arXiv subjects

Eric Loubeau

Publications and source records attributed to Eric Loubeau.

14 recordsLinked to original sources

arXiv2026

Flows of geometric structures II

We advance the general theory of flows of tensorial $\mathrm{H}$-structures, focusing on non-isometric flows and on the case $\mathrm{H}=\mathrm{SU}(m)\subset\mathrm{SO}(2m)$. After developing the relevant $\mathrm{SU}(m)$ algebra, we compare two natural evolutions: the unrestricted negative gradient flow of the intrinsic-torsion energy and a Ricci-harmonic flow. We prove short-time existence and uniqueness for the Ricci-harmonic $\mathrm{H}$-flow, with arbitrary lower-order torsion-quadratic terms, for every closed subgroup $\mathrm{H}\subset\mathrm{SO}(n)$. For groups for which the projection to $\mathfrak{h}^\perp$ defines a $4$-form, including $\{1\}$, $\mathrm{SU}(2)$, $\mathrm{G}_2$, and $\mathrm{Spin}(7)$, we express the negative gradient flow in Ricci-harmonic form up to explicit lower-order torsion terms and prove short-time existence and uniqueness by a modified DeTurck argument. We treat the genuinely different $\mathrm{SU}(m)$ case by a separate principal-symbol computation, proving short-time existence and uniqueness for the unrestricted negative gradient flow of $\mathrm{SU}(m)$-structures. The same computation identifies the natural negative gradient flow of $\mathrm{U}(m)$-structures as a borderline case, which cannot be made strictly parabolic by first-order diffeomorphism gauges. For the modified Ricci-harmonic flow, we derive heat-type evolution equations for the intrinsic torsion, a doubling-time estimate and Shi-type derivative estimates for $(|\mathrm{Rm}|^2+|\nabla T|^2+|T|^4)^{1/2}$, and a finite-time continuation criterion. In dimension six, we translate the formalism into the standard torsion forms of an $\mathrm{SU}(3)$-structure and describe, to highest order, the corresponding family of second-order quasilinear $\mathrm{SU}(3)$-flows.

math.DG↗

arXiv2026

Topology of isometric classes and flows of geometric structures

We revisit flows of tensorial $H$-structures for closed and connected Lie subgroups $H\leqslant\mathrm{SO}(n)$, focusing on the topology of isometric classes. We prove that the natural map assigning to an $H$-structure its induced Riemannian metric is surjective and satisfies a parametric homotopy lifting property. Since the space of Riemannian metrics is contractible, the full space of $H$-structures is homotopy equivalent to any fixed isometric class. For parallelizable manifolds, especially flat tori, these classes reduce to mapping spaces into $\mathrm{SO}(n)/H$. We discuss almost Hermitian, $\mathrm{SU}(m)$, $\mathrm{G}_2$, and $\mathrm{Spin}(7)$ structures on flat tori, showing that their isometric classes and moduli modulo orientation-preserving diffeomorphisms may have infinitely many connected components. We relate this topology to the variational theory of the intrinsic torsion energy. On the unrestricted space of $H$-structures, the functional is scale-degenerate in dimensions $n>2$: its infimum is zero on every nonempty path component, and its only critical points are torsion-free structures. Inside fixed isometric classes this homothetic escape direction is absent. We reinterpret finite-time singularity formation as concentration in nontrivial isometric homotopy classes with zero energy infimum, and contrast this with cohomological classes, such as $\mathrm{U}(3)$-structures on the flat $6$-torus, which have positive lower bounds and admit smooth harmonic representatives from holomorphic maps into $\mathbb{CP}^3$. Finally, we revisit analytical aspects of our earlier work: we prove a lifting principle for metric-dependent flows, reinterpret the Ricci $H$-flow, derive a general evolution identity for isometric flows, and extend the harmonic-flow theory beyond the original structural assumptions.

math.DG↗

arXiv2024

A Weitzenb\"ock formula on Sasakian holomorphic bundles

This work seeks to advance the understanding of the smooth structure of the moduli space of self-dual contact instantons (SDCI) on Sasakian 7-manifolds M. A neighborhood of a smooth point of M is locally modeled on the first cohomological group of an elliptic complex (1.4). There is a cohomological obstruction to the smoothness for the moduli space, in terms of a second basic cohomological group, in this paper we study conditions under which this obstruction disappears, by computing a Weitzenb\"ock formula and using a Bochner-type method to obtain a vanishing theorem. Given an SDCI on a Sasakian bundle E, we find sufficient conditions for the vanishing of the obstruction in the positivity of a couple of operators R and F depending on the curvatures of the connection and the Riemann curvature of the Sasakian metric g. In particular, we find that if M is transversely Ricci positive and F positive, the moduli space of SDCI must be smooth. However, in general, the operator F is not positive definite and we describe bundles over the Stiefel manifold for which it is the case. Finally, we show that when the energy of the curvature is less than the first non-zero eigenvalue of RicT the obstruction vanishes.

math.DG↗

arXiv2022

Flows of geometric structures

We develop an abstract theory of flows of geometric $H$-structures, i.e., flows of tensor fields defining $H$-reductions of the frame bundle, for a closed and connected subgroup $H\subset SO(n)$, on any connected and oriented $n$-manifold with sufficient topology to admit such structures. The first part of the article sets up a unifying theoretical framework for deformations of $H$-structures, by way of the natural infinitesimal action of $\mathrm{GL}(n,\mathbb{R})$ on tensors combined with various bundle decompositions induced by $H$-structures. We compute evolution equations for the intrinsic torsion under general flows of $H$-structures and, as applications, we obtain general Bianchi-type identities for $H$-structures, and, for closed manifolds, a general first variation formula for the $L^2$-Dirichlet energy functional $\mathcal{E}$ on the space of $H$-structures. We then specialise the theory to the negative gradient flow of $\mathcal{E}$ over isometric $H$-structures, i.e., their harmonic flow. The core result is an almost monotonocity formula along the flow for a scale-invariant localised energy, similar to the classical formulae by Chen-Struwe for the harmonic map heat flow. This yields an $\varepsilon$-regularity theorem and an energy gap result for harmonic structures, as well as long-time existence for the flow under small initial energy, relative to the $L^\infty$-norm of initial torsion, in the spirit of Chen-Ding. Moreover, below a certain energy level, the absence of a torsion-free isometric $H$-structure in the initial homotopy class imposes the formation of finite-time singularities. These seemingly contrasting statements are illustrated by examples on flat $n$-tori, so long as $[\mathbb{S}^n,SO(n)/H]$ contains more than one element and the universal cover of $SO(n)/H$ is a sphere; e.g. when $n=7$ and $H=\rm G_2$, or $n=8$ and $H=\rm Spin(7)$.

math.DG↗

arXiv2021

Harmonic flow of $\mathrm{Spin}(7)$-structures

We formulate and study the isometric flow of $\mathrm{Spin}(7)$-structures on compact $8$-manifolds, as an instance of the harmonic flow of geometric structures. Starting from a general perspective, we establish Shi-type estimates and a correspondence between harmonic solitons and self-similar solutions for arbitrary isometric flows of $H$-structures. We then specialise to $H=\mathrm{Spin}(7)\subset\mathrm{SO}(8)$, obtaining conditions for long-time existence, via a monotonicity formula along the flow, which actually leads to an $\varepsilon$-regularity theorem. Moreover, we prove Cheeger--Gromov and Hamilton-type compactness theorems for the solutions of the harmonic flow, and we characterise Type-$\mathrm{I}$ singularities as being modelled on shrinking solitons.We also establish a Bryant-type description of isometric $\mathrm{Spin}(7)$-structures, based on squares of spinors, which may be of independent interest.

math.DG↗

arXiv2021

Harmonic $Sp(2)$-invariant $G_2$-structures on the $7$-sphere

We describe the $10$-dimensional space of $Sp(2)$-invariant $G_2$-structures on the homogeneous $7$-sphere $S^7=Sp(2)/Sp(1)$ as $\mathbb{R}^+\times Gl^+(3,\mathbb{R})$. In those terms, we formulate a general Ansatz for $G_2$-structures, which realises representatives in each of the $7$ possible isometric classes of homogeneous $G_2$-structures. Moreover, the well-known nearly parallel round and squashed metrics occur naturally as opposite poles in an $S^3$-family, the equator of which is a new $S^2$-family of coclosed $G_2$-structures satisfying the harmonicity condition $div T=0$. We show general existence of harmonic representatives of $G_2$-structures in each isometric class through explicit solutions of the associated flow and describe the qualitative behaviour of the flow. We study the stability of the Dirichlet gradient flow near these critical points, showing explicit examples of degenerate and nondegenerate local maxima and minima, at various regimes of the general Ansatz. Finally, for metrics outside of the Ansatz, we identify families of harmonic $G_2$-structures, prove long-time existence of the flow and study the stability properties of some well-chosen examples.

math.DG↗

arXiv2020

Harmonic vector fields on extended 3-dimensional Riemannian Lie groups

Given two Riemannian manifolds $(B,g_B)$ and $(F,g_F)$, we give harmonicity conditions for vector fields on the Riemannian warped product $B\times_fF$, with $f:B \longrightarrow ]0,+\infty[$, using a characteristic variational condition. Then, we apply this to the case $B=\mathbb{R}$ and $F$ is a three-dimensional connected Riemannian Lie group $G$ equipped with a left-invariant metric, to determine harmonic vector fields on $\mathbb{R}\times_fG$. We give examples of harmonic vector fields on $G$ which are not left-invariant and determine harmonic vector fields on $\mathbb{R}\times_fG$. We conclude with some examples of vector fields on $\mathbb{R}\times_fG$ which are harmonic maps.

math.DG↗

arXiv2020

Unique Continuation Property for Biharmonic Hypersurfaces in Spheres

We study properties of non-minimal biharmonic hypersurfaces of spheres. The main result is a CMC Unique Continuation Theorem for biharmonic hypersurfaces of spheres. We then deduce new rigidity theorems to support the Conjecture that biharmonic submanifolds of Euclidean spheres must be of constant mean curvature.

math.DG↗

arXiv2019

Harmonic flow of geometric structures

We give a twistorial interpretation of geometric structures on a Riemannian manifold, as sections of homogeneous fibre bundles, following an original insight by Wood (2003). The natural Dirichlet energy induces an abstract harmonicity condition, which gives rise to a geometric gradient flow. We establish a number of analytic properties for this flow, such as uniqueness, smoothness, short-time existence, and some sufficient conditions for long-time existence. This description potentially subsumes a large class of geometric PDE problems from different contexts. As applications, we recover and unify a number of results in the literature: for the isometric flow of ${\rm G}_2$-structures, by Grigorian (2017, 2019), Bagaglini (2019), and Dwivedi-Gianniotis-Karigiannis (2019); and for harmonic almost complex structures, by He (2019) and He-Li (2019). Our theory also establishes original properties regarding harmonic flows of parallelisms and almost contact structures.

math.DG↗

arXiv2018

Bochner-Simons formulas and the rigidity of biharmonic submanifolds

We find some integral formulas of Simons and Bochner type and use them to study biharmonic and biconservative submanifolds in space forms. We obtain rigidity results that in the biharmonic case represent partial answers to two well-known conjectures on such submanifolds in spheres.

math.DG↗

arXiv2016

Biharmonic tori in spheres

We consider proper-biharmonic flat tori with constant mean curvature (CMC) in spheres and find necessary and sufficient conditions for certain rectangular tori and square tori to admit full CMC proper-biharmonic immersions in $\mathbb{S}^n$, as well as the explicit expressions of some of these immersions.

math.DG↗

arXiv2007

Harmonic morphisms between Weyl spaces and twistorial maps II

We define, on smooth manifolds, the notions of almost twistorial structure and twistorial map, thus providing a unified framework for all known examples of twistor spaces. The condition of being harmonic morphisms naturally appears among the geometric properties of submersive twistorial maps between low-dimensional Weyl spaces endowed with a nonintegrable almost twistorial structure due to Eells and Salamon. This leads to the twistorial characterisation of harmonic morphisms between Weyl spaces of dimensions four and three. Also, we give a thorough description of the twistorial maps with one-dimensional fibres from four-dimensional Weyl spaces endowed with the almost twistorial structure of Eells and Salamon.

math.DG↗

arXiv1996

Pluriharmonic Morphisms

Pluriharmonic maps form an important class of harmonic maps which includes holomorphic maps. We study their morphisms, in particular the inter-relationships between $(1,1)$-geodesic, pluriharmonic and $\pm$holomorphic maps. Then we characterise pluriharmonic morphisms between Hermitian manifolds. We make a special study of the situation where the target is K{ä}hler, pluriharmonic morphisms being particularly well understood for this case.

dg-ga↗

arXiv1996

Pseudo Harmonic Morphisms

We study a geometrical condition (PHWC) which is weaker than horizontal weak conformality. In particular, we show that harmonic maps satisfying this condition, which will be called {\em pseudoharmonic morphisms}, include harmonic morphisms and can be described as pulling back certain germs to certain other germs. Finally, we construct a canonical f-structure associated to every map satisfying (PHWC) and find conditions on this f-structure to ensure the harmonicity of the map.

dg-ga↗