Plücker Relations on Schur Functions

We present a set of algebraic relations among Schur functions which are a multi-time generalization of the discrete Hirota relations known to hold among the Schur functions of rectangular partitions. We prove the relations as an application of a technique for turning Plücker relations into statements about Schur functions and other objects with similar definitions as determinants. We also give a quantum analogue of the relations which incorporates spectral parameters. Our proofs are mostly algebraic, but the relations have a clear combinatorial side, which we discuss.     

Vanishing cotangent cohomology for Plücker algebras

We use representation theory and Bott’s theorem to show vanishing of higher cotangent cohomology modules for the homogeneous coordinate ring of Grassmannians in the Plücker embedding. As a by-product, we answer a question of Wahl about the cohomology of the square of the ideal sheaf for the case of Plücker relations. We obtain slightly weaker vanishing results for the cotangent cohomology of the coordinate rings of isotropic Grassmannians.     

Poisson Brackets after Jacobi and Plücker

We construct a symplectic realization and a bi-Hamiltonian formulation of a 3-dimensional system whose solution are the Jacobi elliptic functions. We generalize this system and the related Poisson brackets to higher dimensions. These more general systems are parametrized by lines in projective space. For these rank 2 Poisson brackets the Jacobi identity is satisfied only when the Plücker relations hold. Two of these Poisson brackets are compatible if and only if the corresponding lines in projective space intersect. We present several examples of such systems.     

Differential geometry of grassmannians and the Plücker map

Using the Plücker map between grassmannians, we study basic aspects of classic grassmannian geometries. For ‘hyperbolic’ grassmannian geometries, we prove some facts (for instance, that the Plücker map is a minimal isometric embedding) that were previously known in the ‘elliptic’ case.     

A characteristic property of elliptic Plücker transformations

We discuss elliptic Plücker transformations of three-dimensional elliptic spaces. These are permutations on the set of lines such that any two related (orthogonally intersecting or identical) lines go over to related lines in both directions. It will be shown that for classical elliptic 3-spaces a bijection of its lines is already a Plücker transformation, if related lines go over to related lines. Moreover, if the ground field admits only surjective monomorphisms, then bijection can be replaced by injection.Dedicated to Walter Benz on the occasion of his 65th birthday     

Poincaré polynomials of moduli spaces of stable maps into flag manifolds

By using the Bialynicki-Birula decomposition and holomorphic Lefschetz formula, we calculate the Poincaré polynomials of the moduli spaces in low degrees.     

Local Aronson–Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds

In this work we derive local gradient and Laplacian estimates of the Aronson–Bénilan and Li–Yau type for positive solutions of porous medium equations posed on Riemannian manifolds with a lower Ricci curvature bound. We also prove similar results for some fast diffusion equations. Inspired by Perelman's work we discover some new entropy formulae for these equations.     

线性子空间关系运算的Plücker坐标表示

本文讨论了线性子空间Plücker的线性表示,给出了Plücker关系式的矩阵形式,并由之导出了子空间关联关系、零化子空间、和子空间与交子空间Plücker坐标的矩阵表达式.     

Two questions on the exterior geometry of the plücker embeddings of the Grassmann manifolds

The Plücker model of the Grassmann manifold G _p,p+q ⁺ is considered. The structure of intersections of G _p,p+q ⁺ with tangent spaces of G _p,p+q ⁺ regarded as subspaces of the ambient exterior algebra is described. An explicit formula for the second fundamental form of G _2,4 ⁺ as of a hypersurface in the five-dimensional sphere is given. The level sets of the normal curvature functions for this hypersurface are studied. Bibliography: 3 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 5–12. Translated by N. Yu. Netsvetaev.     

Symmetric orbits of orthogonal Plücker actions and triviality of their second order envelopes

The orbits of Lie groups acting in Euclidean spaces by isometries are extrinsically symmetric iff they are parallel, i.e. satisfy % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSijHikaaa!3764!\[\mathbb{Z}\]h = 0. Submanifolds characterized by the integrability condition \-R · h = 0 of this system % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSijHikaaa!3764!\[\mathbb{Z}\]h = 0 are called semi-parallel; they are the second order envelopes of the symmetric orbits. Let the orbit set of an action of SO(n, R) in E ^{% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% aIXaaabaGaaGOmaaaaaaa!3773!\[\frac{1}{2}\]n(n–1)} contain a Plücker submanifold. It is proved that 1) the only symmetric orbits are Plücker orbits and for n = 2 > 4 the unitary orbits, 2) each of their second order envelopes is trivial, i.e. is a single orbit or its open part.Partially supported by ESF Grant 139/305     

Sectional Curvatures and the Separation Set of the Complex Projective Space in Its Plücker Model

The Plücker embedding of the complex projective space in the Grassmannian of bivectors is used for proving several theorems on the relationship between the complex structure of and its Riemannian geometry. It is shown that the separation set of in the Plücker model is the face of for a certain calibration. Bibliography: 11 titles.     

Gross–Sobolev spaces on path manifolds: uniqueness and intertwining by Itô maps

Conditions are given under which the solution map $\text{I}$ of a stochastic differential equation on a Riemannian manifolds M intertwines the differentiation operator d on the path space of M and that of the canonical Wiener space, . A uniqueness property of d on the path space follows. Results are also given for higher derivatives and covariant derivatives. To cite this article: K.D. Elworthy, X.-M. Li, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Enumerative meaning of mirror maps for toric Calabi–Yau manifolds

We prove that the inverse of a mirror map for a toric Calabi–Yau manifold of the form _KY$K_Y$, where Y$Y$ is a compact toric Fano manifold, can be expressed in terms of generating functions of genus 0 open Gromov–Witten invariants defined by Fukaya–Oh–Ohta–Ono (2010)  [15]. Such a relation between mirror maps and disk counting invariants was first conjectured by Gross and Siebert (2011)  [24, Conjecture 0.2 and Remark 5.1] as part of their program, and was later formulated in terms of Fukaya–Oh–Ohta–Ono’s invariants in the toric Calabi–Yau case in Chan et al. (2012)  [8, Conjecture 1.1].     

Moment maps associated with holomorphic isometric actions on K?hler manifolds

We give a new formula which writes down the moment map for holomorphic isometric action on a K?hler manifold using the motion of a charged particle when the action of is Hamiltonian.     

An isomorphism theorem for Teichmüller curves

It is proved that for any Fuchsian group Г such that H/Г is a hyperbolic Riemann surface, the Teichmuller curve V(Г) has a unique complex manifold structure so that the natural projection of the Bers fiber space F(Г) onto V(Г) is holomorphic with local holomorphic sections. An isomorphism theorem for Teichmuller curves is deduced, which generalizes a classical result that the Teichmuller curve V(Г) depends only on the type of Г and not on the orders of the elliptic elements of Г when H/Г is a compact hyperbolic Riemann surface.     

Spectral estimates for Schrödinger and Dirac-type operators on Riemannian manifolds

Supported by funds of M.U.R.S.T. (Italy). The author is grateful to S. Gallot for his encouragement and for helpful discussions and to G. Besson for some interesting remarks     

Whitney’s formulas for curves on surfaces

The classical Whitney formula relates the number of times an oriented plane curve cuts itself to its rotation number and the index of a base point. In this paper we generalize Whitney’s formula to curves on an oriented punctured surface Σ _{m, n} , obtaining a family of identities indexed by elements of π ₁(Σ _{m, n} ). To define analogs of the rotation number and the index of a base point of a curve γ, we fix an arbitrary vector field on Σ _{m, n} . Similar formulas are obtained for non-based curves.     

Gamma-convergence and the emergence of vortices for Ginzburg–Landau on thin shells and manifolds

We analyze the Ginzburg–Landau energy in the presence of an applied magnetic field when the superconducting sample occupies a thin neighborhood of a bounded, closed manifold in ${\mathbb R^3}$ . We establish Γ-convergence to a reduced Ginzburg–Landau model posed on the manifold in which the magnetic potential is replaced in the limit by the tangential component of the applied magnetic potential. We then study the limiting problem, constructing two-vortex critical points when the manifold ${\mathcal{M}}$ is a simply connected surface of revolution and the applied field is constant and vertical. Finally, we calculate that the exact asymptotic value of the first critical field H _c1 is simply (4π/(area of ${\mathcal{M}}$ )) ln κ for large values of the Ginzburg–Landau parameter κ. Merging this with the Γ-convergence result, we also obtain the same asymptotic value for H _c1 in 3d valid for large κ and sufficiently thin shells.