Abstract:
We study low-dimensional problems in topology and geometry via a study of contact and Cauchy-Riemann ($CR$) structures. A contact structure is called spherical if it admits a compatible spherical $CR$ structure. We will talk about spherical contact structures and our analytic tool, an evolution equation of $CR$ structures. We argue that solving such an equation for the standard contact 3-sphere is related to the Smale conjecture in 3-topology. Furthermore, we propose a contact analogue of Ray-Singer's analytic torsion. This ''contact torsion'' is expected to be able to distinguish among ''spherical space forms'' $\{\Gamma\backslash S^{3}\}$ as contact manifolds. We also propose the study of a certain kind of monopole equation associated with a contact structure. In view of the recently developed theory of contact homology algebras, we will discuss its overall impact on our study.

Abstract:
In the present article, we develop the analysis of the following nonlinear elliptic system of equations $$ \bar\partial^\pi w = 0, \, d(w^*\lambda \circ j) = 0 $$ first introduced by Hofer, associated to each given contact triad $(M,\lambda,J)$ on a contact manifold $(M,\xi)$. We directly work with this elliptic system on the contact manifold without involving the symplectization process. We establish the local a priori $C^k$ coercive pointwise estimates for all $k \geq 2$ in terms of $\|dw\|_{C^0}$ by doing tensorial calculations on contact manifold itself using the contact triad connection introduced by present the authors. Equipping the punctured Riemann surface $(\dot \Sigma,j)$ with a cylindrical K\"ahler metric and isothermal coordinates near every puncture, we prove the asymptotic (subsequence) convergence to the `spiraling' instantons along the `rotating' Reeb orbit for any solution $w$, not necessarily for $w^*\lambda \circ j$ being exact (i.e., allowing non-zero `charge' $Q \neq 0$), with bounded gradient $\|d w\|_{C^0} < C$ and finite $\pi$-harmonic energy. For nondegenerate contact forms, we employ the `three-interval method' to prove the exponential convergence to a closed Reeb orbit when $Q = 0$. (The Morse-Bott case using this method is treated in a sequel.) In particular, this also provides a new proof of the exponential convergence result even for the case of pseudo-holomorphic curves in symplectization, which was established by Hofer-Wyosocki-Zehnder and by Bourgeois.

Abstract:
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 problem of filling a CR manifold by a complex hyperbolic manifold, and more generally by a Kahler-Einstein or an Einstein metric.

Abstract:
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.

Abstract:
This is a sequel to the papers [OW1] (arXiv:1212.4817), [OW2] (arXiv:1212.5186). In [OW1], the authors introduced a canonical affine connection on $M$ associated to the contact triad $(M,\lambda,J)$. In [OW2], they used the connection to establish a priori $W^{k,p}$-coercive estimates for maps $w: \dot \Sigma \to Q$ satisfying $\overline{\partial^\pi} w= 0, \, d(w^*\lambda \circ j) = 0$ \emph{without involving symplectization}. We call such a pair $(w,j)$ a contact instanton. In this paper, we first prove a canonical neighborhood theorem of the clean manifold $Q$ foliated by Reeb orbits of Morse-Bott contact form. Then using a general framework of the 3-interval method, we establish exponential decay estimates for contact instantons $(w,j)$ of the triad $(M,\lambda,J)$, with $\lambda$ a Morse-Bott contact form and $J$ a CR-almost complex structure adapted to $Q$, under the condition that the asymptotic charge of $(w,j)$ at the associated puncture vanishes. We also apply the 3-interval method to the symplectization case and provide an alternative approach via tensorial calculations to exponential decay estimates in the Morse-Bott case for the pseudoholomorphic curves on the symplectization of contact manifolds. This was previously established by Bourgeois [Bou] (resp. by Bao [Ba]), by using a special coordinate, for the cylindrical (resp. for the asymptotically cylindrical) ends.

Abstract:
This article sketches various ideas in contact geometry that have become useful in low-dimensional topology. Specifically we (1) outline the proof of Eliashberg and Thurston's results concerning perturbations of foliatoins into contact structures, (2) discuss Eliashberg and Weinstein's symplectic handle attachments, and (3) briefly discuss Giroux's insights into open book decompositions and contact geometry. Bringing these pieces together we discuss the construction of ``symplectic caps'' which are a key tool in the application of contact/symplectic geometry to low-dimensional topology.

Abstract:
We show that for dynamically convex contact forms in three dimensions, the cylindrical contact homology differential d can be defined by directly counting holomorphic cylinders for a generic almost complex structure, without any abstract perturbation of the Cauchy-Riemann equation. We also prove that d^2 = 0. Invariance of cylindrical contact homology in this case can be proved using S^1-dependent almost complex structures, similarly to work of Bourgeois-Oancea; this will be explained in another paper.

Abstract:
If we use the form z = (cos + I sin ) and set f(z) = f (rei ) = u(r, ) + iv(r, ), then the Cauuchy-Riemann equations are In this study, we establish the conjugate forms of the above Cauchy-Riemann differential equations in polar coordinate. That is; if we use the conjugate polar form =r (cos + I sin ) and set f( ) = f (re-i ) = u(r, ) + iv(r, - ), then the conjugate polar form Cauuchy-Riemann equations are which is a `reflection` of Cauchy-Riemann Differential Equations in Polar coordinate.

Abstract:
A joint generalization of real smooth as well of complex manifolds are the Cauchy-Riemann manifolds. The main objective of the paper is to inroduce a class of symmetric CR manifolds containing both classes of Riemannian and Hermitian symmetric spaces. It turns out that the classical requirement of isolated fixed points for the symmetries is no longer adequate, because it would imply the Levi-flatness. Among all symmetric CR-manifolds we distinguish a large subclass consisting of all Shilov boundaries of bounded symmetric domains. For this class we calculate the polynomial and the rational convex hulls Both hulls are canonically stratified into real-analytic CR-submanifolds such that the (unique) stratum of the highest dimension is complex for the polynomial and Levi-flat for the rational convex hull. It is also proved that the CR-functions extend continuously to the rational convex hull such that the extension is CR on every stratum.