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Rukmini Dey

Publications and source records attributed to Rukmini Dey.

At least 19 recordsLinked to original sources
arXiv2026

Decompositions of ZMC Graphs and Euler-Ramanujan type identities

In this paper, we study finite and infinite decomposition formulas for zero mean curvature (ZMC) graphs in Euclidean, Lorentz--Minkowski, and isotropic (3)-spaces. We first derive new Euler--Ramanujan-type identities that decompose the conjugate of Scherk's first minimal surface into dilated catenoids. Using Weierstrass factorisation and power series methods, we then obtain infinite decompositions for a broad class of isotropic ZMC graphs into helicoids, logarithmoids of revolution, and Enneper surfaces. These results are extended to wider families of ZMC surfaces arising from the L\'opez--Ros transformation, Bonnet rotation, and a one-parameter family of metric deformations. We also establish finite decomposition formulas, including analogues of Scherk tower decompositions in Euclidean and isotropic settings, and prove a characterisation theorem for finite decompositions of isotropic minimal surfaces. Finally, we discuss applications to lamellar structures.

math.DG↗
arXiv2024

CMC surfaces of revolution, Elliptic curves, Weierstrass-$\wp$ functions, and Algebraicity

This paper establishes an interesting connection between the family of CMC surfaces of revolution in $\mathbb E_1^3$ and some specific families of elliptic curves. As a consequence of this connection, we show in the class of spacelike CMC surfaces of revolution in the $\mathbb E_1^3$, only spacelike cylinders and standard hyperboloids are algebraic. We also show that a similar connection exists between CMC surfaces of revolution in $\mathbb E^3$ and elliptic curves. Further, we use this to reestablish the fact that the only CMC algebraic surfaces of revolution in $\mathbb E^3$ are spheres and right circular cylinders.

math.DG↗
arXiv2024

RKHS, Berezin and Odzijewicz-type quantizations on arbitrary compact smooth manifold

In this article we define Berezin-type and Odzijewicz-type quantizations on compact smooth manifolds. The method is we embed the smooth manifold of real dimension $n$ into ${\mathbb C}P^n$ and induce the quantizations from there. The standard way by which reproducing kernel Hilbert spaces are defined on submanifolds gives a way to define the pullback coherent states. In Berezin-type quantization the Hilbert space of quantization is the pullback (by the embedding) of the Hilbert space of geometric quantization of ${\mathbb C}P^n$. In the Odzijewicz-type quantization one has to consider a tensor product of the geometric quantization line bundle with holomorphic $n$-forms. In the Berezin case, the operators that are quantized are those induced from the ambient space ${\mathbb C}P^n$. The Berezin-type quantization exhibited here is a generalization of an earlier work of the author and Ghosh. In both Berezin and Odzijewicz-type quantizations we first exhibit this quantization explicitly on ${\mathbb C}P^n$ and we induce the quantization on the smooth compact embedded manifold from ${\mathbb C}P^n$.

math-ph↗
arXiv2023

Berezin-type quantization on even-dimensional compact manifolds

In this article we show that a Berezin-type quantization can be achieved on a compact even dimensional manifold $M^{2d}$ by removing a skeleton $M_0$ of lower dimension such that what remains is diffeomorphic to $R^{2d}$ (cell decomposition) which we identify with $C^d$ and embed in $ CP^d$. A local Poisson structure and Berezin-type quantization are induced from $ CP^d$. Thus we have a Hilbert space with a reproducing kernel. The symbols of bounded linear operators on the Hilbert space have a star product which satisfies the correspondence principle outside a set of measure zero. This construction depends on the diffeomorphism. One needs to keep track of the global holonomy and hence the cell decomposition of the manifold. As an example, we illustrate this type of quanitzation of the torus. We exhibit Berezin-Toeplitz quantization of a complex manifold in the same spirit as above.

math-ph↗
arXiv2022

Coadjoint orbits and Kähler structure: examples from coherent states

Do co-adjoint orbits of Lie groups support a Kähler structure? We study this question from a point of view derived from coherent states. We examine three examples of Lie groups: the Weyl-Heisenberg group, $\mathrm{SU(2)}$ and $\mathrm{SU(1,1)}$. In cases, where the orbits admit a Kähler structure, we show that coherent states give us a Kähler embedding of the orbit into projective Hilbert space. In contrast, squeezed states, (which like coherent states, also saturate the uncertainty bound) only give us a symplectic embedding. We also study geometric quantisation of the co-adjoint orbits of the group $\mathrm{SUT(2,\mathbb{R})}$ of real, special, upper triangular matrices in two dimensions. We glean some general insights from these examples. Our presentation is semi-expository and accessible to physicists.

math-ph↗
arXiv2022

Pullback Coherent States, Squeezed States and Quantization

In this semi-expository paper, we define certain Rawnsley-type coherent and squeezed states on an integral Kähler manifold (after possibly removing a set of measure zero) and show that they satisfy some properties which are akin to maximal likelihood property, reproducing kernel property, generalised resolution of identity property and overcompleteness. This is a generalization of a result by Spera. Next we define the Rawnsley-type pullback coherent and squeezed states on a smooth compact manifold (after possibly removing a set of measure zero) and show that they satisfy similar properties. Finally we show a Berezin-type quantization involving certain operators acting on a Hilbert space on a compact smooth totally real embedded submanifold of $U$ of real dimension $n$, where $U$ is an open set in ${\mathbb C}{\rm P}^n$. Any other submanifold for which the criterion of the identity theorem holds exhibit this type of Berezin quantization. Also this type of quantization holds for totally real submanifolds of real dimension $n$ of a general homogeneous Kähler manifold of real dimension $2n$ for which Berezin quantization exists. In the appendix we review the Rawnsley and generalized Perelomov coherent states on ${\mathbb C}{\rm P}^n$ (which is a coadjoint orbit) and the fact that these two types of coherent states coincide.

math.DG↗
arXiv2021

Local interpolation for minimal surfaces

Let $a: I\to \mathbb{R}^3 $ be a real analytic curve satisfying some conditions. In this article, we show that for any real analytic curve $l:I\to \mathbb R^3$ close to $a$ (in a sense which is precisely defined in the paper) there exists a translation of $l$, and a minimal surface which contains both $ a $ and the translated $l$.

math.DG↗
arXiv2021

Finite Decomposition of Minimal surfaces, Maximal surfaces, Timelike Minimal surfaces and Born-Infeld solitons

We show that the height function of Scherk's second surface decomposes into a finite sum of scaled and translated versions of itself, using an Euler Ramanujan identity. A similar result appears in R. Kamien's work on liquid crystals where he shows (using an Euler-Ramanujan identity) that the Scherk's first surface decomposes into a finite sum of scaled and translated versions of itself. We give another finite decomposition of the height function of the Scherk's first surface in terms of translated helicoids and scaled and translated Scherk's first surface. We give some more examples, for instance a (complex) maximal surface and a (complex) BI soliton. We then show, using the Weierstrass-Enneper representation of minimal (maximal) surfaces, that one can decompose the height function of a minimal (maximal) surface into finite sums of height functions of surfaces which, upon change of coordinates, turn out to be minimal (maximal) surfaces, each minimal (maximal) w.r.t. to its own new coordinates. We then exhibit a general property of minimal surfaces, maximal surfaces, timelike minimal surfaces and Born-Infeld soliton surfaces that their local height functions $z=Z(x,y)$ split into finite sum of scaled and translated versions of functions of the same form. Upto scaling these new functions are height functions of the minimal surfaces, maximal surfaces, timelike minimal surfaces and Born-Infeld soliton surfaces respectively. Lastly, we exhibit a foliation of ${\mathbb R}^3$ minus certain lines by shifted helicoids (which appear in one of the Euler-Ramanujan identities).

math.DG↗
arXiv2021

Interpolation by maximal and minimal surfaces

In this article, we use the inverse function theorem for Banach spaces to interpolate a given real analytic spacelike curve $a$ in Lorentz-Minkowski space $\mathbb{L}^3$ to another real analytic spacelike curve $c$, which is ``close" enough to $a$ in a certain sense by constructing a maximal surface containing them. Next we apply the same method to interpolate two given real analytic curve $a$ in Euclidean space $\mathbb{E}^3$ and a real analytic curve $c$, which is also ``close" enough to ``a" in a certain sense with a minimal surface. Throughout this study, the Bj\"orling problem and Schwarz's solution to it play pivotal roles.

math.DG↗
arXiv2019

On Euler-Ramanujan formula, Dirichlet series and minimal surfaces

In this paper, we rewrite two forms of an Euler-Ramanujan identity in terms of certain Dirichlet series and derive functional equation of the latter. We also use the Weierstrass-Enneper representation of minimal surfaces to obtain some identities involving these Dirichlet series and one complex parameter.

math.NT↗
arXiv2017

Geometric quantization of finite Toda systems and coherent States

Adler had shown in 1979 that the Toda system can be given a coad- joint orbit description. We quantize the Toda system by viewing it as a single orbit of a multiplicative group of lower triangular matrices of determinant one with pos- itive diagonal entries. We get a unitary representation of the group with square integrable polarized sections of the quantization as the module . We find the Rawnsley coherent states after a completion of the above space of sections. We also find non-unitary finite dimensional quantum Hilbert spaces for the system.

math.DG↗
arXiv2017

Existence of maximal surface containing given curve and special singularity

We give a different formulation for describing maximal surfaces in Lorentz-Minkowski space, $\mathbb{L}^3$, using the identification of $\mathbb L^3$ with $\mathbb C\times \mathbb R$. Further we give a different proof for the singular Björling problem for the case of closed real analytic null curve. As an application, we show the existence of maximal surface which contains a given curve and has a special singularity.

math.DG↗
arXiv2017

Geometric Quantization of the Hitchin System

This paper is about geometric quantization of the Hitchin system. We quantize a Kahler form on the Hitchin moduli space (which is half the first Kahler form defined by Hitchin) by considering the Quillen bundle as the prequantum line bundle and modifying the Quillen metric using the Higgs field so that the curvature is proportional to the Kahler form. We show that this Kahler form is integral and the Quillen bundle descends as a prequantum line bundle on the moduli space. It is holomorphic and hence one can take holomorphic square integrable sections as the Hilbert space of quantization of the Hitchin moduli space.

math.DG↗
arXiv2017

Born-Infeld solitons, Maximal surfaces and Ramanujan's identities

We show that a Born-Infeld soliton can be realised either as a spacelike minimal graph or timelike minimal graph over a timelike plane or a combination of both away from singular points. We also obtain some exact solutions of the Born-Infeld equation from already known solutions to the maximal surface equation. Further we present a method to construct a one-parameter family of complex solitons from a given one parameter family of maximal surfaces. Finally, using Ramanujan's Identities and the Weierstrass-Enneper representation of maximal surfaces, we derive further non-trivial identities.

math.DG↗
arXiv2016

Generalized Seiberg-Witten equations on Riemann surface

In this paper we consider twice-dimensionally reduced, generalized Seiberg-Witten equations, defined on a compact Riemann surface. A novel feature of the reduction technique is that the resulting equations produce an extra "Higgs field". Under suitable regularity assumptions, we show that the moduli space of gauge-equivalent classes of solutions to the reduced equations, is a smooth Kahler manifold and construct a pre-quantum line bundle over the moduli space of solutions.

math.DG↗