arXiv subjects

Alexander Pigazzini

Publications and source records attributed to Alexander Pigazzini.

At least 19 recordsLinked to original sources

arXiv2026

Yau's conjecture for the stacked Clifford tori of Wiygul

We prove Yau's conjecture $\lambda_{1}=2$ for the stacked Clifford tori of Wiygul: for all integers $N\ge2$, $k,\ell\ge1$ and every sufficiently large $m$, the closed embedded minimal surface of genus $k\ell m^{2}(N-1)+1$ in the round three-sphere which resembles $N$ parallel copies of the Clifford torus joined by small catenoidal tunnels has first Laplace eigenvalue $2$. For $N\ge3$ these surfaces are chains rather than doublings, and the even--odd decomposition on which all previous verifications for gluing constructions rest is not available. The reflection lemma of Choe and Soret reduces the problem to the sector of functions invariant under the symmetry group of the construction, and we show that the lowest nonzero eigenvalue of that sector equals $4+O(m^{-1})$. The value $4$ is the outcome of an exact identity: the limiting waist ratios of the construction form the Perron vector of the adjacency operator of the line graph of a path, so that the spectral gap of the path cancels against the total conductance of the tunnels prescribed by the balancing conditions, and what survives is the coefficient of the Jacobi operator of the Clifford torus. The analytic input consists of a conformally invariant channel inequality on a cylinder and of a Poincar\'e inequality on a perforated torus. The properties of the construction on which the argument rests are isolated in a reduction theorem for closed surfaces decomposed into blocks joined by families of thin channels along the edges of a finite graph, subject to a symmetry assumption and to Poincar\'e and trace inequalities on the blocks.

math.DG↗

arXiv2026

A degenerate free boundary minimal annulus and non-uniqueness in spherical caps beyond the hemisphere

Let $\{\Sigma_a\}$, $a\in(0,1/2)$, be de Oliveira's family of embedded free boundary minimal annuli of revolution in geodesic balls $B(R(a))\subset\mathbb{S}^3$, $R(a)>\pi/2$. We prove that $R$ is real-analytic, tends to $\pi/2$ at both ends, and therefore folds: it has an interior maximum $R_*>\pi/2$ and is not injective. Hence each $B(\rho)$ with $\pi/2<\rho \pi/2$. The exact identity $\dim K_0^{ev}(\Sigma_a)=\mathbf{1}_{\{R'=0\}}(a)$ detects the degeneration; it follows from the symmetry-free relation $\partial_\eta\varphi_a-\operatorname{ct}_\kappa(r(a))\varphi_a=r'(a)A_a(\eta,\eta)$, a Robin defect identity, which requires of the ambient only that the barrier family be umbilic and which extends to capillary boundary conditions at constant contact angle.

math.DG↗

arXiv2026

First Chen Inequality for CR-Warped Product Submanifolds of a Complex Space Form and Applications

In this paper, the first Chen inequality is proved for CR-warped product submanifolds in complex space forms. This inequality involves intrinsic invariants (a leaf-wise $\delta$-invariant and the sectional curvature) controlled by an extrinsic one (the mean curvature vector), which provides an answer to Problem [1]. We carefully distinguish the leaf-wise $\delta$-invariant of a factor (used in the bound) from the intrinsic Chen invariant of the same factor, the two being related, on the totally real factor, by the Bishop--O'Neill formula. The bound is sharp and is uniform in the sign of the holomorphic sectional curvature $c$. As a geometric application, we derive necessary conditions for the immersed CR-warped product submanifold to be minimal in a complex space form, providing a partial answer to a well-known problem proposed by S.S. Chern (Problem [2]). For further research directions, we address a couple of open problems (Problem [3]} and Problem [4]).

math.DG↗

arXiv2026

Analytic local resolution of Medvedev's Morse index conjecture for the critical spherical catenoid in $\mathbb{H}^3$

Let $\Sigma_a\subset B^3(r(a))\subset\mathbb{H}^3$ ($a>1/2$) be the critical spherical catenoid of the Mori family, a free boundary minimal surface in the geodesic ball. The Medvedev conjecture [15] states ind$(\Sigma_a)=4$ for all $a>1/2$. We study its strong form: ind$(\Sigma_a)=4$ and nul$(\Sigma_a)=2$. The nullity condition nul$(\Sigma_a)=2$ combines the mode-$|k|=1$ result $\text{nul}_R(\Sigma_a)|_{|k|=1}=2$ of [17, Cor. 4.4] with vanishing kernel in modes $|k|=0,|k|\ge2$; the latter, not in [17], is established here for $a\in(1/2,1/2+\delta_0)$. The main result is the analytic local resolution of the strong Medvedev conjecture: $\exists\delta_0>0$ s.t. ind$(\Sigma_a)=4$, nul$(\Sigma_a)=2$ for all $a\in(1/2,1/2+\delta_0)$. This follows from the expansion $H(a):=\sinh r(a)/K(a)=\sigma_*\cosh\sigma_*+C_0(a-\frac12)+O((a-\frac12)^2)$ as $a\to(1/2)^+$, with $C_0=\frac{\sigma_*\cosh\sigma_*(\sinh^2\sigma_*-1)(3\sinh^2\sigma_*-2)}{12\sinh^2\sigma_*}$, where $\sigma_*>0$ the unique positive root of $\sigma=\coth\sigma$, and $C_0>0$ by $\sigma_*>\log(1+\sqrt2)$. The proof proceeds via three reductions: $(i)$ the strong Medvedev conjecture is equivalent to $\mu_0^{\mathrm{even}}(2)>0$ $(E)$ and $\mu_2(0)>0$ with non-degeneracy in mode $0$ $(F)$; $(ii)$ $\mu_2(0)>0$ reduces, via a Sturm shooting-count argument, to $\phi_a>0$ of the parametric Jacobi field on the principal branch; $(iii)$ $\phi_a>0$ reduces, under $B(s_0(a))^2>2K(a)^2$ $(G)$, to $H'(a)>0$ via a const. Wronskian and Sturm separation. Aux results: Picone identity (base $f_*$) closing unconditionally the odd radial sector for $|k|\ge2$; a second Picone identity (base $B$) proving $(E)$ unconditionally on $(1/2,1]$ and, via Hardy estimates, on $(1/2,A_*]$ ($A_*>1$); analytic closure of $(G)$ on $(1/2,1]$ via strict concavity of a transcendental function; an alternative proof of ind$(\Sigma_a)\ge4$ via Lorentz ambient coordinates.

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

Robin nullity in mode $|k|=1$ and asymptotic radius of the critical spherical catenoid

Medvedev proved that the critical spherical catenoid $Sigma_a$, the rotationally symmetric free boundary minimal annulus in a geodesic ball $B^3(r(a))$ in $H^3$ from the family of Mori and do Carmo-Dajczer, has Morse index at least 4, and conjectured equality. For each $a>1/2$ we establish three analytic results about $Sigma_a$. (I) Robin nullity and index in mode $|k|=1$. The Robin nullity of the Jacobi operator $L_{\Sigma_a}=\Delta_g+(|II|^2-2)$ in angular Fourier mode $|k|=1$ equals $2$, with kernel spanned by the Killing--Jacobi fields associated to the rotations $L_{12},L_{13}\in\mathfrak{so}(3,1)$ that fix the geodesic axis of $\Sigma_a$ and send $\partial B^3(r(a))$ to itself. The radial profile admits the closed form $f_*(s)=\partial_s\Phi_a^0(s,0)=\frac{d}{ds}[A(s)\cosh\varphi(s)]=\sinh r(s)\cdot r'(s)$, where $r(s)$ is the geodesic distance from $p_0=(1,0,0,0)$. By Sturm--Liouville theory, the Robin Morse index of $\Sigma_a$ in mode $|k|=1$ also equals $2$, refining the lower bound of Medvedev. (II) Asymptotic radius. The boundary radius satisfies $r(a)=\tfrac{3}{2}\log a+d_\infty+o(1)$ as $a\to\infty$, with $d_\infty=\log[\sqrt{2}\,\Gamma(1/4)^2/\pi^{3/2}]=\log[2\sqrt{2\pi}/\Gamma(3/4)^2]$. The closed form for $d_\infty$ follows from a Beta-function evaluation of $I_\infty=\int_0^{\infty}\cosh(2t)^{-3/2}\,dt$. (III) Degenerate limit. As $a\to(1/2)^+$, $r(a)=c_*\sqrt{a-1/2}\,(1+o(1))$ with $c_*=\sigma_*\cosh\sigma_*$, where $\sigma_*$ is the unique positive fixed point of $\sigma=\coth\sigma$. The proof of (I) follows the mode-by-mode strategy of Devyver for the Euclidean critical catenoid, with $\mathfrak{so}(3,1)$ replacing $\mathfrak{so}(3)$, supplemented by the closed-form identification $f_*=\partial_s\Phi^0$ specific to the hyperbolic ambient. The proof of (II) is a Laplace-type asymptotic analysis of the implicit free boundary condition.

math.DG↗

arXiv2026

Robin nullity and asymptotic geometry of the critical hyperbolic catenoid

For each parameter $a>1$, the critical hyperbolic catenoid $\Sigma_a$ is a rotationally symmetric, free boundary minimal annulus in a geodesic ball $B^3(r(a))\subset\mathbb{H}^3$. The Morse index of $\Sigma_a$ is at least $4$ by Medvedev [7], who conjectures equality. In this paper we identify a new geometric and spectral phenomenon for the family $\{\Sigma_a\}_{a>1}$, which we call "parameter-criticality", and study its consequences for the Robin spectrum. Specifically, we prove two main results: (I) Parameter-criticality (Theorem 1.5). The boundary radius $r(a)$ is non-monotone on $(1,\infty)$: it satisfies $r'(1^+)<0$ and $r(a)=\frac{3}{2}\log a+d_\infty+o(1)$ as $a\to\infty$ with $d_\infty=\log[\Gamma(1/4)/\Gamma(3/4)]-\frac{1}{2}\log(2\pi)$ (Theorem 1.4). Hence there exists a parameter-critical value $a^\sharp\in(1,\infty)$ with $r'(a^\sharp)=0$. (II) Robin nullity jump (Theorem 1.6). At every such $a^\sharp$, the Robin nullity of $\Sigma_{a^\sharp}$ satisfies $\text{nul}(L_{\Sigma_{a^\sharp}})\geq 3$, with an additional kernel element in mode $k=0$ generated by the parametric variation field $j_a=\langle\partial_a\Phi_a,\nu\rangle_L|_{a=a^\sharp}$, which we show is non-vanishing at the catenoid neck via the closed-form $j_a(0)=1/(2\sqrt{a^2-1})$. The argument requires the limit $r_0:=\lim_{a\to 1^+}r(a)$ characterized as the unique positive solution of the transcendental equation $\tanh(r_0)\,\tanh(2r_0/\sqrt{3})=\sqrt{3}/2$ (Theorem 1.3), giving a clean parametrization of the degeneration $\Sigma_a\to\Sigma_1$. The Robin nullity of $\Sigma_a$ in mode $|k|=1$ is shown to equal $2$ (Proposition 1.1); this extends to the hyperbolic setting the mode-by-mode Fourier decomposition technique of Devyver [2] for the Euclidean critical catenoid, and is used in the proof of (II) to identify the extra kernel as a mode-$k=0$ phenomenon.

math.AP↗

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

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 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 $[\omega] \in H^3(M;\mathbb{R})$ 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 $S^2\times \Sigma_g$, 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

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 $\mathcal{T}_\omega$, 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

Introduction of the G$_2$-Ricci Flow: Geometric Implications for Spontaneous Symmetry Breaking and Gauge Boson Masses

This work introduces the G$_2$-Ricci flow on seven-dimensional manifolds with non-zero torsion and explores its physical implications. By extending the Ricci flow to manifolds with G$_2$ structures, we study the evolution of solitonic solutions and their role in spontaneous symmetry breaking in gauge theories. In particular, this model proposes that the masses of the W and Z bosons are determined not by an external scalar field, as in the Higgs mechanism, but by the intrinsic geometric torsion of the manifold. Furthermore, a possible connection between the geometry of extra dimensions and the curvature of our spacetime is explored, with implications for the experimentally observed positive cosmological constant. This approach provides an innovative interpretation of fundamental interactions in theoretical physics, opening new possibilities for studying extra dimensions and the geometry of G$_2$-manifolds.

physics.gen-ph↗

arXiv2025

Positive biorthogonal curvature on $S^2 \times T^2$ via affine connection

We address the long-standing problem of the existence of a Riemannian metric on $S^2\times T^2$ with strictly positive biorthogonal curvature ($ K_{\text{biort}}(\sigma) > 0 $). This work tackles this challenge within a weaker, yet geometrically consistent, framework by introducing an affine connection, topologically motivated, on $ S^2 \times T^2 $ with antisymmetric torsion. Crucially, this torsion is calibrated via non-trivial cohomology classes in $ H^3(S^2 \times T^2; \mathbb{R}) \cong \mathbb{R}^2 $, an approach that allows overcoming topological constraints such as $ \chi = 0 $. We demonstrate that this construction, while not requiring metric compatibility (though retaining the metric ( $g$ ) for norms and orthogonality), successfully yields strictly positive biorthogonal curvature across the manifold.

math.DG↗

arXiv2025

A note on the theoretical approach to Grassmannians and Pl\"ucker coordinates for additive skew-symmetric pairwise comparisons matrices

Symmetry and antisymmetry are fundamental concepts in many strict sciences. Pairwise comparisons (PC) matrices are fundamental tools for representing pairwise relations in decision making. In this theoretical study, we present a novel framework that embeds additive skew-symmetric PC matrices into the Grassmannian manifold $G(2, n)$. This framework leverages Pl\"ucker coordinates to provide a rigorous geometric interpretation of their structure. Our key result demonstrates that the algebraic consistency condition $a_{ij} + a_{jk} - a_{ki} = 0$ is equivalent to the geometric consistency of $2$-planes in $G(2, n)$, satisfying the Pl\"ucker relations. This connection reveals that the algebraic properties of PC matrices can be naturally understood through their geometric representation. Additionally, we extend this framework by interpreting PC matrices as differential $2$-forms, providing a new perspective on their consistency as a closedness condition. Our framework of linear algebra, differential geometry, and algebraic geometry, placing PC matrices in a broader mathematical context. Rather than proposing a practical alternative to existing methods, our study aims to offer a theoretical foundation for future research by exploring new insights into higher-dimensional geometry and the geometric modeling of pairwise comparisons.

math.DG↗

arXiv2024

Analysis of a special type of soliton on Kenmotsu manifolds

In this paper, we aim to investigate the properties of an almost $*$-Ricci-Bourguignon soliton (almost $*-$R-B-S for short) on a Kenmotsu manifold (K-M). We start by proving that if a Kenmotsu manifold (K-M) obeys an almost $*-$R-B-S, then the manifold is $\eta$-Einstein. Furthermore, we establish that if a $(\kappa, -2)'$-nullity distribution, where $\kappa<-1$, has an almost $*$-Ricci-Bourguignon soliton (almost $*-$R-B-S), then the manifold is Ricci flat. Moreover, we establish that if a K-M has almost $*$-Ricci-Bourguignon soliton gradient and the vector field $\xi$ preserves the scalar curvature $r$, then the manifold is an Einstein manifold with a constant scalar curvature given by $r=-n(2n-1)$. Finaly, we have given en example of a almost $*-$R-B-S gradient on the Kenmotsu manifold.

physics.gen-ph↗

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↗

arXiv2023

A K-Theory approach to characterize admissible physical manifolds

We will classify physically admissible manifold structures by the use of Waldhausen categories. These categories give rise to algebraic K-Theory. Moreover, we will show that a universal K-spectrum is necessary for a physical manifold being admissible. Application to the generalized structure of D-branes are also provided. This might give novel insights in how the manifold structure in String and M-Theory looks like.

math.GN↗

arXiv2022

General inequalities and new shape operator inequality for contact CR-warped product submanifolds in cosymplectic space form

We establish two main inequalities; one for the norm of the second fundamental form and the other for the matrix of the shape operator. The results obtained are for cosymplectic manifolds and, for these, we show that the contact warped product submanifolds naturally possess a geometric property; namely $\mathcal{D}_1$-minimality which, by means of the Gauss equation, allows us to obtain an optimal general inequality. For sake of generalization, we state our hypotheses for nearly cosymplectic manifolds, then we obtain them as particular cases for cosymplectic manifolds. For the other part of the paper, we derived some inequalities and applied them to construct and introduce a shape operator inequality for cosimpleptic manifolds involving the harmonic series. As further research directions, we have addressed a couple of open problems arose naturally during this work and which depend on its results.

math.DG↗