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Yann Rollin

Publications and source records attributed to Yann Rollin.

18 recordsLinked to original sources
arXiv2024

Isotropic maps and moment map flow

We consider the moduli space of isotropic maps from a closed surface $Σ$ to a symplectic affine space and construct a Kähler moment map geometry, on a space of differential forms on $Σ$, such that the isotropic maps correspond to certain zeroes of the moment map. The moment map geometry induces a modified moment map flow, whose fixed point set correspond to isotropic maps. This construction can be adapted to the polyhedral setting. In particular, we prove that the polyhedral modified moment map flow induces a strong deformation retraction from the space of polyhedral maps onto the space of polyhedral isotropic maps.

math.DG↗
arXiv2024

Symplectic maps and hyperKähler moment map geometry

We obtain a correspondence between the group of symplectic diffeomorphisms of a 4-dimensional real torus and the vanishing locus of a certain hyperKähler moment map. This observation gives rise to a new flow, called the modified moment map flow. The construction can be adapted to the polyhedral setting, for which we prove a Duistermaat type theorem. This paper lays out the ground work for some effective polyhedral symplectic geometry and for a potential Morse-Bott theory, with applications to the topology of the space of symplectic maps of the 4-torus.

math.SG↗
arXiv2022

Polyhedral approximation by Lagrangian and isotropic tori

We prove that every smoothly immersed 2-torus of $\mathbb{R}^4$ can be approximated, in the C0-sense, by immersed polyhedral Lagrangian tori. In the case of a smoothly immersed (resp. embedded) Lagrangian torus of $\mathbb{R}^4$, the surface can be approximated in the C1-sense by immersed (resp. embedded) polyhedral Lagrangian tori. Similar statements are proved for isotropic 2-tori of $\mathbb{R}^{2n}$.

math.SG↗
arXiv2019

Discrete geometry and isotropic surfaces

We consider smooth isotropic immersions from the 2-dimensional torus into $R^{2n}$, for $n \geq 2$. When $n = 2$ the image of such map is an immersed Lagrangian torus of $R^4$. We prove that such isotropic immersions can be approximated by arbitrarily $C^0$-close piecewise linear isotropic maps. If $n \geq 3$ the piecewise linear isotropic maps can be chosen so that they are piecewise linear isotropic immersions as well. The proofs are obtained using analogies with an infinite dimensional moment map geometry due to Donaldson. As a byproduct of these considerations, we introduce a numerical flow in finite dimension, whose limit provide, from an experimental perspective, many examples of piecewise linear Lagrangian tori in $R^4$. The DMMF program, which is freely available, is based on the Euler method and shows the evolution equation of discrete surfaces in real time, as a movie.

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

ALE scalar-flat Kähler metrics on non-compact weighted projective spaces

We construct new explicit toric scalar-flat K{ä}hler ALE metrics on weighted projective spaces of non-compact type, which we use to obtain smooth extremal K{ä}hler metrics on appropriate resolutions of orbifolds. In particular, we obtain new extremal metrics certain resolutions of weighted projective spaces of compact type.

math.DG↗
arXiv2013

Smoothing singular extremal Kähler surfaces and minimal Lagrangians

We consider smoothings of a complex surface with singularities of class T and no nontrivial holomorphic vector field. Under an hypothesis of non degeneracy of the smoothing at each singular point, we prove that if the singular surface admits an extremal metric, then the smoothings also admit extremal metrics in nearby Kähler classes. In addition, we construct small Lagrangian stationary spheres which represent Lagrangian vanishing cycles for surfaces close to the singular one.

math.DG↗
arXiv2013

K-stability and parabolic stability

Parabolic structures with rational weights encode certain iterated blowups of geometrically ruled surfaces. In this paper, we show that the three notions of parabolic polystability, K-polystability and existence of constant scalar curvature Kähler metrics on the iterated blowup are equivalent, for certain polarizations close to the boundary of the Kähler cone.

math.DG↗
arXiv2013

Deformations of extremal toric manifolds

Let $X$ be a compact toric extremal Kähler manifold. Using the work of Székelyhidi, we provide a combinatorial criterion on the fan describing $X$ to ensure the existence of complex deformations of $X$ that carry extremal metrics. As an example, we find new CSC metrics on 4-points blow-ups of $\C¶^1\times\C¶^1$.

math.DG↗
arXiv2012

Deformation of extremal metrics, complex manifolds and the relative Futaki invariant

Let (X,Ω) be a closed polarized complex manifold, g be an extremal metric on X that represents the Kähler class Ω, and G be a compact connected subgroup of the isometry group Isom(X,g). Assume that the Futaki invariant relative to G is nondegenerate at g. Consider a smooth family $(M \to B)$ of polarized complex deformations of (X,Ω)\simeq (M_0,Θ_0) provided with a holomorphic action of G with trivial action on B. Then for every t\in B sufficiently small, there exists an h^{1,1}(X)-dimensional family of extremal Kaehler metrics on M_t whose Kähler classes are arbitrarily close to Θ_t. We apply this deformation theory to show that certain complex deformations of the Mukai-Umemura 3-fold admit Kaehler-Einstein metrics.

math.DG↗
arXiv2009

Legendrian knots and monopoles

We prove a generalization of Bennequin's inequality for Legendrian knots in a 3-dimensional contact manifold (Y,xi), under the assumption that Y is the boundary of a 4-dimensional manifold M and the version of Seiberg-Witten invariants introduced by Kronheimer and Mrowka [Invent. Math. 130 (1997) 209-255] is nonvanishing. The proof requires an excision result for Seiberg-Witten moduli spaces; then the Bennequin inequality becomes a special case of the adjunction inequality for surfaces lying inside M.

math.DG↗
arXiv2007

Constant scalar curvature Kaehler surfaces and parabolic polystability

A complex ruled surface admits an iterated blow-up encoded by a parabolic structure with rational weights. Under a condition of parabolic stability, one can construct a Kaehler metric of constant scalar curvature on the blow-up according to math.DG/0412405. We present a generalization of this construction to the case of parabolically polystable ruled surfaces. Thus we can produce numerous examples of Kaehler surfaces of constant scalar curvature with circle or toric symmetry.

math.DG↗
arXiv2007

Wormholes in ACH Einstein manifolds

We give a new construction of Einstein and Kaehler-Einstein manifolds which are asymptotically complex hyperbolic, inspired by the work of Mazzeo-Pacard in the real hyperbolic case. The idea is to develop a gluing theorem for 1-handle surgery at infinity, which generalizes the Klein construction for the complex hyperbolic metric.

math.DG↗
arXiv2004

Non-minimal scalar-flat Kaehler surfaces and parabolic stability

A new construction is presented of scalar-flat Kaehler metrics on non-minimal ruled surfaces. The method is based on the resolution of singularities of orbifold ruled surfaces which are closely related to rank-2 parabolically stable holomorphic bundles. This rather general construction is shown also to give new examples of low genus: in particular, it is shown that CP^2 blown up at 10 suitably chosen points, admits a scalar-flat Kaehler metric; this answers a question raised by Claude LeBrun in 1986 in connection with the classification of compact self-dual 4-manifolds.

math.DG↗
arXiv2002

Rigidité d'Einstein du plan hyperbolique complexe

We prove that every Einstein metric on the unit ball B^4 of C^2, asymptotic to the Bergman metric, is equal to it up to a diffeomorphism. We need a solution of Seiberg--Witten equations in this infinite volume setting. Therefore, and more generally, if M^4 is a manifold with a CR-boundary at infinity, an adapted spinc-structure which has a non zero Kronheimer--Mrowka invariant and an asymptotically complex hyperbolic Einstein metric, we produce a solution of Seiberg--Witten equations with an strong exponential decay property.

math.DG↗
arXiv2001

Kahler surfaces of finite volume and Seiberg-Witten equations

Let M=P(E) be a ruled surface. We introduce metrics of finite volume on M whose singularities are parametrized by a parabolic structure over E. Then, we generalise results of Burns--de Bartolomeis and LeBrun, by showing that the existence of a singular Kahler metric of finite volume and constant non positive scalar curvature on M is equivalent to the parabolic polystability of E; moreover these metrics all come from finite volume quotients of $H^2 \times CP^1$. In order to prove the theorem, we must produce a solution of Seiberg-Witten equations for a singular metric g. We use orbifold compactifications $\overline M$ on which we approximate g by a sequence of smooth metrics; the desired solution for g is obtained as the limit of a sequence of Seiberg-Witten solutions for these smooth metrics.

math.DG↗