- coordinates. We also introduce a notion of a canonical Hermitian complex connection in the CR structure of a CR almost cosymplectic manifold. As an e…
Let $X$ be a CR manifold with transversal, proper CR $G$-action. We show that $X/G$ is a complex space such that the quotient map is a CR map. Moreover the quotient is universal, i.e. every invariant CR map into a complex manifold factorises uniquely over a holomorphic map on $X/G$. We then use this result and complex geometry to proof an embedding theorem for (non-compact) strongly pseudoconvex CR manifolds with transversal $G \rtimes S^1$-action. The methods of the proof are applied to obtain a projective embedding theorem for compact CR manifolds.
We consider a contact manifold with a pseudo-Riemannian metric and define a contact vector field intrinsically associated to this pair of structures. We call this new differential invariant the contact Riemannian curl. On a Riemannian manifold, Killing vector fields are those that annihilate the metric; a Killing $1$-form is obtained from a Killing vector field by lowering indices. We show that the contact Riemannian curl vanishes if the metric is of constant curvature and the contact structure is defined by a Killing $1$-form. We also show that the contact Riemannian curl has a strong similar…
We consider a closed three-dimensional contact sub-Riemannian manifold. The objective of this note is to provide a precise description of the sub-Riemannian geodesics with large initial momenta: we prove that they "spiral around the Reeb orbits", not only in the phase space but also in the configuration space. Our analysis is based on a normal form along any Reeb orbit due to Melrose.
We study isometries in the contact sub-pseudo-Riemannian geometry. In particular we give an upper bound on the dimension of the isometry group of a general sub-pseudo-Riemannian manifold and prove that the maximal dimension is attained for the left invariant structures on the Heisenberg group.
We construct a canonically defined affine connection in sub-Riemannian contact geometry. Our method mimics that of the Levi-Civita connection in Riemannian geometry. We compare it with the Tanaka-Webster connection in the three-dimensional case.
We investigate contact Lie groups having a left invariant Riemannian or pseudo-Riemannian metric with specific properties such as being bi-invariant, flat, negatively curved, Einstein, etc. We classify some of such contact Lie groups and derive some obstruction results to the existence of left invariant contact structures on Lie groups
We define an ADM-like mass, called p-mass, for an asymptotically flat pseudohermitian manifold. The p-mass for the blow-up of a compact pseudohermitian manifold (with no boundary) is identified with the first nontrivial coefficient in the expansion of the Green function for the CR Laplacian. We deduce an integral formula for the p-mass, and we reduce its positivity to a solution of Kohn's equation. We prove that the p-mass is non-negative for (blow-ups of) compact 3-manifolds of positive Tanaka-Webster class and with non-negative CR Paneitz operator. Under these assumptions, we also characteri…
We consider local geometry of sub-pseudo-Riemannian structures on contact manifolds. We construct fundamental invariants of the structures and show that the structures give rise to Einstein-Weyl geometries in dimension 3, provided that certain additional conditions are satisfied.
The conjecture of D.Blair says that there are no nonflat Riemannian metrics of nonpositive curvature compatible with a contact structure. We prove this conjecture for a certain class of contact structures on closed 3-dimensional manifolds and construct a local counterexample. We also prove that a hyperbolic metric on $\mathbb{R}^3$ cannot be compatible with any contact structure.
For a Riemannian $G$-structure, we compute the divergence of the vector field induced by the intrinsic torsion. Applying the Stokes theorem, we obtain the integral formula on a closed oriented Riemannian manifold, which we interpret in certain cases. We focus on almost harmitian and almost contact metric structures.
In this paper we obtain the structure equation of a contact-complex Riemannian submersion and give some applications of this equation in the study of almost cosymplectic manifolds with Kaehler fibres.
It is shown that the redshift between two Cauchy surfaces in a globally hyperbolic spacetime equals the ratio of the associated contact forms on the space of light rays of that spacetime.
There are considered 4-dimensional pseudo-Riemannian spaces with inner products of signature (3,1) and (2,2). The objects of investigation are space-like and time-like hyperspheres in the respective cases. These hypersurfaces are equipped with almost contact B-metric structures. The constructed manifolds are characterized geometrically.
In this paper we consider surfaces of class $C^1$ with continuous prescribed mean curvature in a three-dimensional contact sub-Riemannian manifold and prove that their characteristic curves are of class $C^2$. This regularity result also holds for critical points of the sub-Riemannian perimeter under a volume constraint. All results are valid in the first Heisenberg group $\mathbb{H}^1$.
We consider the horospherical transform and its inversion in 3 examples of hyperboloids. We want to illustrate via these examples the fact that the horospherical inversion formulas can be directly extracted from the classical Radon inversion formula. In a more broad context, this possibility reflects the fact that the harmonic analysis on symmetric spaces (Riemannian as well as pseudo-Riemannian ones) is equivalent (homologous), up to the Abelian Fourier transform, to the similar problem in the flat model. On the technical level it is important that we work not with the usual horospherical tra…
Contact Riemannian manifolds, whose complex structures are not necessarily integrable, are generalization of pseudohermitian manifolds in CR geometry. The Tanaka-Webster-Tanno connection plays the role of the Tanaka-Webster connection of a pseudohermitian manifold. Conformal transformations and the Yamabe problem are also defined naturally in this setting. By constructing the special frames and the normal coordinates on a contact Riemannian manifold, we prove that if the complex structure is not integrable, its Yamabe invariant on a contact Riemannian manifold is always less than the Yamabe in…
In this paper a conformal classification of three dimensional left-invariant sub-Riemannian contact structures is carried out; in particular we will prove the following dichotomy: either a structure is locally conformal to the Heisenberg group $\mathbb H_3$, or its conformal classification coincides with the metric one. If a structure is locally conformally flat, then its conformal group is locally isomorphic to $SU(2,1)$.
In this paper we consider a set $E\subsetΩ$ with prescribed mean curvature $f\in C(Ω)$ and Euclidean Lipschitz boundary $\partial E=Σ$ inside a three-dimensional contact sub-Riemannian manifold $M$. We prove that if $Σ$ is locally a regular intrinsic graph, the characteristic curves are of class $C^2$. The result is shape and improves the ones contained in \cite{MR2583494} and \cite{GalRit15}.
We compare different notions of curvature on contact sub-Riemannian manifolds. In particular we introduce canonical curvatures as the coefficients of the sub-Riemannian Jacobi equation. The main result is that all these coefficients are encoded in the asymptotic expansion of the horizontal derivatives of the sub-Riemannian distance. We explicitly compute their expressions in terms of the standard tensors of contact geometry. As an application of these results, we obtain a sub-Riemannian version of the Bonnet-Myers theorem that applies to any contact manifold.
In this paper we study the small time asymptotics for the heat kernel on a sub-Riemannian manifold, using a perturbative approach. We then explicitly compute, in the case of a 3D contact structure, the first two coefficients of the small time asymptotics expansion of the heat kernel on the diagonal, expressing them in terms of the two basic functional invariants $χ$ and $κ$ defined on a 3D contact structure.
We prove several relations between spectrum and dynamics including wave trace expansion, sharp/improved Weyl laws, propagation of singularities and quantum ergodicity for the sub-Riemannian (sR) Laplacian in the four dimensional quasi-contact case. A key role in all results is played by the presence of abnormal geodesics and represents the first such appearance of these in sub-Riemannian spectral geometry.
In this paper we establish a Gaffney type inequality, in $W^{\ell,p}$-Sobolev spaces, for differential forms on sub-Riemannian contact manifolds without boundary, having bounded geometry (hence, in particular, we have in mind non-compact manifolds). Here $p\in]1,\infty[$ and $\ell=1,2$ depending on the order of the differential form we are considering. The proof relies on the structure of the Rumin's complex of differential forms in contact manifolds, on a Sobolev-Gaffney inequality proved by Baldi-Franchi in the setting of the Heisenberg groups and on some geometric properties that can be pro…
We discuss contact geometry naturally related with optimal control problems (and Pontryagin Maximum Principle). We explore and expand the observations of [Ohsawa, 2015], providing simple and elegant characterizations of normal and abnormal sub-Riemannian extremals.
We are concerned with stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds. Employing the Riemannian approximations to the sub-Riemannian manifold which make use of the Reeb vector field, we obtain a second order partial differential operator on the surface arising as the limit of Laplace-Beltrami operators. The stochastic process associated with the limiting operator moves along the characteristic foliation induced on the surface by the contact distribution. We show that for this stochastic process elliptic characteristic points are inaccessible, while hyperb…
We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is defined by a straightforward discretisation of the Cauchy-Riemann equation. A lot of classical results in Riemann theory have a discrete counterpart, Hodge star, harmonicity, Hodge theorem, Weyl's lemma, Cauchy integral formula, existence of holomorphic forms with prescribed holonomies. Giving a geometrical meaning to the construction on a Riemann surface, we define a notion of critical…
Properties of the Cauchy-Riemann-Fueter equation for maps between quaternionic manifolds are studied. Spaces of solutions in case of maps from a K3-surface to the cotangent bundle of a complex projective space are computed. A relationship between harmonic spinors of a generalized nonlinear Dirac operator and solutions of the Cauchy-Riemann-Fueter equation are established.
