On the first eigenvalue of invariant Kähler metrics

Panelli, Francesco; Podestà, Fabio

doi:10.1007/s00209-015-1495-7

Cited by 4 publications

(4 citation statements)

References 13 publications

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“…In [8] we recast the method of Hersch-Bouguignon-Li-Yau in terms of momentum mapping and applied it when M is an arbitrary compact Hermitian symmetric space, ĝ is the symmetric metric, G = Aut(M ) and the functions fj are the components of the momentum mapping µ : M → k * for K := Isom(M, ĝ). (Related papers include [2], [7], [34] and [40]).…”

Section: Polystability and Semi-stabilitymentioning

confidence: 99%

Stability of measures on Kähler manifolds

Biliotti

Ghigi

2017

Advances in Mathematics

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Let (M, ω) be a Kähler manifold and let K be a compact group that acts on M in a Hamiltonian fashion. We study the action of K C on probability measures on M . First of all we identify an abstract setting for the momentum mapping and give numerical criteria for stability, semi-stability and polystability. Next we apply this setting to the action of K C on measures. We get various stability criteria for measures on Kähler manifolds. The same circle of ideas gives a very general surjectivity result for a map originally studied

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Section: Polystability and Semi-stabilitymentioning

confidence: 99%

Stability of measures on Kähler manifolds

Biliotti

Ghigi

2017

Advances in Mathematics

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“…In this work we examine the Ricci flow, on compact homogeneous spaces with simple spectrum of isotropy representation, in terms of Graev [31,32], or of monotypic isotropy representation, in terms of Buzano [18], or Pulemotov and Rubinstein [52]. Nowadays, such spaces are of special interest due to their rich applications in the theory of homogeneous Einstein metrics, prescribed Ricci curvature, Ricci iteration, Ricci flow and other (see [4,53,10,15,31,8,9,18,32,24,11,50,52,30]). Here, we focus on flag manifolds M = G/K of a compact simple Lie group G and examine the dynamical system induced by the vector field corresponding to the homogeneous Ricci flow equation.…”

Section: Introductionmentioning

confidence: 99%

Ancient solutions of the homogeneous Ricci flow on flag manifolds

Anastassiou

Chrysikos

2021

Extr. Math.

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For any flag manifold M=G/K of a compact simple Lie group G we describe non-collapsing ancient invariant solutions of the homogeneous unnormalized Ricci flow. Such solutions pass through an invariant Einstein metric on M, and by [13] they must develop a Type I singularity in their extinction finite time, and also to the past. To illustrate the situation we engage ourselves with the global study of the dynamical system induced by the unnormalized Ricci flow on any flag manifold M=G/K with second Betti number b2(M) = 1, for a generic initial invariant metric. We describe the corresponding dynamical systems and present non-collapsed ancient solutions, whose α-limit set consists of fixed points at infinity of MG. Based on the Poincaré compactification method, we show that these fixed points correspond to invariant Einstein metrics and we study their stability properties, illuminating thus the structure of the system’s phase space.

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“…The first eigenvalue of the Laplacian operator, that we denote by λ 1 (M, g), is one of the most natural and studied Riemannian invariants. There has been a considerable amount of work devoted to estimating the first eigenvalue in terms of other geometric quantities associated to (M, g), see for instance [3,4,6,7,17,54,51,59,67]. More precisely one would like to study the quantity λ 1 (M, g)Vol(M, g) n/2 , which is scale invariant.…”

Section: A Remark On Eigenvalue Estimatesmentioning

confidence: 99%

“…In [8] we recast the method of Hersch-Bourguignon-Li-Yau in terms of momentum map and applied it when M is an arbitrary Hermitian symmetric space, g 0 is the symmetric metric, G = Aut(M) and the functions are the components of the momentum map µ : M −→ k for K := Isom(M, g 0 ). Related paper are [3,4,54,59]. This problem was recently solved for a large class of Kähler manifolds see [12].…”

Section: A Remark On Eigenvalue Estimatesmentioning

confidence: 99%

Satake-Furstenberg compactifications and gradient map

Biliotti¹

2020

Preprint

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Let G be a real semisimple Lie group with finite center and let g = k⊕p be a Cartan decomposition of its Lie algebra. Let K be a maximal compact subgroup of G with Lie algebra k and let τ be an irreducible representation of G on a complex vector space V . We denote by µp : P(V ) −→ p the G-gradient map and by O the unique closed orbit of G in P(V ) which is a K-orbit [33,40]. We prove that up to equivalence the set of irreducible representations of parabolic subgroups of G induced by τ are completely determined by the facial structure of the polar orbitope E = conv(µp(O)). Moreover, any parabolic subgroup of G admits a unique closed orbit which is well-adapted to O and µp respectively. These results are new also in the complex reductive case. The connection between E and τ provides a geometrical description of the Satake compactifications without root data. In this context the properties of the Bourguignon-Li-Yau map are also investigated. Given a measure γ on O, we construct a map Ψγ from the Satake compactification of G/K associated to τ and E . If γ is a K-invariant measure then Ψγ is an homeomorphism of the Satake compactification and E . Finally, we prove that for a large class of measures the map Ψγ is surjective.

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On the first eigenvalue of invariant Kähler metrics

Cited by 4 publications

References 13 publications

Stability of measures on Kähler manifolds

Stability of measures on Kähler manifolds

Ancient solutions of the homogeneous Ricci flow on flag manifolds

Satake-Furstenberg compactifications and gradient map

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