Journées Nancéiennes de Géométrie 2016
mardi 19 et mercredi 20 janvier 2016




Mardi 19 janvier 2016

Café, thé d'accueil
Pierre Jammes (Université de Nice)
Multiplicity of eigenvalues on surfaces, old and new

According to a theorem of Cheng, the multiplicity of the k-th eigenvalue of a Schrödinger operator on a compact surface of genus g is bounded by a constant depending only on k and g. For k=2, Colin de Verdière conjectured that this bound is related to the chromatic number of the surface. I will survey this subject, and present results obtained recently for the multiplicity of Steklov eigenvalues on surfaces with boundary.
Andrei Moroianu (Université de Versailles-Saint Quentin)
The holonomy problem for locally conformally Kähler metrics

A locally conformally Kähler (lcK) manifold is a complex manifold $(M,J)$ together with a $J$-compatible Riemannian metric $g$ which has the property that around every point of $M$ there exists a locally defined Kähler metric belonging to the conformal class of $g$. In this talk I will discuss some recent progress towards the classification of compact lcK manifolds with reduced holonomy obtained in collaboration with Farid Madani and Mihaela Pilca. In particular, I will describe all manifolds admitting two non-homothetic Kähler metrics in the same conformal class.
Georges Habib (Université Libanaise)
Eigenvalue Estimate for the basic Laplacian on manifolds with foliated boundary

In this paper, we give a sharp lower bound for the first eigenvalue of the basic Laplacian acting on basic $1$-forms defined on a compact manifold whose boundary is endowed with a Riemannian flow. The limiting case gives rise to a particular geometry of the flow and the boundary. Namely, the flow is a local product and the boundary is $\eta$-umbilical. This allows to characterize the quotient of $\mathbb{R}\times B'$ by some group $\Gamma$ as being the limiting manifold. Here $B'$ denotes the unit closed ball. Finally, we deduce several rigidity results describing the product $\mathbb{S}^1\times \mathbb{S}^n$ as the boundary of a manifold. This is a joint work with Fida El Chami, Ola Makhoul and Roger Nakad.

Mercredi 20 janvier 2016

Volker Branding (Technische Universität Wien)
On the nodal set of solutions to spinorial equations on closed surfaces

We derive estimates on the nodal set of solutions to several spinorial equations on closed surfaces. With the help of these estimates we obtain various non-existence results. We apply this technique to eigenspinors of the classical Dirac operator, twistor spinors, solutions to a nonlinear Dirac equation and the twisted Dirac operator that appears in the context of Dirac-harmonic maps.
Gaël Cousin (Université de Strasbourg)
Foliations of the plane induced by simple derivations

I will present a joint work with Luis Gustavo Mendes and Ivan Pan. Simple derivations of $\mathbb{C}[x,y]$ correspond to polynomial vector fields on $\mathbb{C}^2$ that possess neither invariant algebraic curves nor singularities. We study the foliations of $\mathbb{C}\mathrm{P}^2$ arising from these vector fields and give their position in the Kodaira-type classification of foliated surfaces due to Brunella, McQuillan and Mendes.
Nicolina Istrati (Université Paris Diderot)
Twisted holomorphic symplectic forms

It is a fact based on Yau's theorem that compact hyperkähler manifolds are exactly the Kähler manifolds admitting a holomorphic symplectic form. This talk will be about the the twisted analogue, and we will show that a compact Kähler manifold admitting a non-degenerate holomorphic 2-form valued in a line bundle is a finite cyclic cover of a hyperkähler manifold. Moreover, we will see that the converse is not generally true but needs additional hypotheses on the structure of the fundamental group of the manifold in order to hold.

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