Laplacian eigenvalue functionals and metric deformations on compact manifolds

Soufi, Ahmad El; Ilias, Saïd

doi:10.1016/j.geomphys.2007.09.008

Cited by 58 publications

(96 citation statements)

References 24 publications

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“…We should point out that although the characterization (1.11) is similar to the Riemannian case [11] and sub-Riemannian case [4] (see also [2] for a similar result in Kähler case), our case exhibits an important difference. Precisely, it is proved in [11] that the criticality of the λ k (∆)-functional of the Laplacian is characterized by the existence of a finite collection of λ k (∆)-eigenfunctions f j such that the sum of squares f 2 1 + f 2 2 + · · · + f 2 d is constant on the manifold. In our characterization, the identity (1.11) contains not only the sum of squared norms of the eigenfunctions, but also their first order derivatives.…”

Section: Introductionsupporting

confidence: 56%

“…Proof. The proof use similar arguments as in [11] and [4]. For the implication (i) ⇒ (ii), recall that for each k, the convex cone C k ⊂ L 2 (M, θ; R) are defined by…”

Section: Eigenvalue Ratio Functionalsmentioning

confidence: 96%

“…However, a difficult in our situation comes from the fact that b is generally a complex operator and hence the eigenfunctions are not necessarily real-valued. This makes the method treating the Laplacian [11] and sub-Laplacian [3] cases breaks down in the Kohn Laplacian case. Nevertheless, the characterization (1.11) still leads to the following Corollary 1.5.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Eigenvalues of the sub-Laplacian and deformations of contact structures on a compact CR manifold

Aribi

Dragomir

Soufi

2015

Differential Geometry and its Applications

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Given a compact strictly pseudoconvex CR manifold $M$, we study the differentiability of the eigenvalues of the sub-Laplacian $\Delta_{b, theta}$ associated with a compatible contact form (i.e. a pseudo-Hermitian structure) $\theta$ on $M$, under conformal deformations of $\theta$. As a first application, we show that the property of having only simple eigenvalues is generic with respect to $\theta$, i.e. the set of structures $\theta$ such that all the eigenvalues of $\Delta_{b,\theta}$ are simple, is residual (and hence dense) in the set of all compatible positively oriented contact forms on $M$. In the last part of the paper, we introduce a natural notion of critical pseudo-Hermitian structure of the functional $\theta \to \lambda_k(\theta)$, where $\lambda_k(\theta)$ is the $k$-th eigenvalue of the subì-Laplacian $\Delta_{b, \theta}$ , and obtain necessary and sufficient conditions for a pseudo-Hermitian structure to be critical

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Section: Introductionsupporting

confidence: 56%

“…Proof. The proof use similar arguments as in [11] and [4]. For the implication (i) ⇒ (ii), recall that for each k, the convex cone C k ⊂ L 2 (M, θ; R) are defined by…”

Section: Eigenvalue Ratio Functionalsmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

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Eigenvalues of the sub-Laplacian and deformations of contact structures on a compact CR manifold

Aribi

Dragomir

Soufi

2015

Differential Geometry and its Applications

Self Cite

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“…Theorem 1.5 (Nadirashvili [32], El Soufi and Ilias, [15], see also [25]). If g is conformally extremal for the functionalλ i (M, g) then there exists a harmonic map Ψ : (M, g) → S n whose components are λ i -eigenfunctions.…”

Section: 2mentioning

confidence: 99%

On the Yang–Yau inequality for the first Laplace eigenvalue

Karpukhin

2019

Geom. Funct. Anal.

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In a seminal paper published in 1980, P. C. Yang and S.-T. Yau proved an inequality bounding the first eigenvalue of the Laplacian on an orientable Riemannian surface in terms of its genus γ and the area. The equality in Yang-Yau's estimate is attained for γ = 0 by an old result of J. Hersch and it was recently shown by S. Nayatani and T. Shoda that it is also attained for γ = 2. In the present article we combine techniques from algebraic geometry and minimal surface theory to show that Yang-Yau's inequality is strict for all genera γ > 2. Previously this was only known for γ = 1. In the second part of the paper we apply Chern-Wolfson's notion of harmonic sequence to obtain an upper bound on the total branching order of harmonic maps from surfaces to spheres. Applications of these results to extremal metrics for eigenvalues are discussed.

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“…Впрочем, известно, что для аналитических деформаций g t исходной метрики g существуют левые и правые производные по t, см. [3], [12], [13]. Этот факт является мотивацией для следующего определения.…”

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Немаксимальность Экстремальных Метрик На Торе И Бутылке Клейна

Карпухин¹,

Karpukhin²

2013

Математический сборник

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Немаксимальность экстремальных метрик на торе и бутылке КлейнаТеорема Эль Суфи и Илиаса устанавливает соответствие между ми-нимальными подмногообразиями в сферах и метриками, экстремальными для собственных значений оператора Лапласа-Бельтрами. В недавних ра-ботах с помощью этого соответствия были построены новые явные приме-ры экстремальных метрик. В настоящей работе изучаются свойства этих метрик и доказывается, что ни одна из них не является максимальной.Библиография: 24 названия.Ключевые слова: экстремальные метрики, биполярная поверхность, торы Оцуки, тау-поверхности Лоусона.

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Laplacian eigenvalue functionals and metric deformations on compact manifolds

Cited by 58 publications

References 24 publications

Eigenvalues of the sub-Laplacian and deformations of contact structures on a compact CR manifold

Eigenvalues of the sub-Laplacian and deformations of contact structures on a compact CR manifold

On the Yang–Yau inequality for the first Laplace eigenvalue

Немаксимальность Экстремальных Метрик На Торе И Бутылке Клейна

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