arXiv subjects

Andrea Loi

Publications and source records attributed to Andrea Loi.

At least 19 recordsLinked to original sources

arXiv2026

The Fundamental Semigroup and Associative Lifts of Topological Semigroups

We introduce the fundamental semigroup $Λ(S)$ of a nonempty path-connected topological semigroup $S$, defined using right translations and change-of-basepoint isomorphisms on its fundamental group. We prove that this construction is closed under composition and independent of the chosen basepoint up to isomorphism. It is also invariant under homotopy equivalences that preserve multiplication up to homotopy. We characterize exactly when the fundamental semigroup is a group and identify that group with an inner automorphism group. For a connected, locally path-connected and semilocally simply connected topological semigroup, we show that multiplication admits a continuous associative lift to the universal cover if and only if $Λ(S)$ contains an idempotent. Every such idempotent can be realized by an associative lift, and any two associative lifts are isomorphic over $S$ through a deck transformation. We obtain existence results when the fundamental group has finite inner automorphism group or contains no subgroup isomorphic to Thompson's group $F$, and when the underlying space is a finite-dimensional CW complex. In contrast, we construct a path-connected countable CW topological semigroup with cells in arbitrarily high dimensions whose multiplication admits no continuous associative lift to its universal cover.

math.AT↗

arXiv2026

Complex Hyperbolic Immersions of Cheng--Yau Metrics on Thullen Domains

We classify the homotheties of the complete Cheng--Yau Kähler--Einstein metrics on the two-dimensional Thullen domains $M(μ)$, $μ>0$, that admit Kähler immersions into $\mathbb{CH}^{\infty}$. With $\Ric(g_{\mathrm{CY}})=-3g_{\mathrm{CY}}$, an immersion of $c\,g_{\mathrm{CY}}$ exists exactly when $0<μ\leqμ_+$ and $0 1$ is characterized by an explicit equation, and its endpoint is included. Every admissible metric has a proper global Kähler embedding. The same parameter interval classifies immersibility into flat and projective Hilbert space, where every positive homothety admits a proper global embedding. Beyond the wall, all three targets are excluded at every positive scale. The proof combines an algebraic normalization of the Cheng--Yau equation, all-order coefficient estimates, and complex continuation of a scalar flow. For $μ\neq1$, the resulting complete nonhomogeneous examples disprove proposed Kähler--Einstein rigidity conjectures in the complex-hyperbolic and flat Hilbert settings. We also prove a complementary rigidity theorem for Bergman--Hartogs domains and construct complete $η$-Einstein Sasakian manifolds with nonhomogeneous Riemannian metrics and proper global Sasakian embeddings into the infinite-dimensional Heisenberg space form.

math.DG↗

arXiv2026

Projectively induced Kähler--Einstein surfaces

We classify compact Kähler--Einstein surfaces whose metric is induced by a holomorphic isometric immersion into a finite-dimensional complex projective space. No symmetry assumption and no bound on the codimension are imposed. We prove that the only such surfaces are \[ (\PP^2,m g_{\FS}) \quad\text{and}\quad \bigl(\PP^1\times\PP^1,m(g_{\FS}\oplus g_{\FS})\bigr), \qquad m\in\mathbb Z_{>0}, \] realized respectively by the Veronese and Segre--Veronese embeddings. The main new ingredient is a codimension-independent exclusion of the entire Fano-index-one branch, combining a common anticanonical root construction with Gram--Gauss rank estimates and, in degree five, an equivariant curvature argument. Consequently, every connected compact Kähler--Einstein surface whose metric is induced by a holomorphic isometric immersion into a finite-dimensional complex projective space is homogeneous.

math.DG↗

arXiv2026

Monodromy of Walrasian Equilibrium Prices: A Fixed-Preference Endowment-Redistribution Construction

Suppose an economy moves continuously along a closed path of fundamentals and returns exactly to its initial state. A regular Walrasian equilibrium price followed continuously along that path need not return to itself. For every number of commodities $L\ge3$, we construct a closed one-parameter family of pure-exchange economies in which all preferences are fixed, only three individual endowments vary, and aggregate resources remain constant. Along the loop, two regular equilibrium-price branches are exchanged while a third returns to itself; at the end, every economic primitive is exactly as it was initially. The construction uses at most $2L+2$ consumers. We also provide an exact realization with at most $2L$ consumers in which all endowments are fixed and only one preference varies; the two families generate identical aggregate excess demand. The key realization problem is parametric: classical aggregate-realization results guarantee an economy at each fixed parameter value, but not a continuous loop of realizing primitives. We construct such a loop explicitly. The resulting monodromy is structurally stable under sufficiently small $C^1$ perturbations. It is impossible with one or two commodities, making three commodities the sharp threshold. Thus local regularity guarantees local continuation but not a globally path-independent identity of an equilibrium branch, with implications for global comparative statics under multiplicity and for equilibrium selection based on continuation.

econ.TH↗

arXiv2026

Radial Projectively Induced Canonical Kähler Metrics: Rigidity and Classification

We study radial Kähler metrics on domains of $\mathbb{C}^n$, $n\geq 2$, admitting a Kähler immersion into a finite- or infinite-dimensional complex projective space. We classify those with constant non-negative scalar curvature: up to a linear change of coordinates, they are positive integer multiples of the Fubini-Study metric, the flat metric, or, in complex dimension two, generalized Burns-Simanca metrics. We also prove that every radial projectively induced Kähler-Einstein metric has constant holomorphic sectional curvature and is therefore a Fubini-Study, flat, or complex hyperbolic metric. Finally, we show that a radial infinitely projectively induced extremal Kähler metric has unbounded maximal radial domain if and only if it is scalar-flat.

math.DG↗

arXiv2026

On Simply Connected Simple Lie Skew Braces with Nilpotent Multiplicative Group

We prove that a simply connected simple Lie skew brace with nilpotent multiplicative Lie group must be one-dimensional and abelian. Equivalently, if $(G,\cdot,\circ)$ is a simply connected Lie skew brace with nilpotent multiplicative Lie group and $\dim G>1$, then $(G,\cdot,\circ)$ is not simple. Thus, in the simply connected Lie setting, nilpotency of the multiplicative group is incompatible with simplicity in every dimension greater than one. The proof is carried out at the post-Lie algebra level. First, if the additive Lie algebra is solvable, then its nilradical is automatically an ideal of the associated post-Lie algebra. Second, when both Lie algebras underlying an integrable post-Lie structure are nilpotent, one always obtains a proper post-Lie ideal with trivial quotient. To pass from infinitesimal ideals to global ideals of the Lie skew brace, we show that trivial post-Lie quotients give rise to homomorphisms onto abelian trivial Lie skew braces, whose kernels yield connected closed ideals.

math.GR↗

arXiv2026

Sorting and Global Uniqueness in Two-Good HARA Economies with Many Patience Types

We study global uniqueness of competitive equilibrium in two-good pure-exchange economies with heterogeneous impatience types and a common HARA Bernoulli utility. The paper connects the CRRA sorting result of \citet{GeanakoplosWalsh2018} with the line of HARA uniqueness results developed in \citet{LoiMatta2022,LoiMatta2024}. In the CRRA case, ordered endowments provide a sorting mechanism for uniqueness. In the HARA case, uniqueness is known to hold for arbitrary endowments under the curvature bound $γ\le I/(I-1)$, where $I$ is the number of impatience types. For two types, the curvature restriction can be removed under a monotone sorting condition linking patience and endowment composition. The present paper shows that this high-curvature HARA sorting mechanism is not specific to the two-type case. Our main result proves global uniqueness for any finite number of impatience types and any $γ>1$. If types can be ordered so that more patient agents hold weakly more of the first good and weakly less of the second, then the equilibrium price is globally unique. Thus the paper extends the two-type high-curvature HARA result to a genuinely multi-type setting and complements the arbitrary-endowment low-curvature result by replacing the low-curvature restriction with an economically interpretable sorting restriction. In the CRRA subcase ($b=0$), the ordered-endowment condition coincides with that of \citet{GeanakoplosWalsh2018}, and our corollary recovers their uniqueness result. The contribution of the present paper is therefore not the sorting condition itself but its reach: the same ordered heterogeneity in patience and endowment composition rules out multiplicity throughout the shifted HARA case ($b>0$), for any finite number of types and any $γ>1$, through a global coefficient-ratio argument.

econ.TH↗

arXiv2026

Curvature, Minimality and Uniqueness of Equilibrium

For a smooth pure exchange economy with fixed aggregate resources, we study two geometric conditions on the equilibrium manifold $E(r)$ endowed with the metric induced from its Euclidean ambient space. First, for arbitrary numbers of commodities and consumers, we prove that intrinsic flatness forces equilibrium prices to be locally constant. Together with Balasko's uniqueness--constancy criterion, this yields a necessary and sufficient condition: $E(r)$ is intrinsically flat if and only if the normalized equilibrium price is unique for every economy with aggregate resources $r$. This extends the curvature--uniqueness theorem of \cite{LoiMatta2018} and completes the higher-dimensional direction pursued in \cite{LoiMattaUccheddu2023}. Second, in the two-commodity case, we show that minimality of $E(r)$ already forces local constancy of the price map. Under the uniform-distribution interpretation of \cite{LoiMatta2021}, this gives the minimal-entropy/uniqueness equivalence without the additional asymptotic assumption used there. Both arguments rely on the same local parametrization of $E(r)$ and avoid the explicit construction of a normal frame.

econ.TH↗

arXiv2026

Solvability and Rigidity for Topological Skew Braces

We study compact and locally compact topological analogues of the Byott--Vendramin solvability problem for finite skew braces, asking whether solvability of the additive group forces solvability of the multiplicative group. Our main theorem proves an affirmative result in the connected locally compact Hausdorff setting: if $B=(B,\cdot,\circ)$ is a connected locally compact Hausdorff topological skew brace and the additive group $(B,\cdot)$ is solvable, then the multiplicative group $(B,\circ)$ is solvable. The proof proceeds by reducing the additive group to a solvable Lie quotient and then applying an affine-action theorem: a connected Lie group acting transitively and affinely on a connected solvable Lie group, with solvable stabilizer identity component, is itself solvable. We further show that the Hausdorff, local compactness, and connectedness hypotheses are essential by constructing counterexamples when each is omitted. In the compact connected Hausdorff case with abelian additive group, we obtain a stronger rigidity phenomenon: the two group laws coincide.

math.GR↗

arXiv2026

Topological and differentiable aspects of Clifford semigroups

This paper investigates the interplay between algebraic structure, topology, and differentiability in Clifford semigroups. The study is developed along three main themes. First, in the compact Hausdorff setting, we provide an explicit construction of a compatible metric for the Bowman topology. Second, we address Hilbert-fifth-type questions by establishing criteria under which the maximal subgroups are forced to be Lie groups. Finally, we prove a structural rigidity theorem: $C^1$-regularity at the idempotents implies that the idempotent semilattice is discrete.

math.GN↗

arXiv2026

On simple compact Lie skew braces

We study simplicity of Lie skew braces from both global and infinitesimal perspectives. After reviewing the correspondence between connected Lie skew braces, simply transitive affine actions, and post-Lie algebras, we investigate ideals and rigidity phenomena. Our main result concerns compact connected Lie skew braces. We prove that any compact connected simple Lie skew brace is either the trivial Lie skew brace on $S^1$, or both of its underlying Lie groups are simple and the brace is trivial or almost trivial. Consequently, apart from the exceptional $S^1$ case, simplicity of a compact connected Lie skew brace is equivalent to simplicity of either underlying Lie group. We also show that every connected compact solvable Lie skew brace is trivial. Finally, we construct a noncompact example demonstrating that this rigidity phenomenon does not hold in general: there exists a connected simply connected simple Lie skew brace whose additive and multiplicative Lie groups are both solvable.

math.GR↗

arXiv2026

Structural and rigidity properties of Lie skew braces

We investigate structural and rigidity properties of \emph{Lie skew braces} (LSBs), objects essentially known in the literature as \emph{post--Lie groups}, obtained by endowing a manifold with two compatible group laws that share the same identity element. LSBs extend skew left braces, which are central to the study of non-involutive set-theoretic solutions of the Yang--Baxter equation, to the smooth category. Our first main result shows that, for every connected LSB $(G,\cdot,\circ)$, linearity (in the simply-connected case) and solvability carry over from $(G,\cdot)$ to $(G,\circ)$, whereas the converse direction is rigid: if $(G,\circ)$ is nilpotent (respectively, semisimple) then $(G,\cdot)$ is forced to be solvable (respectively, isomorphic to $(G,\circ)$). Our second results provides two ``flexibility'' statements: every non-linear simply connected Lie group $(G, \cdot )$ admits an LSB $(G,\cdot,\circ)$ such that $(G,\circ)$ is linear, and every simply connected solvable Lie group $(G, \circ )$ supports a LSB $(G,\cdot,\circ)$ such that $(G,\cdot)$ is nilpotent. A third result provides a complete existence table for non-trivial LSBs across the six standard Lie-group classes, abelian, nilpotent (non-abelian), solvable (non-nilpotent), simple, semisimple (non-simple) and mixed type, identifying precisely when a LSB can be built and when only the trivial or no structure occurs. Both the explicit constructions and the properties established in our theorems rely on a factorisation technique for Lie groups, on the correspondence between LSBs and regular subgroups of the affine group $\operatorname{Aff}(G,\cdot)$, which renders LSB theory equivalent to simply transitive affine actions, and on the theory of post-Lie algebras together with their integrability properties.

math.GR↗

arXiv2026

On the Bergman metric of symmetric spaces

We study bounded domains $Ω\subset\mathbb{C}^n$ whose Bergman metric is locally symmetric, i.e. its Riemannian curvature tensor is parallel with respect to the Levi-Civita connection. Following the strategy developed in \cite{UnifThm2}, we obtain two rigidity results. If the Bergman metric of $Ω$ is complete, then $Ω$ is (globally) symmetric. If instead $Ω$ is pseudoconvex, then $Ω$ is biholomorphic to $\widetildeΩ\setminus E$, where $\widetildeΩ\subset\mathbb{C}^n$ is a bounded symmetric domain and $E\subset\widetildeΩ$ is relatively closed and pluripolar. The proofs combine the structure theory of Hermitian symmetric spaces with Calabi's theory of Kähler immersions into the infinite dimensional complex projective space (in particular, rigidity and the hereditary property of the diastasis), together with analytic and pluripotential tools based on extension properties of square-integrable holomorphic functions and the Bergman kernel.

math.CV↗

arXiv2025

The polydisk theorem for Hartogs domains over symmetric domains

We extend the polydisk theorem of [21], originally established for classical Cartan-Hartogs domains, to Hartogs domains over arbitrary (possibly reducible and exceptional) bounded symmetric domains. We further establish a dual counterpart of this result. As an application, we show that the dual of a Hartogs domain over a bounded symmetric domain admits no totally geodesic immersion into any compact Riemannian manifold, thereby broadening the rigidity phenomena obtained in [13].

math.DG↗

arXiv2025

On the Bergman metric of Cartan-Hartogs domains

We study the Bergman metric and introduce the Bergman dual on Cartan-Hartogs (CH) domains. For a bounded domain D in C^n with Bergman kernel K_D, we define the Bergman dual of (D, g_D) as (D*, g_D*), where D* is the maximal domain on which the modified kernel K_D*(z, zbar) = K_D(z, -zbar) is positive, and g_D* is the Kahler metric obtained from K_D*. For a Cartan-Hartogs domain M_{Omega, mu} we prove the equivalence of: (i) M_{Omega, mu} is biholomorphic to the unit ball; (ii) its Bergman metric is a Kahler-Ricci soliton; (iii) after rescaling by a constant factor, the Bergman dual is finitely projectively induced. Conditions (i) and (ii) are Bergman-metric analogues of classical rigidity for Kahler-Einstein metrics (related to Yau's problem and Cheng's conjecture) and to recent rigidity for Kahler-Ricci solitons. Condition (iii) emphasizes the duality viewpoint, inspired by bounded symmetric domains and their compact duals. We also compare our results with other canonical metrics on CH domains, namely g_{Omega, mu} and hat g_{Omega, mu}, and discuss open problems about the maximal domain on which the Bergman dual is defined.

math.CV↗

arXiv2025

Universal embeddings of flag manifolds and rigidity phenomena

We prove a universal embedding theorem for flag manifolds: every flag manifold admits a holomorphic isometric embedding into an irreducible classical flag manifold. This result generalizes the classical celebrated embedding theorems of Takeuchi [30] and Nakagawa-Takagi [27]. Using this embedding, we establish new rigidity phenomena for holomorphic isometries between homogeneous Kähler manifolds. As a first immediate consequence we show the triviality of a Kähler-Ricci soliton submanifod of $C \times Ω$, where $C$ is a flag manifold and $Ω$ is a homogeneous bounded domain. Secondly, we show that no \emph{weak-relative} relationship can occur among the fundamental classes of homogeneous Kähler manifolds: flat spaces, flag manifolds, and homogeneous bounded domains. Two Kähler manifolds are said to be \emph{weak relatives} if they share, up to local isometry, a common Kähler submanifold of complex dimension at least two. Our main result precisely shows that if $E$ is (possibly indefinite) flat, $C$ is a flag manifold, and $Ω$ is a homogeneous bounded domain, then: $E$ is not weak relative to $C\timesΩ$; $C$ is not weak relative to $E\timesΩ$; $Ω$ is not weak relative to $E\times C$. This extends, in two independent directions, the rigidity theorem of Loi-Mossa [22]: we pass from \emph{relatives} to the more flexible notion of \emph{weak relatives} and dispense with the earlier ''special'' restriction on the flag-manifold factor. This result also unifies previous rigidity results from the literature, e.g., [5, 6, 7, 9, 12, 13, 32].

math.DG↗

arXiv2024

Kähler duality and projective embeddings

Motivated by the duality theory between Hermitian symmetric spaces of noncompact and compact types, we introduce and examine the concept of Kähler duality between domains of $\mathbb C^n$.

math.DG↗

arXiv2024

Any Sasakian structure is approximated by embeddings into spheres

We show that, for any given $q\geq 0$, any Sasakian structure on a closed manifold $M$ is approximated in the $C^{q}$-norm by structures induced by CR embeddings into weighted Sasakian spheres. In order to obtain this result, we also strengthen the approximation of an orbifold Kähler form by projectively induced ones given by Ross and Thomas in [21] in the $C^0$-norm to a $C^{q}$-approximation.

math.DG↗