arXiv subjects

Marie-Amélie Lawn

Publications and source records attributed to Marie-Amélie Lawn.

13 recordsLinked to original sources

arXiv2026

The Lebrun--Salamon conjecture is true in dimension up to 56

We prove the LeBrun--Salamon conjecture for positive quaternionic Kähler manifolds of quaternionic dimension n <= 14. To do so, we first show that the isometry group Isom(M) of any minimal counterexample M has rank at most one, hence dimension at most three. On the other hand, we consider a characteristic number which equals dim Isom(M) - (n+3) if n is even and dim Isom(M) - (n+1) if n is odd, and show that this index is a sum of positive quantities when n <= 14, yielding a contradiction. The toolset used to show positivity enlarges in stages. Projection curvature moments suffice through n=8; positive-semidefinite curvature moments extend the argument through n=10; and signed scalar orbital moments give the new inequalities needed for n=11,12. Scalar orbital moments alone fail at n=13; combining them with Hodge--Riemann positivity for squares of cohomology classes extends the proof to n=13,14. We give exact separating obstructions explaining the changes of method.

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arXiv2026

Newman--Penrose formalism in $3$-dimensional trans-Sasakian manifolds

We study $3$-dimensional trans-Sasakian manifolds using the Newman--Penrose formalism. In this framework, the geometry of the structure vector field is encoded by scalar spin coefficients: acceleration, shear, expansion, and twist. A central observation is that, in dimension $3$, the trans-Sasakian condition is equivalent to the characteristic vector field defining a shear-free geodesic congruence, or equivalently a conformal foliation by geodesics. Thus, the Newman--Penrose equations provide a direct scalar formulation of the conformal foliations studied by Baird and Wood in the theory of harmonic morphisms. Within this framework, we derive curvature and Laplacian identities for trans-Sasakian manifolds and their main subclasses, including formulae for the Ricci tensor, scalar curvature, Einstein condition, rough Laplacian, divergence and harmonicity of the characteristic vector field, together with several illustrative examples. As an application, we consider trans-Sasakian structures compatible with fixed homogeneous metrics of type ${\Bbb E}(κ,τ)$. We prove a rigidity result: in the non-space-form cases, the Newman--Penrose equations force the characteristic vector field to be vertical. Hence, for $τ\neq0$ and $κ\neq4τ^2$, every compatible trans-Sasakian structure is the canonical vertical $α$-Sasakian structure, while for $τ=0$ and $κ\neq0$, it is vertical and cosymplectic. In particular, these non-space-form homogeneous metrics admit no proper compatible trans-Sasakian structures.

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arXiv2025

New Examples of Translating Solitons in Generalised Robertson-Walker Geometries

Translators can be regarded as submanifolds which satisfy the mean curvature flow equation when evolving by translations along a distinguished vector field of the ambient space. We study translators in Generalised Robertson-Walker spacetimes, due to their importance as Lorentzian manifolds, and because they admit a natural conformal Killing timelike vector field carrying substantial geometric information, which will play the role of this translating vector field. We identify three one-parameter families of warping functions for which these objects exist. As a first example of this notion of translator, we classify the analogues of the classical Grim Reapers within this context.

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arXiv2025

Killing Mean Curvature Solitons from Riemannian Submersions

We present a new general construction of examples of mean curvature solitons on manifolds admitting a nowhere-vanishing Killing vector field. Using Riemannian submersion techniques, we reduce the problem from a PDE to an ODE. As an application, we obtain new examples of rotators in hyperbolic space.

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arXiv2025

Generalised Spin$^r$ Structures on Homogeneous Spaces

Spinorial methods have proven to be a powerful tool to study geometric properties of spin manifolds. Our aim is to continue the spinorial study of manifolds that are not necessarily spin. We introduce and study the notion of $G$-invariance of spin$^r$ structures on a manifold $M$ equipped with an action of a Lie group $G$. For the case when $M$ is a homogeneous $G$-space, we prove a classification result of these invariant structures in terms of the isotropy representation. As an example, we study the invariant spin$^r$ structures for all the homogeneous realisations of the spheres.

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arXiv2023

Spinorial representation of submanifolds in a product of space forms

We present a method giving a spinorial characterization of an immersion in a product of spaces of constant curvature. As a first application we obtain a proof using spinors of the fundamental theorem of immersion theory in that spaces. We also study special cases: we recover previously known results concerning immersions in $\mathbb{S}^2\times\mathbb{R}$ and we obtain new spinorial characterizations of immersions in $\mathbb{S}^2\times\mathbb{R}^2$ and in $\mathbb{H}^2\times\mathbb{R}.$ We then study the theory of $H=1/2$ surfaces in $\mathbb{H}^2\times\mathbb{R}$ using this spinorial approach, obtain new proofs of some of its fundamental results and give a direct relation with the theory of $H=1/2$ surfaces in $\mathbb{R}^{1,2}.$

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arXiv2023

Invariant Spinors on Homogeneous Spheres

Using the characterization of the spin representation in terms of exterior forms, we give a complete classification of invariant spinors on the nine homogeneous realizations of the sphere $S^n$. In each of the cases we determine the dimension of the space of such spinors, give their explicit description, and study the underlying related geometric structures depending on the metric. We recover some known results in the Sasaki and 3-Sasaki cases and find several new examples: in particular we give the first known examples of generalized Killing spinors with four distinct eigenvalues.

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arXiv2022

G-invariant Spin Structures on Spheres

We examine which of the compact connected Lie groups that act transitively on spheres of different dimensions leave the unique spin structure of the sphere invariant. We study the notion of invariance of a spin structure and prove this classification in two different ways; through examining the differential of the actions and through representation theory.

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arXiv2019

Associated Families of Surfaces in Warped Products and Homogeneous Spaces

We classify Riemannian surfaces admitting associated families in three dimensional homogeneous spaces with four-dimensional isometry groups and in a wide family of (semi-Riemannian) warped products, with an extra natural condition (namely, rotating structure vector field). We prove that, provided the surface is not totally umbilical, such families exist in both cases if, and only if, the ambient manifold is a product and the surface is minimal. In particular, there exists no associated families of surfaces with rotating structure vector field in the Heisenberg group.

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arXiv2015

A Fundamental Theorem for submanifolds of multiproducts of real space forms

We prove a Bonnet theorem for isometric immersions of submanifolds into the products of an arbitrary number of simply connected real space forms. Then, we prove the existence of associated families of minimal surfaces in such products. Finally, in the case of $\mathbb{S}^2\times\mathbb{S}^2$, we give a complex version of the main theorem in terms of the two canonical complex structures of $\mathbb{S}^2\times\mathbb{S}^2$.

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arXiv2012

Associated Families of Immersions of Three Dimensional CR Manifolds in Euclidean Spaces

We consider isometric immersions in arbitrary codimension of three-dimensional strongly pseudoconvex pseudo-hermitian CR manifolds into the Euclidean space $\mathbb{R}^n$ and generalize in a natural way the notion of associated family. We show that the existence of such deformations turns out to be very restrictive and we give a complete classification.

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arXiv2012

Isometric and CR pluriharmonic immersions of three dimensional CR manifolds in Euclidean spaces

Using a bigraded differential complex depending on the CR and pseudohermitian structure, we give a characterization of three-dimensional strongly pseudoconvex pseudo-hermitian CR-manifolds isometrically immersed in Euclidean space $\mathbb{R}^n$ in terms of an integral representation of Weierstrass type. Restricting to the case of immersions in $\mathbb{R}^4$, we study harmonicity conditions for such immersions and give a complete classification of CR-pluriharmonic immersions.

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arXiv2008

Isometric Immersions of Hypersurfaces in 4-dimensional Manifolds via Spinors

We give a spinorial characterization of isometrically immersed hypersurfaces into 4-dimensional space forms and product spaces $\M^3(κ)\times\R$, in terms of the existence of particular spinor fields, called generalized Killing spinors or equivalently solutions of a Dirac equation. This generalizes to higher dimensions several recent results for surfaces by T. Friedrich, B. Morel and the two authors. The main argument is the interpretation of the energy-momentum tensor of a generalized Killing spinor as the second fundamental form, possibly up to a tensor depending on the ambient space. As an application, we deduce a non-existence result for hypersurfaces in the 4-dimensional Euclidean space.

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