arXiv subjects

Eveline Legendre

Publications and source records attributed to Eveline Legendre.

At least 19 recordsLinked to original sources

arXiv2026

The toric CR Yamabe problem

We study the equivariant CR Yamabe problem for a fixed-point-free torus action on a co-oriented compact contact manifold. Such examples arise naturally in the study of Sasakian manifolds with constant scalar curvature. In the toric case, we show that this problem is equivalent to a kind of boundary value problem for an elliptic PDE on a pair of functions defined on a convex, polyhedral domain. We provide examples of solutions and prove that many contact toric manifolds admit compatible CR structures with distinct signs of CR Yamabe invariants.

math.DG↗

arXiv2025

The CR Yamabe invariant and constant scalar curvature Sasaki metrics

We propose a new approach to the existence of constant transversal scalar curvature Sasaki structures drawing on ideas and tools from the CR Yamabe problem, establishing a link between the CR Yamabe invariant, the existence of Sasaki structures of constant transversal scalar curvature, and the K-stability of Sasaki manifolds. Assuming that the Sasaki-Reeb cone contains a regular vector field, we show that if the CR Yamabe invariant of a compact Sasaki manifold attains a specific value determined by the geometry of the Reeb cone, then the Sasaki manifold is K-semistable. Under the additional assumption of non-positive average scalar curvature, the CR Yamabe invariant attains this topological value if the manifold admits approximately constant scalar curvature Sasaki structures, and we also show a partial converse. As an application, we provide a new numerical criterion for the K-semistability of polarised compact complex manifolds.

math.DG↗

arXiv2025

From Kähler Ricci solitons to Calabi-Yau Kähler cones

We show that if $X$ is a smooth Fano manifold which caries a Kähler Ricci soliton, then the canonical cone of the product of $X$ with a complex projective space of sufficiently large dimension is a Calabi--Yau cone. This can be seen as an asymptotic version of a conjecture by Mabuchi and Nikagawa. This result is obtained by the openness of the set of weight functions $v$ over the momentum polytope of a given smooth Fano manifold, for which a $v$-soliton exists. We discuss other ramifications of this approach, including a Licherowicz type obstruction to the existence of a Kähler Ricci soliton and a Fujita type volume bound for the existence of a $v$-soliton.

math.DG↗

arXiv2024

Twins in K{ä}hler and Sasaki geometry

We introduce the notions of weighted extremal K{ä}hler twins together with the related notion of extremal Sasaki twins. In the Kähler setting this leads to a generalization of the twinning phenomenon appearing among LeBrun's strongly Hermitian solutions to the Einstein-Maxwell equations on the first Hirzebruch surface \cite{Leb16} to weighted extremal metrics on Hirzebruch surfaces in general. We discover that many twins appear and that this can be viewed in the Sasaki setting as a case where we have more than one extremal ray in the Sasaki cone even when we do not allow changes within the isotopy class. We also study extremal Sasaki twins directly in the Sasaki setting with a main focus on the toric Sasaki case.

math.DG↗

arXiv2023

The Einstein-Hilbert functional and the Donaldson-Futaki invariant

Given a Kaehler manifold polarised by a holomorphic ample line bundle, we consider the circle bundle associated to the polarisation with the induced transversal holomorphic structure. The space of contact structures compatible with this transversal structure is naturally identified with a bundle, of infinite rank, over the space of Kaehler metrics in the first Chern class of the polarisation. We show that the Einstein-Hilbert functional of the associated Tanaka-Webster connections is a functional on this bundle, whose critical points are constant scalar curvature Sasaki structures. In particular, when the group of automorphisms is discrete, these critical points correspond to constant scalar curvature Kaehler metrics in the first Chern class of the polarisation. We show that the Einstein--Hilbert functional satisfies some monotonicity properties along some one-parameter families of CR-contact structures that are naturally associated to test configurations, and that its limit on the central fibre of a test configuration is related to the Donaldson-Futaki invariant through an expansion in terms of an extra real parameter. As a by-product, we obtain an original proof that the existence of constant scalar curvature Kaehler metrics on a polarized manifold implies K-semistability. We also show that the limit of the Einstein-Hilbert functional on the central fibre coincides with the ratio of the equivariant index characters pole coefficients of the central fibre.

math.DG↗

arXiv2023

Existence and Non-Existence of Constant Scalar Curvature and Extremal Sasaki Metrics

We discuss the existence and non-existence of constant scalar curvature, as well as extremal, Sasaki metrics. We prove that the natural Sasaki-Boothby-Wang manifold over the admissible projective bundles over local products of non-negative CSC Kähler metrics, as described in https://link-springer-com.libproxy.unm.edu/article/10.1007/s00222-008-0126-x, always has a constant scalar curvature (CSC) Sasaki metric in its Sasaki-Reeb cone. Moreover, we give examples that show that the extremal Sasaki--Reeb cone, defined as the set of Sasaki--Reeb vector fields admitting a compatible extremal Sasaki metric, is not necessarily connected in the Sasaki--Reeb cone, and it can be empty even in the non-Gorenstein case. We also show by example that a non-empty extremal Sasaki--Reeb cone need not contain a (CSC) Sasaki metric which answers a question posed in https://mathscinet-ams-org.libproxy.unm.edu/mathscinet-getitem?mr=4420789. The paper also contains an appendix where we explore the existence of Kähler metrics of constant weighted scalar curvature, as defined in https://londmathsoc-onlinelibrary-wiley-com.libproxy.unm.edu/doi/full/10.1112/plms.12255, on admissible manifolds over local products of non-negative CSC Kähler metrics.

math.DG↗

arXiv2021

Valuative stability of polarised varieties

Fujita and Li have given a characterisation of K-stability of a Fano variety in terms of quantities associated to valuations, which has been essential to all recent progress in the area. We introduce a notion of valuative stability for arbitrary polarised varieties, and show that it is equivalent to K-stability with respect to test configurations with integral central fibre. The numerical invariant governing valuative stability is modelled on Fujita's beta invariant, but includes a term involving the derivative of the volume. We give several examples of valuatively stable and unstable varieties, including the toric case. We also discuss the role that the delta invariant plays in the study of valuative stability and K-stability of polarised varieties.

math.AG↗

arXiv2020

Weighted K-stability of polarized varieties and extremality of Sasaki manifolds

We use the correspondence between extremal Sasaki structures and weighted extremal Kahler metrics defined on a regular quotient of a Sasaki manifold, established by the first two authors, and Lahdili's theory of weighted K-stability in order to define a suitable notion of (relative) weighted K-stability for compact Sasaki manifolds of regular type. We show that the (relative) weighted K-stability with respect to a maximal torus is a necessary condition for the existence of a (possibly irregular) extremal Sasaki metric. We also compare weighted K-stability to the K-stability of the corresponding polarized affine cone (introduced by Collins-Szekelyhidi), and prove that they agree on the class of test configurations we consider. As a byproduct, we strengthen the obstruction to the existence of a scalar-flat Kahler cone metric from the K-semistability to the K-stability on these test configurations. We use our approach to give a characterization of the existence of a compatible extremal Sasaki structure on a principal circle bundle over an admissible ruled manifold, expressed in terms of the positivity of a single polynomial of one variable over a given interval.

math.DG↗

arXiv2020

Localizing the Donaldson-Futaki invariant

We use the equivariant localization formula to prove that the Donaldson-Futaki invariant of a compact smooth (K{ä}hler) test configuration coincides with the Futaki invariant of the induced action on the central fiber when this fiber is smooth or have orbifold singularities. We also localize the Donaldson-Futaki invariant of the deformation to the normal cone.

math.DG↗

arXiv2020

Toric Sasaki-Einstein metrics with conical singularities

We show that any toric Kähler cone with smooth compact cross-section admits a family of Calabi-Yau cone metrics with conical singularities along its toric divisors. The family is parametrized by the Reeb cone and the angles are given explicitly in terms of the Reeb vector field. The result is optimal, in the sense that any toric Calabi-Yau cone metric with conical singularities along the toric divisor (and smooth elsewhere) belongs to this family. We also provide examples and interpret our results in terms of Sasaki-Einstein metrics.

math.DG↗

arXiv2019

Some Open Problems in Sasaki Geometry

This paper has been submitted to the Proceedings of the Australian-German Workshop on Differential Geometry in the Large held at the mathematical research institute MATRIX in Creswick, Victoria, Australia, Feb.2-Feb.14, 2019. We describe and discuss 2 important open problems in Sasaki geometry.

math.DG↗

arXiv2018

A note on extremal toric almost Kähler metrics

An almost Kähler structure is {\it extremal} if the Hermitian scalar curvature is a Killing potential [29]. When the almost complex structure is integrable it coincides with extremal Kähler metric in the sense of Calabi [8]. We observe that the existence of an extremal {\it toric} almost Kähler structure of involutive type implies uniform K-stability and we point out the existence of a formal solution of the Abreu equation for any angle along the invariant divisor. Applying the recent result of Chen--Cheng [10] and He [27], we conclude that the existence of a compatible extremal toric almost Kähler structure of involutive type on a compact symplectic toric manifold is equivalent to its relative uniform $K$--stability (in a toric sense). As an application, using [5], we get the existence of an extremal toric Kähler metric in each Kähler class of $\mathbb{P}(\mathcal{O}\oplus \mathcal{O}(k_1) \oplus \mathcal{O}(k_2))$.

math.DG↗

arXiv2018

Reducibility in Sasakian Geometry

The purpose of this paper is to study reducibility properties in Sasakian geometry. First we give the Sasaki version of the de Rham Decomposition Theorem; however, we need a mild technical assumption on the Sasaki automorphism group which includes the toric case. Next we introduce the concept of {\it cone reducible} and consider $S^3$ bundles over a smooth projective algebraic variety where we give a classification result concerning contact structures admitting the action of a 2-torus of Reeb type. In particular, we can classify all such Sasakian structures up to contact isotopy on $S^3$ bundles over a Riemann surface of genus greater than zero. Finally, we show that in the toric case an extremal Sasaki metric on a Sasaki join always splits.

math.DG↗

arXiv2017

Toric contact geometry in arbitrary codimension

We define toric contact manifolds in arbitrary codimension and give a description of such manifolds in terms of a kind of labelled polytope embedded into a grassmannian, analogous to the Delzant polytope of a toric symplectic manifold.

math.DG↗

arXiv2017

Levi-Kahler reduction of CR structures, products of spheres, and toric geometry

We study CR geometry in arbitrary codimension, and introduce a process, which we call the Levi-Kahler quotient, for constructing Kahler metrics from CR structures with a transverse torus action. Most of the paper is devoted to the study of Levi-Kahler quotients of toric CR manifolds, and in particular, products of odd dimensional spheres. We obtain explicit descriptions and characterizations of such quotients, and find Levi-Kahler quotients of products of 3-spheres which are extremal in a weighted sense introduced by G. Maschler and the first author.

math.DG↗

arXiv2017

An application of the Duistertmaat--Heckman Theorem and its extensions in Sasaki Geometry

Building on an idea laid out by Martelli--Sparks--Yau, we use the Duistermaat-Heckman localization formula and an extension of it to give rational and explicit expressions of the volume, the total transversal scalar curvature and the Einstein--Hilbert functional, seen as functionals on the Sasaki cone (Reeb cone). Studying the leading terms we prove they are all proper. Among consequences we get that the Einstein-Hilbert functional attains its minimal value and each Sasaki cone possess at least one Reeb vector field with vanishing transverse Futaki invariant.

math.DG↗

arXiv2016

Hamiltonian stationary Lagrangian fibrations

Hamiltonian stationary Lagrangian submanifolds (HSLAG) are a natural generalization of special Lagrangian manifolds (SLAG). The latter only make sense on Calabi-Yau manifolds whereas the former are defined for any almost Kähler manifold. Special Lagrangians, and, more specificaly, fibrations by special Lagrangians play an important role in the context of the geometric mirror symmetry conjecture. However, these objects are rather scarce in nature. On the contrary, we show that HSLAG submanifolds, or fibrations, arise quite often. Many examples of HSLAG fibrations are provided by toric Käh-ler geometry. In this paper, we obtain a large class of examples by deforming the toric metrics into non toric almost Kähler metrics, together with HSLAG submanifolds.

math.DG↗

arXiv2016

Toric aspects of the first eigenvalue

In this paper we study the smallest non-zero eigenvalue $λ_1$ of the Laplacian on toric Kähler manifolds. We find an explicit upper bound for $λ_1$ in terms of moment polytope data. We show that this bound can only be attained for $\mathbb{CP}^n$ endowed with the Fubini-Study metric and therefore $\mathbb{CP}^n$ endowed with the Fubini-Study metric is spectrally determined among all toric Kähler metrics. We also study the equivariant counterpart of $λ_1$ which we denote by $λ_1^T$. It is the the smallest non-zero eigenvalue of the Laplacian restricted to torus-invariant functions. We prove that $λ_1^T$ is not bounded among toric Kähler metrics thus generalizing a result of Abreu-Freitas on $S^2$. In particular, $λ_1^T$ and $λ_1$ do not coincide in general.

math.DG↗