arXiv subjects

Eveline Legendre

Publications and source records attributed to Eveline Legendre.

24 records · Page 2Linked to original sources

arXiv2015

The Einstein-Hilbert functional and the Sasaki-Futaki invariant

We show that the Einstein-Hilbert functional, as a functional on the space of Reeb vector fields, detects the vanishing Sasaki-Futaki invariant. In particular, this provides an obstruction to the existence of a constant scalar curvature Sasakian metric. As an application we prove that K-semistable polarized Sasaki manifold has vanishing Sasaki-Futaki invariant. We then apply this result to show that under the right conditions on the Sasaki join manifolds of [7] a polarized Sasaki manifold is K-semistable if only if it has constant scalar curvature.

math.DG↗

arXiv2013

Toric Kähler-Einstein metrics and convex compact polytopes

We show that any compact convex simple lattice polytope is the moment polytope of a Kähler-Einstein orbifold, unique up to orbifold covering and homothety. We extend the Wang-Zhu Theorem \cite{WZ} giving the existence of a Kähler-Ricci soliton on any toric monotone manifold on any compact convex simple labelled polytope satisfying the combinatoric condition corresponding to monotonicity. We obtain that any compact convex simple polytope $P\subset \bR^n$ admits a set of inward normals, unique up to dilatation, such that there exists a symplectic potential satisfying the Guillemin boundary condition (with respect to these normals) and the Kähler-Einstein equation on $P\times \bR^n$. We interpret our result in terms of existence of singular Kähler-Einstein metrics on toric manifolds.

math.DG↗

arXiv2012

Toric Generalized Kähler-Ricci Solitons with Hamiltonian 2-form

We show that the generalized Kähler-Ricci soliton equation on 4-dimensional toric Kähler orbifolds reduces to ODEs assuming there is a Hamiltonian 2-form. This leads to an explicit resolution of this equation on labeled triangles and convex labeled quadrilaterals. In particular, we give the explicit expression of the Kähler-Ricci solitons of weighted projective planes as well as new examples.

math.DG↗

arXiv2010

Toric geometry of convex quadrilaterals

We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric Kähler-Einstein and toric Sasaki-Einstein metrics constructed in [6,22,14]. As a byproduct, we obtain a wealth of extremal toric (complex) orbi-surfaces, including Kähler-Einstein ones, and show that for a toric orbi-surface with 4 fixed points of the torus action, the vanishing of the Futaki invariant is a necessary and sufficient condition for the existence of Kähler metric with constant scalar curvature. Our results also provide explicit examples of relative K-unstable toric orbi-surfaces that do not admit extremal metrics.

math.DG↗

arXiv2010

Existence and non uniqueness of constant scalar curvature toric Sasaki metrics

We study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least 5. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which the transversal Futaki invariant (restricted to the Lie algebra of the torus) vanishes. Using existence result of [25], we show that a co-oriented compact toric contact 5-manifold whose moment cone has 4 facets admits a finite number of rays of transversal homothetic compatible toric Sasaki metrics with constant scalar curvature. We point out a family of well-known toric contact structures on $S^2\times S^3$ admitting two non isometric and non transversally homothetic compatible toric Sasaki metrics with constant scalar curvature.

math.DG↗

arXiv2007

Extrema of low eigenvalues of the Dirichlet-Neumann Laplacian on a disk

We study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of Dirichlet-Neumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact 1-parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded number of parts on which Dirichlet and Neumann conditions are imposed alternately, the first eigenvalue is maximized by the uniformly distributed partition.

math.SP↗