arXiv subjects

Diego Artacho

Publications and source records attributed to Diego Artacho.

8 recordsLinked to original sources

arXiv2026

Stability of Einstein 4-manifolds satisfying a chiral curvature condition

Let $(M,g)$ be a compact oriented Einstein four-manifold with Einstein constant $E$ and let $\widehat{R}^+$ denote the action of the Riemann curvature tensor on self-dual two-forms. We show that if $\widehat{R}^+ < 0$, then $g$ is strictly linearly stable for the Einstein-Hilbert functional, thus giving a chiral criterion for stability. Our result is stronger than the one by Fine-Krasnov-Singer, who conclude local rigidity from $\widehat{R}^+ < 0$ by proving stability for a different action functional. Our proof proceeds by showing that $(M,g)$ admits a natural spin$^h$ structure carrying a non-zero parallel spin$^h$-spinor. We then apply a lower bound on the Lichnerowicz Laplacian on traceless symmetric two-tensors in the presence of such a spinor.

math.DG↗

arXiv2025

On the stability of Einstein metrics carrying a special twisted spinor

We prove linear semi-stability for a large class of Einstein metrics of non-positive scalar curvature. More precisely, we show that any Einstein $n$-manifold with non-positive scalar curvature carrying a parallel twisted pure spin$^r$ spinor is linearly semi-stable, under mild restrictions on $n$ and $r$. We thus extend the parallel spin and spin$^c$ stability results of Dai--Wang--Wei. As an application, our result implies linear semi-stability for all negative quaternion-K{ä}hler manifolds of dimension greater than $8$.

math.DG↗

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.

math.DG↗

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.

math.DG↗

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.

math.DG↗

arXiv2025

The Geometry of Generalised Spin$^r$ Spinors on Projective Spaces

In this paper, we adapt the characterisation of the spin representation via exterior forms to the generalised spin$^r$ context. We find new invariant spin$^r$ spinors on the projective spaces $\mathbb{CP}^n$, $\mathbb{HP}^n$, and the Cayley plane $\mathbb{OP}^2$ for all their homogeneous realisations. Specifically, for each of these realisations, we provide a complete description of the space of invariant spin$^r$ spinors for the minimum value of $r$ for which this space is non-zero. Additionally, we demonstrate some geometric implications of the existence of special spin$^r$ spinors on these spaces.

math.DG↗

arXiv2025

Invariant Spinors on Flag Manifolds

In this note, we characterise the existence of non-trivial invariant spinors on maximal flag manifolds associated to complex simple Lie algebras. This characterisation is based on the combinatorial properties of their set of positive roots. We also give some bounds for the dimension of the space of invariant spinors in each case.

math.DG↗

arXiv2025

Generalised Killing Spinors on Three-Dimensional Lie Groups

We present a complete classification of invariant generalised Killing spinors on three-dimensional Lie groups. We show that, in this context, the existence of a non-trivial invariant generalised Killing spinor implies that all invariant spinors are generalised Killing with the same endomorphism. Notably, this classification is independent of the choice of left-invariant metric. To illustrate the computational methods underlying this classification, we also provide the first known examples of homogeneous manifolds admitting invariant generalised Killing spinors with $n$ distinct eigenvalues for each $n > 4$.

math.DG↗