arXiv subjects
Publications and source records attributed to Pradip Kumar.
This article investigates the global structure of maximal surfaces (space-like immersions with zero mean curvature) in Lorentz-Minkowski $3$-space $\mathbb{E}^3_1$, especially focusing on the interplay between genus, the number of singular components--loci, and simple ends. We construct complete maximal maps with arbitrarily many singular components for any genus $p\geq 0$ with simple ends.
Let $S_g$ be a closed oriented surface of genus $g\ge2$. For a reductive representation $\rho:\pi_1(S_g)\to\PU(2,1)$, let $E_\rho$ be the energy function on Teichm\"uller space associated to equivariant harmonic maps into $\CH^2$. For every positive integer $d$ with $3\nmid d$, all sufficiently large $h$, and every $g>h$, we construct an irreducible reductive representation \[ \rho_{g,h,d}:\pi_1(S_g)\to\PU(2,1) \] with \[ \tau(\rho_{g,h,d})=2h-2-\frac{2d}{3}\notin\mathbb Z, \qquad \operatorname{Crit}(E_{\rho_{g,h,d}})=\varnothing. \] Consequently, the associated branched-minimal-surface forgetful map is not surjective in these nonintegral Toledo components.
For a regular semisimple Higgs bundle with a smooth spectral curve, we prove that, over the \etale\ locus, the Atiyah class of the underlying bundle is induced by the Atiyah class of the spectral line bundle and takes values in the centralizer of the Higgs field. Further, when the discriminant is reduced, we construct a logarithmic refinement across the branch divisor: the Atiyah class extends as a class with logarithmic poles and values in a natural regularized centralizer sheaf.
We extend Atiyah's holomorphic jet bundle formalism to holomorphic vector bundles over noncommutative algebras endowed with a bigraded differential calculus truncated at bidegree $(1,1)$; such structures are referred to as noncommutative complex curves. For a holomorphic vector bundle $(E,\,\overline{\nabla}_E)$ over such an algebra $\mathcal{A}$, we construct a canonical holomorphic structure $\overline{\nabla}_J$ on the first jet module $J_E^1\,$, making the jet sequence \[ 0\longrightarrow \Omega^{1,0}(\mathcal{A})\otimes_{\mathcal A}E \longrightarrow J_E^1 \longrightarrow E \longrightarrow 0 \] exact in the holomorphic category. The assignment $(E,\,\overline\nabla_E)\,\rightsquigarrow\,(J_E^1,\,\overline\nabla_J)$ defines an endofunctor on the category of holomorphic vector bundles over $\mathcal{A}$. We define the notion of holomorphic connection in this setting and prove that a holomorphic vector bundle admits a holomorphic connection if and only if the above jet sequence splits in the holomorphic category, or equivalently, if and only if its Atiyah class vanishes. This yields a noncommutative analogue of Atiyah's classical correspondence for Riemann surfaces. Finally, we specialize to the quantum projective line $\mathbb{CP}_q^1\,$ and determine when $\overline{\nabla}_J$ defines a bimodule connection, assuming that $\overline{\nabla}_E$ does so.
Let $X$ be a smooth projective complex curve, $P\subset X$ a reduced effective divisor, and $X^{0}=X\setminus P$. We study logarithmic $V$-twisted Higgs bundles arising from a logarithmic Hecke compactification of a rank-two bundle on $X^{0}$. We show that a pair of induced logarithmic line-twisted fields lifts uniquely exactly under explicit local Hecke conditions, and that the lift is integrable precisely when the fields commute. Fixing the compactified spectral curve $Y$, we classify such Higgs bundles by pairs $(F,\,\vartheta)$, where $F$ is a rank-one torsion-free sheaf on $Y$ and $\vartheta$ satisfies a marked spectral condition on a finite subscheme $Z\subset Y$. This gives a logarithmic extension of the compact rank-two spectral correspondence of~\cite{ABK} to the punctured case. On the line-bundle locus, the moduli stack is canonically equivalent to $\mathrm{Pic}^{d}(Y)\times A_Z$.
We develop an abstract framework for coupled period--realization of meromorphic $1$--forms on punctured Riemann surfaces. A configuration datum $C$ gives the combinatorics and determines a restricted character domain $\Delta_C\subset\mathrm{Hom}(\Gamma_{g,n},{\mathbb C})^2$ with a scale--fixed slice $\Delta_C^{\mathrm{sc}}$. Assuming Teichm\"uller--regularity, degeneration detection, and pushability, we prove that there is point in $\Delta_C^{\mathrm{sc}}$ which corresponds to a surface carrying two meromorphic differentials realizing any prescribed restricted pair. This abstracts the Weber--Wolf extremal--length minimization method while constructing minimal surfaces.
Given a noncompact Riemann surface $\Sigma_0\,=\, \Sigma \setminus P$, where $P$ is a finite subset of a compact connected Riemann surface $\Sigma$, and a reductive representation $\rho\,:\,\pi_1(\Sigma_0)\,\longrightarrow\, \mathrm{PU}(2,1)$, we prove that any finite--energy $\rho$--equivariant conformal minimal immersion is proper around every cusp if and only if the peripheral holonomy of $\rho$ is parabolic. Assuming parabolic peripheral holonomy, we give an explicit parametrization of complete finite--energy immersions in the mixed case in terms of tame parabolic $\mathrm{PU}(2,1)$--Higgs bundles with nilpotent residues and satisfying concrete parabolic slope inequalities. We also discuss complete ends and construct explicit families of $\rho$ equivariant proper $\mathbb{CH}^2$ $n$--noids on $\mathbb{CP}^1\setminus P$ for $|P|\,\ge\, 5$.
Take a holomorphic Lie algebroid $(V,\, \phi)$ on a compact connected Riemann surface $X$ such that the anchor map $\phi$ is not surjective. Let $P$ be a parabolic subgroup of a complex reductive affine algebraic group $G$ and $E_P\, \subset\, E_G$ a holomorphic reduction of structure group, to $P$, of a holomorphic principal $G$--bundle $E_G$ on $X$. We prove that $E_P$ admits a holomorphic Lie algebroid connection for $(V,\,\phi)$ if the reduction $E_P$ is infinitesimally rigid. If $E_P$ is the Harder--Narasimhan reduction of $E_G$, then it is shown that $E_P$ admits a holomorphic Lie algebroid connection for $(V,\,\phi)$. In particular, for any point $x_0\,\in\, X$, the Harder--Narasimhan reduction $E_P$ admits a logarithmic connection that is nonsingular on the complement $X\setminus\{x_0\}$.
It has recently been observed that, in contrast to the classical case, holomorphic structures on line bundles over the quantum projective line are not uniquely determined by degree. We formulate a fixed-point-theoretic framework for the analysis of flat $\bar\partial$-connections that define holomorphic structures on line bundles over the quantum projective line. Within this framework, we relate the existence of invertible solutions to the gauge equation associated with holomorphic structures precisely to the existence of fixed points, lying in the open unit ball, of certain nonlinear maps acting on an appropriate Banach space.
We prove the existence of complete minimal surfaces in $\mathbb{R}^3$ of arbitrary genus $p\, \ge\, 1$ and least total absolute curvature with precisely two ends -- one catenoidal and one Enneper-type -- thereby solving, affirmatively, a problem posed by Fujimori and Shoda. These surfaces, which are called \emph{Angel surfaces}, generalize some examples numerically constructed earlier by Weber. The construction of these minimal surfaces involves extending the orthodisk method developed by Weber and Wolf \cite{weber2002teichmuller}. A central idea in our construction is the notion of \emph{partial symmetry}, which enables us to introduce controlled symmetry into the surface.
Given a holomorphic Lie algebroid $(V, \phi)$ on a compact connected Riemann surface $X$, we give a necessary and sufficient condition for a holomorphic vector bundle $E$ on $X$ to admit a holomorphic Lie algebroid connection. If $(V, \phi)$ is nonsplit, then every holomorphic vector bundle on $X$ admits a holomorphic Lie algebroid connection for $(V, \phi)$. If $(V, \phi)$ is split, then a holomorphic vector bundle $E$ on $X$ admits a holomorphic Lie algebroid connection if and only if the degree of each indecomposable component of $E$ is zero.
For any V-twisted Higgs bundle on a compact Riemann surface X, where V is a holomorphic vector bundle of rank two on X, there are two associated Higgs bundles on X, twisted by line bundles, which are constructed using a Hecke transformation on V. We characterize all such pairs of Higgs bundles (twisted by line bundles) given by V-twisted Higgs bundles. Using this characterization, we provide a spectral correspondence for the moduli space, identifying V-twisted Higgs bundles with the direct images of certain rank one torsionfree Higgs sheaves twisted by a line bundle on a spectral covering of the curve X.
We study the optimal green hydrogen production and energy market participation of a renewable-colocated hydrogen producer (RCHP) that utilizes onsite renewable generation for both hydrogen production and grid services. Under deterministic and stochastic profit-maximization frameworks, we analyze RCHP's multiple market participation models and derive closed-form optimal scheduling policies that dynamically allocate renewable energy to hydrogen production and electricity export to the wholesale market. Analytical characterizations of the RCHP's operating profit and the optimal sizing of renewable and electrolyzer capacities are obtained. We use real-time renewable generation and electricity price data from three independent system operators to evaluate the impacts of market prices and environmental policies on RCHP's profitability.
Given a holomorphic Lie algebroid on an m-pointed Riemann surface, we define parabolic Lie algebroid connections on any parabolic vector bundle equipped with parabolic structure over the marked points. An analogue of the Atiyah exact sequence for parabolic Lie algebroids is constructed. For any Lie algebroid whose underlying holomorphic vector bundle is stable, we give a complete characterization of all the parabolic vector bundles that admit a parabolic Lie algebroid connection.
Let $X$ be a compact connected Riemann surface and $(V, \phi)$ a holomorphic Lie algebroid on $X$ such that the holomorphic vector bundle $V$ is stable. We give a necessary and sufficient condition on holomorphic vector bundles $E$ on $X$ to admit a Lie algebroid connection.
The node-opening technique, originally designed for constructing minimal surfaces, is adapted to construct a rich variety of new maxfaces of high genus that are embedded outside a compact set and have arbitrarily many catenoid or planar ends, thus removing the scarcity of examples of maxfaces. The surfaces look like spacelike planes connected by small necks. Among the examples are maxfaces of the Costa--Hoffman--Meeks type. Although very fruitful, the main challenge of this paper is not the construction itself, but the analysis of the positions and natures of singularities on these maxfaces. More specifically, we conclude that the singular set form curves around the waists of the necks. In generic and some symmetric cases, all but finitely many singularities are cuspidal edges, and the non-cuspidal singularities are swallowtails evenly distributed along the singular curves.
We have proven the existence of new higher-genus maxfaces with Enneper end. These maxfaces are not the companions of any existing minimal surfaces, and furthermore, the singularity set is located away from the ends. The nature of the singularities is systematically investigated.
We prove the existence of a genus-zero complete maximal map with a prescribed singularity set and an arbitrary number of simple and complete ends. We also discuss the conditions under which this maximal map can be made into a complete maxface.