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Ali M. Elgindi

Publications and source records attributed to Ali M. Elgindi.

11 recordsLinked to original sources
arXiv2026

A New CR Invariant for Contact 3-Manifolds and Classes of Open Books

This paper introduces a new CR invariant for co-oriented contact structures on closed, orientable 3-manifolds. The invariant, which we denote as $μ_M(ξ)$, takes values in the Picard group of complex line bundles $\Pic_{\C}(M)$. The construction associates to a contact structure $ξ$ and a supporting open book decomposition an embedding into $\C^3$, where the contact structure becomes the holomorphic line field along the binding. Using Stein theory, the induced holomorphic line bundle extends to all of $\C^3$ but we consider only its restriction to $M$. By Giroux's correspondence, we prove this construction is independent of the choice of open book, yielding a well-defined invariant $μ_M(ξ) \in \Pic_{\C}(M)$ over the manifold. As an application, we distinguish two tight contact structures on the 3-torus $\T^3$ by showing their first Chern classes are different.

math.GT↗
arXiv2026

New constructions relating Real and Complex Contact Structure

We establish new connections between real and complex contact geometry via embeddings of 3-manifolds into $\C^3$. We introduce a new \emph{contact wedge} construction combining two transverse real contact structures to make a new \emph{complex contact} structure, subject to obstructions measured by the Nijenhuis tensor and Dolbeault cohomology. Dually, we form a \emph{hedge} construction which extracts real contact structures from complex ones. Applying these tools, we prove that $\C^3$ admits uncountably many complex contact structures.

math.SG↗
arXiv2026

Holomorphic Hulls for Compact 3-Manifolds

We demonstrate the different possible structures for holomorphic hulls for embeddings of compact real 3-manifolds $M \hookrightarrow \mathbb{C}^3$ along the set of complex tangents $γ$. Using our previous work [1], we can construct embeddings with any prescribed link and 2-plane field along it, as well as a prescription of angles at each point that determines the Bishop invariant. We show that elliptic points ($γ< \frac{1}{2}$) produce analytic discs filling a Levi-flat hypersurface, hyperbolic points ($γ> \frac{1}{2}$) add no local hull structure, and parabolic points ($γ= \frac{1}{2}$) may develop a "delicate" Jöricke `onion' structure. We illustrate these phenomena with explicit examples of parabolic knots and links with mixed components.

math.CV↗
arXiv2026

Quantization of Contact 3-Manifolds and the Reeb Gravitational Field

We present a canonical geometric quantization scheme for closed contact 3-manifolds $(M,ξ)$ using the corresponding explicit embedding $M \hookrightarrow \C^3$ as we constructed in our earlier papers. The contact structure becomes holomorphic along the chosen link $L \subset M$ of complex tangents, of which we may also consider this link as the binding of a supporting open book for $ξ$. Stein extension yields a unique holomorphic line bundle $L_ξ\to \C^3$ whose restriction to $L$ will define the quantum Hilbert space $\Hilb_ξ= H^0(L, L_ξ|_L \otimes κ^{1/2})$, which we find will be finite-dimensional. The Reeb vector field $R_α$ of a chosen compatible contact form $α$ is shown to be geodesic with a time Killing Reeb field under further assumption that the manifold is Sasakian, and under this assumption will model Einstein gravity over $M$. This construction depends only on $(M,ξ,α)$ and a choice of supporting open book for $ξ$. This establishes a unified geometric framework in which the contact structure encodes quantum mechanics and whose Reeb field will encode gravity assuming that the manifold is Sasakian. In addition, we use my previous paper in which we get a related invariant $μ_M(ξ) = [L_ξ] \in \Pic_\C(M)$ which is useful in that it distinguishes between different tight contact structures on $\T^3$ in a novel way. We also show how this will have implications for the quantum model and serve as a quantum invariant there-in.

math.GT↗
arXiv2025

Explicit Holomorphic Structures for embeddings of closed 3-manifolds into $\mathbb{C}^3$

Expanding on my former work along with the more recent work of Kasuya and Takase, we demonstrate that for a given link $L \subset M$ which is null-homologous in $H_1(M)$ and for any smooth oriented 2-plane field $η$ over $L$ there exists a smooth embedding $F:M \hookrightarrow \mathbb{C}^3$ so that the set of complex tangents to the embedding is exactly $L$ and at each $x \in L$ the holomorphic tangent space is exactly $η_x$. Furthermore, we demonstrate how the "analyticity" of a complex tangent, as given by the Bishop invariant, may be determined exactly from the angle formed between the holomorphic complex line and the the curve of complex tangents.

math.CV↗
arXiv2015

A Topological Obstruction to the Removal of a Degenerate Complex Tangent and Some Related Homotopy and Homology Groups

In this article, we derive a topological obstruction to the removal of a isolated degenerate complex tangent to an embedding of a 3-manifold into $\mathbb{C}^3$ (without affecting the structure of the remaining complex tangents). We demonstrate how the vanishing of this obstruction is both a necessary and sufficient condition for the (local) removal of the isolated complex tangent. The obstruction is a certain homotopy class of the space $\mathbb{Y}$ consisting of totally real 3-planes in the Grassmanian of real 3-planes in $\mathbb{C}^3$ (=$\mathbb{R}^6$). We further compute additional homotopy and homology groups for the space $\mathbb{Y}$ and of its complement $\mathbb{W}$ consisting of "partially complex" 3-planes in $\mathbb{C}^3$.

math.CV↗
arXiv2015

On the Bishop Invariants of Embeddings of $S^3$ into $\mathbb{C}^3$

The Bishop invariant is a powerful tool in the analysis of real submanifolds of complex space that associates to every (non-degenerate) complex tangent of the embedding a non-negative real number (or infinity). It is a biholomorphism invariant that gives information regarding the local hull of holomorphy of the manifold near the complex tangent. In this paper, we derive a readily applicable formula for the computation of the Bishop invariant for graphical embeddings of 3-manifolds into $\mathbb{C}^3$. We then exhibit some examples over $S^3$ demonstrating the different possible configurations of the Bishop invariant along complex tangents to such embeddings. We will also generate a few more results regarding the behavior of the Bishop invariant in certain situations. We end our paper by analyzing the different possible outcomes from the perturbation of a degenerate complex tangent.

math.CV↗
arXiv2015

Complex Tangencies to Embeddings of Heisenberg Groups and Odd-Dimensional Spheres

The notion of a complex tangent arises for embeddings of real manifolds into complex spaces. It is of particular interest when studying embeddings of real $n$-dimensional manifolds into $\mathbb{C}^n$. The generic topological structure of the set complex tangents to such embeddings $M^n \hookrightarrow \mathbb{C}^n$ takes the form of a (stratified) $(n-2)$-dimensional submanifiold of $M^n$. In this paper, we generalize our results from our previous work for the 3-dimensional sphere and the Heisenberg group to obtain results regarding the possible topological configurations of the sets of complex tangents to embeddings of odd-dimensional spheres $S^{2n-1} \hookrightarrow \mathbb{C}^{2n-1}$ by first considering the situation for the higher dimensional analogues of the Heisenberg group.

math.CV↗
arXiv2015

On the Topological Structure Of Complex Tangencies to Embeddings of $S^3$ into $\mathbb{C}^3$

In the mid-1980's, M. Gromov used his machinery of the $h$-principle to prove that there exists totally real embeddings of $S^3$ into $\mathbb{C}^3$. Subsequently, Patrick Ahern and Walter Rudin explicitly demonstrated such a totally real embedding. In this paper, we consider the generic situation for such embeddings, namely where complex tangents arise as codimension-2 subspaces. We first consider the Heisenberg group $\mathbb{H}$ and generate some interesting results there-in. Then, by using the biholomorphism of $\mathbb{H}$ with the 3-sphere minus a point, we demonstrate that every homeomorphism-type of knot in $S^3$ may arise precisely as the set of complex tangents to an embedding $S^3 \hookrightarrow \mathbb{C}^3$. We also make note of the (non-generic) situation where complex tangents arise along surfaces.

math.CV↗
arXiv2015

Totally Real Perturbations and Non-Degenerate Embeddings of $S^3$

In this article, we demonstrate methods for the local removal and modification of complex tangents to embeddings of $S^3$ into $\mathbb{C}^3$. In particular, given any embedding of $S^3$ and a neighborhood of the complex tangents of the embedding, we show that there exists a ($C^0$-close) totally real embedding which agrees with the original embedding outside the given neighborhood of the complex tangents. We also demonstrate that given any knot type K in $S^3$, either there exists an embedding of $S^3$ which assumes non-degenerate complex tangents exactly along K or there exists a non-degenerate embedding complex tangent along two unlinked copies of K (both cases may hold). We also note possible directions of future investigations.

math.CV↗