Browsing by Browse by FOR 2008 "010111 Real and Complex Functions (incl. Several Variables)"
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Publication CR-Geometry and Shearfree Lorentzian GeometryWe study higher dimensional versions of shearfree null-congruences in conformal Lorentzian manifolds. We show that such structures induce a subconformal structure and a partially integrable almost CR-structure on the leaf space and we classify the Lorentzian metrics that induce the same subconformal structure. In the last section we survey some known applications of the correspondence between almost CR-structures and shearfree null-congruences in dimension 4.1974 5 - Some of the metrics are blocked by yourconsent settings
Publication A criterion for local embeddability of three-dimensional CR structuresWe introduce a CR-invariant class of Lorentzian metrics on a circle bundle over a three-dimensional CR structure, which we call FRT metrics. These metrics generalise the Fefferman metric, allowing for more control of the Ricci curvature, but are more special than the shearfree Lorentzian metrics introduced by Robinson and Trautman. Our main result is a criterion for embeddability of three-dimensional CR structures in terms of the Ricci curvature of the FRT metrics in the spirit of the results by Lewandowski et al. (Class Quantum Gravity 7(11):L241–L246, 1990) and also Hill et al. (Indiana Univ Math J 57(7):3131–3176, 2008. https://doi.org/10.1512/iumj.2008.57.3473).2063 8 - Some of the metrics are blocked by yourconsent settings
Publication A Non-linear Relation for Certain Hypergeometric FunctionsWe describe a family of Gaussian hypergeometric functions that satisfy a nonlinear differential identity.1404 3 - Some of the metrics are blocked by yourconsent settings
Publication Nonlinear Critical Elliptic Problems: Existence and UniquenessIn this thesis, I shall study the existence, local uniqueness and other related subjects for the bubbling solutions of two elliptic problems involving critical Sobolev exponent.
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Publication Normal forms of para-CR hypersurfacesWe consider hypersurfaces of finite type in a direct product space R²×R², which are analogues to real hypersurfaces of finite type in C². We shall consider separately the cases where such hypersurfaces are regular and singular, in a sense that corresponds to Levi degeneracy in hypersurfaces in C². For the regular case, we study formal normal forms and prove convergence by following Chern and Moser. The normal form of such an hypersurface, considered as the solution manifold of a 2nd order ODE, gives rise to a normal form of the corresponding 2nd order ODE. For the degenerate case, we study normal forms for weighted ℓ-jets. Furthermore, we study the automorphisms of finite type hypersurfaces.1596 - Some of the metrics are blocked by yourconsent settings
Publication On the classification of homogeneous affine tube domains with large automorphism groups in arbitrary dimensionsWe classify tube domains in Cn+1 (n ≥ 1) with affinely homogeneous base of their boundary and a.) with positive definite Levi form and b.) with Lorentzian type Levi form and affine isotropy of dimension at least (n−2)(n−3)/2.1103 3 - Some of the metrics are blocked by yourconsent settings
Publication On the Classification of Spherical Rigid CR Manifolds and Sasakian Manifolds in C2We consider spherical hypersurfaces in C2 with a fixed Reeb vector field as 3-dimensional Sasakian manifolds. We establish the correspondence between three different sets of parameters, namely, those arising from representing the Reeb vector field as an automorphism of the Heisenberg sphere, the parameters used in Stanton’s description of rigid spheres, and the parameters arising from the rigid normal forms. We also geometrically describe the moduli space for rigid spheres, and provide geometric distinction between Stanton’s hypersurfaces and those found in [17]. We determine the Sasakian automorphism groups of the rigid spheres, detecting the homogeneous Sasakian manifolds amongst them, and we determine the Sasakian automorphisms of the CR manifolds arising in E. Cartan’s classical list of homogeneous CR hypersur- ´ faces. Furthermore, we relax the condition on the Reeb vector field to allow preservation up to a nonzero dilation, called homothetic Sasakian preservation. Finally, we determine the homogeneous Sasakian manifolds with respect to the homothetic Sasakian preservation.
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Publication Open AccessJournal ArticleOn the generalization of Forelli's theoremThe purpose of this paper is to present a solution to perhaps the final remaining case in the line of study concerning the generalization of Forelli's theorem on the complex analyticity of the functions that are: (i) C∞ smooth at a point, and (ii) holomorphic along the complex integral curves generated by a contracting holomorphic vector field with an isolated zero at the same point.990 - Some of the metrics are blocked by yourconsent settings
Publication Rigid Embeddings of Sasakian Hyperquadrics in Cn+1We give a classification of Sasakian manifolds that are CR-equivalent to hyperquadrics by describing their exact parameter space. For "half" of the parameter space, we find an explicit representation by defining equations. This problem is related to the problem of finding pseudo-Kähler potentials with prescribed Ricci curvature.2162 - Some of the metrics are blocked by yourconsent settings
Publication Open AccessThesis DoctoralShearfree Lorentzian Geometry and CR GeometryWe introduce a CR-invariant class of Lorentzian metrics on a circle bundle over a 3-dimensional CR-structure, which we call FRT-metrics. These metrics generalise the Fefferman metric, allowing for more control of the Ricci curvature, but are more special than the shearfree Lorentzian metrics introduced by Robinson and Trautman. Our main result is a criterion for embeddability of 3-dimensional CR-structures in terms of the Ricci curvature of the FRT-metrics in the spirit of the results by Lewandowski et al. in [37] and also Hill et al. in [25]. We also study higher dimensional versions of shearfree null congruences in conformal Lorentzian manifolds. We show that such structures induce a subconformal structure and a partially integrable almost CR structure on the leaf space and we classify the Lorentzian metrics that induce the same subconformal structure.1481 249 - Some of the metrics are blocked by yourconsent settings
Publication Open AccessJournal ArticleSingular multicontact structuresWe describe the automorphisms of a singular multicontact structure, that is a generalisation of the Martinet distribution. Such a structure is interpreted as a para-CR structure on a hypersurface M of a direct product space R²+ x R²-. We introduce the notion of a finite type singularity analogous to CR geometry and, along the way, we prove extension results for para-CR functions and mappings on embedded para-CR manifolds into the ambient space.1573 1 - Some of the metrics are blocked by yourconsent settings
Publication The zero curvature equation for rigid CR-manifoldsIn this paper, we present the general analytic solution to the zero curvature equation for rigid three-dimensional CR-manifolds. The solutions are uniquely determined by one function and four real parameters.1643