arXiv subjects

Magdalena Toda

Publications and source records attributed to Magdalena Toda.

At least 19 recordsLinked to original sources

arXiv2026

Schwarz-Type Null Curves in $\mathbb{C}^4$: Symmetry and Period Reduction

We study a genus-three hyperelliptic holomorphic null-curve family in $\mathbb{C}^4$, modelled on the algebraic data of the classical Schwarz P/D family, whose real parts define minimal immersions into $\mathbb{R}^4$. For the order-four automorphism $\omega\mapsto i\omega$ of the underlying Schwarz curve, we compute explicitly its action on the four holomorphic Weierstrass $1$-forms and derive the resulting identities for all real period vectors. Consequently, for every lattice invariant under the induced target rotation, torus-period closure can be checked on one representative from each orbit of a symmetry-stable homology generating set. We also identify precisely when the additional parameter produces a nondegenerate codimension-two deformation. The paper does not claim the construction of a new embedded periodic minimal surface in $\mathbb{R}^4$; rather, it provides an explicit symmetry reduction of the period problem associated with this family.

math.GM↗

arXiv2026

Gauss Maps in Hyperbolic Surface Theory:A Unified Perspective

Immersed surfaces in hyperbolic three-space carry several natural Gauss-type maps with distinct geometric roles. The hyperbolic Gauss maps record the ideal endpoints of oriented normal geodesics; the Legendre Gauss lift retains the position-normal data and its contact structure; adjusted Gauss maps arise from gauge normalization and Iwasawa splitting in Weierstrass--Kenmotsu representations; and the conformal Gauss map encodes the mean-curvature sphere congruence in M\"obius geometry. We present these constructions in a common framework, emphasizing their target spaces, analytic properties, and mutual relations. Particular attention is given to the generalized DPW method, the necessity of flatness in adjusted rank-one data, and the harmonic-map characterization of Willmore surfaces. The resulting viewpoint distinguishes the asymptotic, contact, integrable, and conformal information carried by an immersed surface in $\mathbb{H}^{3}(-1)$.

math.DG↗

arXiv2026

The $r^\sharp$ invariant as a discriminant for the survival of the H-flux under T-duality on product manifolds

We show that the cohomological invariant $r^\sharp$, introduced in [1] as a lower bound for the off-diagonal holonomy dimension of metric connections with totally skew torsion on product manifolds, predicts the behaviour of the torsion $3$-form under both dimensional reduction and Buscher T-duality. On a product $M = \Sigma_g \times M_2$ equipped with a product metric, when $r^\sharp = 0$ the parallel-form strata identify a flat circle factor $S^1_\beta \subset M_2$ via the de Rham splitting theorem, and the entire $H$-flux is converted into geometric flux under T-duality along $S^1_\beta$ (the parallel regime); when $r^\sharp = 1$, no such circle factor exists, and the $H$-flux survives T-duality along every flat circle factor as $H$-flux in the dual background (the transversely non-reducible regime). When $M_2 = N \times T^k$ contains a torus factor, we prove that the Bouwknegt--Evslin--Mathai obstruction to successive T-dualities vanishes automatically for $H$-flux of pure bidegree $(2,1)$, that the resulting dualities are non-interfering and order-independent, and that $r^\sharp$ detects the \emph{irreducible kernel} of the $H$-flux: the component that survives T-duality along every flat circle factor and cannot be converted into geometric or non-geometric flux in any duality frame. This provides a metric refinement of topological T-duality: while the latter disregards the Riemannian metric entirely, $r^\sharp$ detects whether the cohomological coupling is aligned with the flat sub-factors identified by the Levi-Civita parallel-form strata.

math.DG↗

arXiv2026

A Weierstrass-Kenmotsu Type Representation for CMC $0\le H<1$ in \$\mathbb{H}^3(-1)$

We develop a Weierstrass-Kenmotsu type representation for conformal immersions of constant mean curvature $0\le H<1$ in hyperbolic $3$-space $\HH$. The construction is based on the Hermitian model of $\HH$, a balanced spectral deformation, and Iwasawa splitting of $\SL$. We show that such immersions arise locally from a rank-one $(1,0)$-form $η$ and a constant complex parameter $λ\in\C^*$ through a flat $\SL$-connection of the form \[ S^{-1}dS=η-λ\,η^*, \] with mean curvature \[ H=\frac{1-|λ|^2}{1+|λ|^2}. \] Conversely, every conformal CMC immersion with $0\le H<1$ is locally obtained from such flat rank-one data. We establish an explicit correspondence with the representation of Aiyama and Akutagawa via a gauge transformation, and interpret the construction in terms of Kokubu's adjusted normal Gauss map. We further discuss the role of the flatness condition, present simple local and cylindrical model examples, and outline aspects of monodromy and numerical implementation within this framework.

math.DG↗

arXiv2026

Explicit Minimal Surface Models in $\mathbb{R}^5$ via Holomorphic Null Curves

We study explicit conformal minimal immersions into $\mathbb{R}^5$ obtained from holomorphic null curves in $\mathbb{C}^5$. Although the general correspondence between conformal minimal immersions in $\mathbb{R}^n$ and holomorphic null data in $\mathbb{C}^n$ is classical, our aim here is different. We isolate the five-dimensional case and develop a concrete, self-contained account that emphasizes explicit formulas, integral-free constructions, and coordinate expressions suitable for computation and visualization. Starting from a Weierstrass-type representation in $\mathbb{R}^5$, we derive a family of conformal minimal immersions depending on a single holomorphic seed function and two real parameters. The resulting formulas allow the immersion and the induced metric to be written in closed form. We then examine polynomial seeds in detail, derive their polar and Cartesian expansions, and discuss the geometric information carried by natural coordinate projections. We reinterpret the construction in the language of moving frames, the generalized Gauss map, and a local DPW-type scheme. This provides a conceptual bridge between explicit holomorphic formulas and the Cartan-integrable-systems viewpoint. The discussion is local and formula-driven; global questions such as periods, completeness, and embeddedness lie beyond the present scope. We also briefly clarify why the complex-analytic structure underlying the representation is essential, and why it cannot be replaced by a naive quaternionic formalism, due to the loss of commutativity, holomorphic structure, and compatibility with the null-curve framework.

math.DG↗

arXiv2025

Local Topological Constraints on Berry Curvature in Spin--Orbit Coupled BECs

We establish a local topological obstruction to flattening Berry curvature in spin-orbit-coupled Bose-Einstein condensates (SOC BECs), valid even when the global Chern number vanishes. For a generic two-component SOC BEC, the extended parameter space $M=T^2_{BZ}\times S^1_{\phi_+}\times S^1_{\phi_-}$ carries a Kaluza-Klein metric $g_M$ and a natural metric connection $\nabla^C$ whose torsion 3-form encodes the synthetic gauge fields. Its harmonic part defines a mixed cohomology class in $(H^2(T^2_{BZ})\otimes H^1(S^1_{\phi_+}))\oplus(H^2(T^2_{BZ})\otimes H^1(S^1_{\phi_-}))$ of mixed tensor rank one. Adapting the Pigazzini-Toda lower bound to the Kaluza-Klein setting through exact pointwise curvature analysis (constant Berry curvatures), we show that the obstruction kernel vanishes and obtain a three-level non-reducibility structure for the physical metric: (i) for the one-parameter family interpolating between the product and physical metrics, $\dim\mathfrak{hol}^{\mathrm{off}}(\nabla^{C_\varepsilon})\ge1$ at every point for all $\varepsilon\in(0,1)$; (ii) at the physical metric, every non-Bismut torsion representative of $[\omega]$ yields $\dim\mathfrak{hol}^{\mathrm{off}}\ge1$ on an open set; (iii) the horizontal-vertical splitting is not invariant under the Riemannian holonomy of the physical metric, with $\dim\mathfrak{hol}^{\mathrm{off}}(\nabla^{\mathrm{LC}})\ge1$ at every point. These bounds prevent the complete gauging-away of Berry phases even at zero net topological charge. The corrected rank $r^\sharp$ detects the robustness of the constraint under phase-reduction protocols: no single phase-locking can eliminate the obstruction, a distinction invisible to the mixed rank $r$ alone. This provides the first cohomological lower bound certifying locally irremovable curvature in SOC BECs beyond the Chern-number paradigm.

math.DG↗

arXiv2025

The higher-order Henneberg-type minimal surfaces family in $\mathbb{R}^4$

We consider a higher-order Henneberg-type minimal surfaces family using the generalized Weierstrass--Enneper representation in four-dimensional space $\mathbb{R}^4$. We derive explicit parametric equations for the surface and determine its differential geometric characteristics, including the normal vector fields $\mathbf{n}_1$ and $\mathbf{n}_2$, as well as the Gauss curvature. Furthermore, by projecting these parametric forms from four to three dimensions, we generate visualizations that reveal the geometric structure of the Henneberg-type minimal surface. In addition, we examine the integral-free form and derive the corresponding algebraic function for this family of surfaces.

math.DG↗

arXiv2025

Cohomologically Calibrated Affine Connections and Forced Irreducibility

We establish a principle of forced geometric irreducibility on product manifolds. We prove that for any product manifold $M=M_1\times M_2$, a cohomologically calibrated affine connection, $\nabla^{\mathcal{C}}$, is necessarily holonomically irreducible, provided its calibration class $[ω] \in H^3(M;\mathbb{R})$ is mixed. The core of the proof relies on Hodge theory; we show that the algebraic structure of the harmonic part of the torsion generates non-zero off-diagonal components in the full Riemann curvature tensor, which cannot be globally cancelled. This non-cancellation is formally proven via an integral argument. We illustrate the main theorem with explicit constructions on $S^2\times Σ_g$, showing that this result holds even in special cases where the Ricci tensor is diagonal, such as the Einstein-calibrated connection. Finally, we briefly discuss speculative analogies between forced irreducibility and quantum entanglement.

math.DG↗

arXiv2025

Cohomological Calibration and Curvature Constraints on Product Manifolds: A Topological Lower Bound

We establish a quantitative relationship between mixed de Rham classes and the geometric complexity of metric connections with totally skew torsion on product manifolds where both factors are compact oriented surfaces. For any cohomologically calibrated connection $\nabla^C$ whose torsion $T$ has pure bidegree with respect to the product decomposition and whose harmonic projection represents a non-trivial mixed class $[\omega]$, we prove that on a non-empty open subset $\mathcal{V} \subset M$, \[ \dim\bigl(\mathfrak{hol}_p^{\mathrm{off}}(\nabla^{C})\bigr)\;\geq\; r^\sharp\;:=\;\operatorname{rank}_{\mathbb{R}}\bigl([\omega]_{\mathrm{mixed}}\bigr)-\dim\mathcal{K}, \] with $\mathcal{K}$ an intrinsically defined obstruction space. The bound is a topological invariant under metric deformations preserving the parallel-form strata and provides an obstruction to the reduction of the holonomy along the product splitting $V_1\oplus V_2$. A counterexample shows the hypothesis is optimal. When the second factor contains a circle factor, we further show that $r^{\sharp}=1$ forces the torsion to survive dimensional reduction along it, so that the failure of the holonomy to preserve the product splitting persists on the reduced product; the mixed rank alone cannot detect this.

math.DG↗

arXiv2025

Cohomologically calibrated affine connections and the Einstein condition on $S^2 \times T^2$

This paper applies the recently developed framework of cohomologically calibrated affine connections to the fundamental problem of constructing non-Riemannian Einstein manifolds. In this framework, the torsion of a connection is intrinsically related to the global topology of the manifold, represented by the de Rham cohomology class specified by a set of real parameters. We focus on the product manifold $S^2 \times T^2$, whose third cohomology group is $H^3(S^2 \times T^2; \mathbb{R}) \cong \mathbb{R}^2$. We analyze how the geometry is modeled by the choice of the torsion tensor $T$ within the family $\mathcal{T}_ω$, defined by the property that each member of this family must have an associated 3-form $T^\flat$ such that it represents the nontrivial cohomology class via Hodge decomposition. Our analysis reveals a dependence on this choice. First, we show that using a torsion tensor, which produces a strictly positive biorthogonal curvature, leads to a non-diagonal Ricci tensor, creating a structural obstacle to any Einstein solution. Conversely, we then show that using the torsion tensor associated with the purelly harmonic 3-form allow us the construction of an explicit non-Riemannian Einstein solution. Our work thus demonstrates that the cohomologically calibrated affine connections allow a family of feasible connections rich enough to allow for several geometries intrinsically justified by the differential topology of the manifold.

math.DG↗

arXiv2025

Red Blood Cells as Elastic Surfaces

We study red blood cells using the Helfrich-Canham functional: due to their lipid bilayer structure, RBCs are naturally modeled using the theory of elastic surfaces. In this study, we demonstrate that Cassinian ovals, except for the limiting case of the round sphere, do not solve the shape equation. We further discuss conditions under which they may serve as effective approximations.

math.DG↗

arXiv2024

Einstein warped-product manifolds and the screened Poisson equation

We study a particular type of Einstein warped-product manifold where the warping function must satisfy the homogeneous version of the screened Poisson equation. Under these assumptions, we show that the dimension of the manifold, the (constant negative) Ricci curvature and the screened parameter are related through a quadratic equation.

math.DG↗

arXiv2022

Instability of Closed $p$-Elastic Curves in $\mathbb{S}^2$

For $p\in\mathbb{R}$, we show that non-circular closed $p$-elastic curves in $\mathbb{S}^2$ exist only when $p=2$, in which case they are classical elastic curves, or when $p\in(0,1)$. In the latter case, we prove that for every pair of relatively prime natural numbers $n$ and $m$ satisfying $m<2n<\sqrt{2}\,m$, there exists a closed spherical $p$-elastic curve with non-constant curvature which winds around a pole $n$ times and closes up in $m$ periods of its curvature. Further, we show that all closed spherical $p$-elastic curves for $p\in(0,1)$ are unstable as critical points of the $p$-elastic energy.

math.DG↗

arXiv2021

On p-Willmore Disks with Boundary Energies

We consider an energy functional on surface immersions which includes contributions from both boundary and interior. Inspired by physical examples, the boundary is modeled as the center line of a generalized Kirchhoff elastic rod, while the interior term is arbitrarily dependent on the mean curvature and linearly dependent on the Gaussian curvature. We study equilibrium configurations for this energy in general among topological disks, as well as specifically for the class of examples known as p-Willmore energies.

math.DG↗

arXiv2021

Stationary Surfaces with Boundaries

This article investigates stationary surfaces with boundaries, which arise as the critical points of functionals dependent on curvature. Precisely, a generalized "bending energy" functional $\mathcal{W}$ is considered which involves a Lagrangian that is symmetric in the principal curvatures. The first variation of $\mathcal{W}$ is computed, and a stress tensor is extracted whose divergence quantifies deviation from $\mathcal{W}$-criticality. Boundary-value problems are then examined, and a characterization of free-boundary $\mathcal{W}$-surfaces with rotational symmetry is given for scaling-invariant $\mathcal{W}$-functionals. In case the functional is not scaling-invariant, certain boundary-to-interior consequences are discussed. Finally, some applications to the conformal Willmore energy and the p-Willmore energy of surfaces are presented.

math.DG↗

arXiv2020

Regarding the Euler-Plateau Problem with Elastic Modulus

We study equilibrium configurations for the Euler-Plateau energy with elastic modulus, which couples an energy functional of Euler-Plateau type with a total curvature term often present in models for the free energy of biomembranes. It is shown that the potential minimizers of this energy are highly dependent on the choice of physical rigidity parameters, and that the area of critical surfaces can be computed entirely from their boundary data. When the elastic modulus does not vanish, it is shown that axially symmetric critical immersions and critical immersions of disk type are necessarily planar domains bounded by area-constrained elasticae. The cases of topological genus zero with multiple boundary components and unrestricted genus with control on the geodesic torsion are also discussed, and sufficient conditions are given which establish the same conclusion in these cases.

math.DG↗

arXiv2020

On the variation of curvature functionals in space forms with application to a generalized Willmore energy

Functionals involving surface curvature are important across a range of scientific disciplines, and their extrema are representative of physically meaningful objects such as atomic lattices and biomembranes. Inspired in particular by the relationship of the Willmore energy to lipid bilayers, we consider a general functional depending on a surface and a symmetric combination of its principal curvatures, provided the surface is immersed in a 3-D space form. We compute the first and second variations of this functional, leading to expressions given entirely in terms of the surface fundamental forms. We then apply the stability criteria afforded by our calculations to a generalization of the Willmore functional, proving a result regarding the stability of spheres.

math.DG↗

arXiv2019

Geometric model of the fracture as a manifold immersed in porous media

In this work, we analyze the flow filtration process of slightly compressible fluids in porous media containing man made fractures with complex geometries. We model the coupled fracture-porous media system where the linear Darcy flow is considered in porous media and the nonlinear Forchheimer equation is used inside the fracture. We develop a model to examine the flow inside fractures with complex geometries and variable thickness, on a Riemannian manifold. The fracture is represented as the normal variation of a surface immersed in $\mathbb{R}^3$. Using operators of Laplace Beltrami type and geometric identities, we model an equation that describes the flow in the fracture. A reduced model is obtained as a low dimensional BVP. We then couple the model with the porous media. Theoretical and numerical analysis have been performed to compare the solutions between the original geometric model and the reduced model in reservoirs containing fractures with complex geometries. We prove that the two solutions are close, and therefore, the reduced model can be effectively used in large scale simulators for long and thin fractures with complicated geometry.

math.AP↗