arXiv subjects

Spiro Karigiannis

Publications and source records attributed to Spiro Karigiannis.

At least 19 recordsLinked to original sources

arXiv2026

Montel's theorem and tautness in calibrated geometry

We relate the hyperbolicity of a calibrated manifold $(X, ϕ)$ to the analytic properties of the space of Smith immersions $\mathrm{SmIm}(B^k, X)$ from the Poincare $k$-ball into $X$. In particular, we establish the following calibrated analogue of a theorem of Royden: if $X$ is $ϕ$-replete, then $R_ϕ$- and $K_ϕ$-hyperbolicity coincide, and either implies the equicontinuity of $\mathrm{SmIm}(B^k, X)$ with respect to the $ϕ$-distance. This yields a Montel theorem for compact $ϕ$-replete calibrated manifolds as an immediate corollary. Our primary technical tool is a new Schwarz lemma for Smith immersions from $B^k$ into $X$, which is of independent interest. In a similar spirit, we also prove a calibrated analogue of Kiernan's theorem to the effect that the $K_ϕ$-hyperbolicity of $X$ is almost equivalent to $\mathrm{SmIm}(B^k, X)$ being a normal family. Finally, we prove that bounded domains in flat euclidean space are $R_ϕ$-hyperbolic for any calibration $ϕ$, and we investigate the hyperbolicity of products and discrete quotients.

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arXiv2026

Non-squeezing and capacities for some calibrated geometries

It was shown by Barron--Shafiee that an analogue of Gromov's non-squeezing theorem holds for affine maps which preserve a power $ω^k$ of the symplectic form $ω$ on $\mathbb{R}^{2n}$. In the present paper, we state and prove in two ways an improved version of their result which is closer to the classical affine non-squeezing theorem. One proof closely follows their argument, and the other consists of a reduction to the classical case by showing that, except for the case $k = n$, every linear map that preserves $ω^k$ must be symplectic or anti-symplectic. We then study when a calibration form satisfies an (affine) non-squeezing theorem. Particular focus is given to the special Lagrangian case, where we are able to establish an affine non-squeezing theorem for the holomorphic volume form $Ω= dz^1 \wedge \cdots \wedge dz^n$. The classical symplectic affine rigidity theorem states roughly that a non-singular linear map is symplectic or anti-symplectic if and only if it preserves the "capacity" of every ellipsoid. We establish an affine special Lagrangian version of this theorem under the necessary assumption that the map is complex-linear. We also discuss some natural future questions.

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arXiv2026

Flows of conformally coclosed $G_2$-structures with dilaton

We study flows of $G_2$-structures guided by the principle of dimensional reduction: natural geometric flows in $G_2$-geometry reduce to natural flows in complex geometry. Our main examples are the $G_2$-Laplacian coflow, which lifts the Kähler--Ricci flow, and a 7-dimensional lift of the anomaly flow on complex threefolds. The $G_2$-lift of the anomaly flow deforms conformally coclosed $G_2$-structures. We compare the $G_2$-anomaly flow to the $G_2$-Laplacian coflow, and investigate short-time existence and fixed points.

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arXiv2026

Hodge theoretic results for nearly Kähler manifolds in all dimensions

We generalize to nearly Kähler manifolds of arbitrary dimensions most of the Hodge-theoretic results for nearly Kähler $6$-manifolds that were established by Verbitsky. In particular, for a compact nearly Kähler manifold of any dimension, the (appropriately defined) Hodge numbers are related to the Betti numbers in the same way as on a compact Kähler manifold. In the $6$-dimensional case, Verbitsky was able to say slightly more using the induced $\mathrm{SU}(3)$ structure. We discuss potential extensions of this to twistor spaces over positive scalar curvature quaternionic-Kähler manifolds, which are a particular class of $(4n+2)$-dimensional nearly Kähler manifolds equipped with a special $\mathrm{SU}(n) \! \cdot \! \mathrm{U}(1)$ structure.

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arXiv2025

A special class of $k$-harmonic maps inducing calibrated fibrations

We consider two special classes of $k$-harmonic maps between Riemannian manifolds which are related to calibrated geometry, satisfying a first order fully nonlinear PDE. The first is a special type of weakly conformal map $u \colon (L^k, g) \to (M^n, h)$ where $k \leq n$ and $α$ is a calibration $k$-form on $M$. Away from the critical set, the image is an $α$-calibrated submanifold of $M$. These were previously studied by Cheng-Karigiannis-Madnick when $α$ was associated to a vector cross product, but we clarify that such a restriction is unnecessary. The second, which is new, is a special type of weakly horizontally conformal map $u \colon (M^n, h) \to (L^k, g)$ where $n \geq k$ and $α$ is a calibration $(n-k)$-form on $M$. Away from the critical set, the fibres $u^{-1} \{ u(x) \}$ are $α$-calibrated submanifolds of $M$. We also review some previously established analytic results for the first class; we exhibit some explicit noncompact examples of the second class, where $(M, h)$ are the Bryant-Salamon manifolds with exceptional holonomy; we remark on the relevance of this new PDE to the Strominger-Yau-Zaslow conjecture for mirror symmetry in terms of special Lagrangian fibrations and to the $\mathrm{G}_2$ version by Gukov-Yau-Zaslow in terms of coassociative fibrations; and we present several open questions for future study.

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arXiv2025

BRIDGES Lectures: Flows of geometric structures, especially $\mathrm{G}_2$-structures

The BRIDGES meeting in gauge theory, extremal structures, and stability was held June 2024 at l'Institut d'Études Scientifiques de Cargèse in Corsica, organized by Daniele Faenzi, Eveline Legendre, Eric Loubeau, and Henrique Sá Earp. The first week was a summer school consisting of four independent but related lecture series by Oscar García Prada, Spiro Karigiannis, Laurent Manivel, and Ruxandra Moraru. The present document consists of notes for the lecture series by Spiro Karigiannis on "Flows of geometric structures, especially $\mathrm{G}_2$-structures". Some assistance in the preparation of these notes by the author was provided by several participants of the summer school. See the Comments field for more information. The main theme is short time existence (STE) and uniqueness for geometric flows. We first introduce geometric structures on manifolds and geometric flows of such structures. We discuss some qualitative features of geometric flows, and consider the notions of strong and weak parabolicity. We focus on the Ricci flow, explaining carefully the DeTurck trick to establish short-time existence and uniqueness, an argument which we then extend to a general class of geometric flows of Riemannian metrics, previewing similar ideas for flows of $\mathrm{G}_2$-structures. Finally, we consider geometric flows of $\mathrm{G}_2$-structures. We review the basics of $\mathrm{G}_2$-geometry and survey several different geometric flows of $\mathrm{G}_2$-structures. In particular, we clarify in what sense STE results for the $\mathrm{G}_2$ Laplacian flow differ from STE results for other geometric flows. We conclude with a summary of some recent results by the author with Dwivedi and Gianniotis, including a classification of all possible heat-type flows of $\mathrm{G}_2$-structures, and a sufficient condition for such a flow to admit STE and uniqueness by a modified DeTurck trick.

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arXiv2025

Holomorphicity of parabolic stable minimal surfaces of high codimension

A classical theorem of Micallef says that if $F \colon (Σ, g) \to \mathbb{R}^4$ is a stable minimal immersion of an oriented $2$-dimensional complete Riemannian manifold (that is parabolic) into $\mathbb{R}^4$, it is necessarily holomorphic with respect to some parallel orthogonal complex structure on $\mathbb{R}^4$. We generalize this theorem by replacing $\mathbb{R}^4$ with $\mathbb{R}^{2 + 2k}$ for any codimension $2k$, under the additional hypothesis that the normal bundle $N Σ$ is equipped with a complex structure that is compatible with the induced metric and parallel with respect to the induced connection. This is a necessary assumption for such a theorem to hold, and it is automatically satisfied in the classical case $k=1$. We also briefly discuss possible further generalizations of such a result to other calibrations and to Smith maps.

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arXiv2025

Flows of $G_2$-structures, II: Curvature, torsion, symbols, and functionals

We continue the investigation of general geometric flows of $G_2$-structures initiated by the third author in "Flows of $G_2$-structures, I." Specifically, we determine the possible geometric flows (up to lower order terms) of $G_2$-structures which are second order quasilinear, by explicitly computing all independent second order differential invariants of $G_2$-structures which are $3$-forms. There are four symmetric $2$-tensors and two vector fields. We do this by deriving explicit computational descriptions of the decompositions of the curvature and the covariant derivative of the torsion into irreducible $G_2$-representations, as well as the decomposition of the $G_2$-Bianchi identity into independent relations. We also show that these six tensors arise as leading order contributions to the Euler-Lagrange equations for the energy functionals of the four independent torsion components, and we establish a $G_2$-analogue of the classical block decomposition of the Riemann curvature operator on oriented $4$-dimensional Riemannian manifolds. Finally, we present a large class of geometric flows of $G_2$-structures which are directly amenable to a deTurck type trick to establish short-time existence and uniqueness, with no initial assumption on the torsion, vastly generalizing an earlier result of Weiss-Witt for the negative gradient flow of the Dirichlet energy. This result is proved through a careful analysis of the principal symbols of the linearizations of these operators, establishing particular linear combinations for which one can prove that the failure of strict parabolicity is due precisely to the diffeomorphism invariance. A detailed introductory section on various foundational results of $G_2$-structure, several of which are not readily available in the literature, should be of wider interest and applicability.

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arXiv2024

Cohomogeneity one solitons for the isometric flow of $G_2$-structures

We consider the existence of cohomogeneity one solitons for the isometric flow of $G_2$-structures on the following classes of torsion-free $G_2$-manifolds: the Euclidean $R^7$ with its standard $G_2$-structure, metric cylinders over Calabi-Yau 3-folds, metric cones over nearly Kähler 6-manifolds, and the Bryant-Salamon $G_2$-manifolds. In all cases we establish existence of global solutions to the isometric soliton equations, and determine the asymptotic behaviour of the torsion. In particular, existence of shrinking isometric solitons on $R^7$ is proved, giving support to the likely existence of type I singularities for the isometric flow. In each case, the study of the soliton equation reduces to a particular nonlinear ODE with a regular singular point, for which we provide a careful analysis. Finally, to simplify the derivation of the relevant equations in each case, we first establish several useful Riemannian geometric formulas for a general class of cohomogeneity one metrics on total spaces of vector bundles which should have much wider application, as such metrics arise often as explicit examples of special holonomy metrics.

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arXiv2024

Extrinsic geometry of calibrated submanifolds

Given a calibration $α$ whose stabilizer acts transitively on the Grassmanian of calibrated planes, we introduce a nontrivial Lie-theoretic condition on $α$, which we call compliancy, and show that this condition holds for many interesting geometric calibrations, including Kähler, special Lagrangian, associative, coassociative, and Cayley. We determine a sufficient condition that ensures compliancy of $α$, we completely characterize compliancy in terms of properties of a natural involution determined by a calibrated plane, and we relate compliancy to the geometry of the calibrated Grassmanian. The condition that a Riemannian immersion $ι\colon L \to M$ be calibrated is a first order condition. By contrast, its extrinsic geometry, given by the second fundamental form $A$ and the induced tangent and normal connections $\nabla$ on $TL$ and $D$ on $NL$, respectively, is second order information. We characterize the conditions imposed on the extrinsic geometric data $(A, \nabla, D)$ when the Riemannian immersion $ι\colon L \to M$ is calibrated with respect to a calibration $α$ on $M$ which is both parallel and compliant. This motivate the definition of an infinitesimally calibrated Riemannian immersion, generalizing the classical notion of a superminimal surface in $\mathbb R^4$.

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arXiv2024

Calibrated Geometry in Hyperkahler Cones, 3-Sasakian Manifolds, and Twistor Spaces

We systematically study calibrated geometry in hyperkähler cones $C^{4n+4}$, their 3-Sasakian links $M^{4n+3}$, and the corresponding twistor spaces $Z^{4n+2}$, emphasizing the relationships between submanifold geometries in various spaces. Our analysis emphasizes the role played by a canonical $\mathrm{Sp}(n)\mathrm{U}(1)$-structure $γ$ on the twistor space $Z$. We observe that $\mathrm{Re}(e^{- i θ} γ)$ is an $S^1$-family of semi-calibrations, and make a detailed study of their associated calibrated geometries. As an application, we obtain new characterizations of complex Lagrangian and complex isotropic cones in hyperkähler cones, generalizing a result of Ejiri and Tsukada. We also generalize a theorem of Storm on submanifolds of twistor spaces that are Lagrangian with respect to both the Kähler-Einstein and nearly-Kähler structures.

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arXiv2023

A variational characterization of calibrated submanifolds

Let $M$ be a fixed compact oriented embedded submanifold of a manifold $\overline{M}$. Consider the volume $\mathcal{V} (\overline{g}) = \int_M \mathsf{vol}_{(M, g)}$ as a functional of the ambient metric $\overline{g}$ on $\overline{M}$, where $g = \overline{g}|_M$. We show that $\overline{g}$ is a critical point of $\mathcal{V}$ with respect to a special class of variations of $\overline{g}$, obtained by varying a calibration $μ$ on $\overline{M}$ in a particular way, if and only if $M$ is calibrated by $μ$. We do not assume that the calibration is closed. We prove this for almost complex, associative, coassociative, and Cayley calibrations, generalizing earlier work of Arezzo-Sun in the almost Kähler case. The Cayley case turns out to be particularly interesting, as it behaves quite differently from the others. We also apply these results to obtain a variational characterization of Smith maps.

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arXiv2022

Cohomologies on almost complex manifolds and the $\partial \bar{\partial}$-lemma

We study cohomologies on an almost complex manifold $(M, J)$, defined using the Nijenhuis-Lie derivations $\mathcal{L}_J$ and $\mathcal{L}_N$ induced from the almost complex structure $J$ and its Nijenhuis tensor $N$, regarded as vector-valued forms on $M$. We show how one of these, the $N$-cohomology $H^{\bullet}_N (M)$, can be used to distinguish non-isomorphic non-integrable almost complex structures on $M$. Another one, the $J$-cohomology $H^{\bullet}_J (M)$, is familiar in the integrable case but we extend its definition and applicability to the case of non-integrable almost complex structures. The $J$-cohomology encodes whether a complex manifold satisfies the $\partial \bar{\partial}$-lemma, and more generally in the non-integrable case the $J$-cohomology encodes whether $(M, J)$ satisfies the $\mathrm{d} \mathcal{L}_J$-lemma, which we introduce and motivate in this paper. We discuss several explicit examples in detail, including a non-integrable example. We also show that $H^k_J$ is finite-dimensional for compact integrable $(M, J)$, and use spectral sequences to establish partial results on the finite-dimensionality of $H^k_J$ in the compact non-integrable case.

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arXiv2022

Observations about the Lie algebra $\mathfrak{g}_2 \subset \mathfrak{so}(7)$, associative $3$-planes, and $\mathfrak{so}(4)$ subalgebras

We make several observations relating the Lie algebra $\mathfrak{g}_2 \subset \mathfrak{so}(7)$, associative $3$-planes, and $\mathfrak{so}(4)$ subalgebras. Some are likely well-known but not easy to find in the literature, while other results are new. We show that an element $X \in \mathfrak{g}_2$ cannot have rank $2$, and if it has rank $4$ then its kernel is an associative subspace. We prove a canonical form theorem for elements of $\mathfrak{g}_2$. Given an associative $3$-plane $P$ in $\mathbb R^7$, we construct a Lie subalgebra $Θ(P)$ of $\mathfrak{so}(7) = Λ^2 (\mathbb R^7)$ that is isomorphic to $\mathfrak{so}(4)$. This $\mathfrak{so}(4)$ subalgebra differs from other known constructions of $\mathfrak{so}(4)$ subalgebras of $\mathfrak{so}(7)$ determined by an associative $3$-plane. These are results of an NSERC undergraduate research project. The paper is written so as to be accessible to a wide audience.

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arXiv2021

Bubble Tree Convergence of Conformally Cross Product Preserving Maps

We study a class of weakly conformal $3$-harmonic maps, called associative Smith maps, from $3$-manifolds into $7$-manifolds that parametrize associative $3$-folds in Riemannian $7$-manifolds equipped with $\mathrm{G}_2$-structures. Associative Smith maps are solutions of a conformally invariant nonlinear first order PDE system, called the Smith equation, that may be viewed as a $\mathrm{G}_2$-analogue of the Cauchy-Riemann system for $J$-holomorphic curves. In this paper, we show that associative Smith maps enjoy many of the same analytic properties as $J$-holomorphic curves in symplectic geometry. In particular, we prove: (i) an interior regularity theorem, (ii) a removable singularity result, (iii) an energy gap result, and (iv) a mean-value inequality. While our approach is informed by the holomorphic curve case, a number of nontrivial extensions are involved, primarily due to the degeneracy of the Smith equation. At the heart of above results is an $\varepsilon$-regularity theorem that gives quantitative $C^{1,β}$-regularity of $W^{1,3}$ associative Smith maps under a smallness assumption on the $3$-energy. The proof combines previous work on weakly $3$-harmonic maps and the observation that the associative Smith equation demonstrates a certain "compensation phenomenon" that shows up in many other geometric PDEs. Combining these analytical properties and the conformal invariance of the Smith equation, we explain how sequences of associative Smith maps with bounded $3$-energy may be conformally rescaled to yield bubble trees of such maps. When the $\mathrm{G}_2$-structure is closed, we prove that both the $3$-energy and the homotopy are preserved in the bubble tree limit. This result may be regarded as an associative analogue of Gromov's Compactness Theorem in symplectic geometry.

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arXiv2021

Twisted-Austere Submanifolds in Euclidean Space

A twisted-austere $k$-fold $(M, μ)$ in $\mathbb R^n$ consists of a $k$-dimensional submanifold $M$ of $\mathbb R^n$ together with a closed $1$-form $μ$ on $M$ such that the `twisted conormal bundle' $N^* M + μ$ is a special Lagrangian submanifold of $\mathbb C^n$. The 1-form $μ$ and the second fundamental form of $M$ must satisfy a particular system of coupled nonlinear second order PDE. We first review these twisted-austere conditions and give an explicit example. Then we focus on twisted-austere 3-folds, giving a geometric description of all solutions when the base $M$ is a cylinder and when $M$ is austere. Finally, we prove that, other than the case of a generalized helicoid in $\mathbb R^5$ discovered by Bryant, there are no other possibilities for the base $M$. This gives a complete classification of twisted-austere $3$-folds in $\mathbb R^n$.

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arXiv2020

Bryant-Salamon $\mathrm{G}_2$ manifolds and coassociative fibrations

Bryant-Salamon constructed three 1-parameter families of complete manifolds with holonomy $\mathrm{G}_2$ which are asymptotically conical to a holonomy $\mathrm{G}_2$ cone. For each of these families, including their asymptotic cone, we construct a fibration by asymptotically conical and conically singular coassociative 4-folds. We show that these fibrations are natural generalizations of the following three well-known coassociative fibrations on $\mathbb R^7$: the trivial fibration by 4-planes, the product of the standard Lefschetz fibration of $\mathbb C^3$ with a line, and the Harvey-Lawson coassociative fibration. In particular, we describe coassociative fibrations of the bundle of anti-self-dual 2-forms over the 4-sphere $\mathcal{S}^4$, and the cone on $\mathbb C \mathbb P^3$, whose smooth fibres are $T^*\mathcal{S}^2$, and whose singular fibres are $\mathbb R^4/\{\pm 1\}$. We relate these fibrations to hypersymplectic geometry, Donaldson's work on Kovalev-Lefschetz fibrations, harmonic 1-forms and the Joyce--Karigiannis construction of holonomy $\mathrm{G}_2$ manifolds, and we construct vanishing cycles and associative "thimbles" for these fibrations.

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arXiv2020

Deformation theory of G_2 conifolds

We consider the deformation theory of asymptotically conical (AC) and of conically singular (CS) $G_2$-manifolds. In the AC case, we show that if the rate of convergence $ν$ to the cone at infinity is generic in a precise sense and lies in the interval $(-4, 0)$, then the moduli space is smooth and we compute its dimension in terms of topological and analytic data. For generic rates $ν< -4$ in the AC case, and for generic positive rates of convergence to the cones at the singular points in the CS case, the deformation theory is in general obstructed. We describe the obstruction spaces explicitly in terms of the spectrum of the Laplacian on the link of the cones on the ends, and compute the virtual dimension of the moduli space. We also present many applications of these results, including: the uniqueness of the Bryant--Salamon AC $G_2$-manifolds via local rigidity and the cohomogeneity one property of AC $G_2$-manifolds asymptotic to homogeneous cones; the smoothness of the CS moduli space if the singularities are modeled on particular $G_2$-cones; and the proof of existence of a "good gauge" needed for desingularization of CS $G_2$-manifolds. Finally, we discuss some open problems.

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