Le volume conforme et ses applications d'après Li et Yau

Soufi, Ahmad El; Ilias, Saïd

doi:10.5802/tsg.11

Cited by 23 publications

(33 citation statements)

References 3 publications

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“…More precisely, one has, for any Riemannian metric g on M , λ 1 (M, g)A(M, g) ≤ 8π(genus(M ) + 1), where A(M, g) stands for the Riemannian area of (M, g) (see [7] for an improvement of this upper bound). In the non-orientable case, the following upper bound follows from Li and Yau's work [16]: λ 1 (M, g)A(M, g) ≤ 24π(genus(M ) + 1).…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle

2006

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Abstract. We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich [15]: on the Klein bottle K, the metric of revolution, 0 ≤ v < π, is the unique extremal metric of the first eigenvalue of the Laplacian viewed as a functional on the space of all Riemannian metrics of given area. The proof leads us to study a Hamiltonian dynamical system which turns out to be completely integrable by quadratures.

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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle

2006

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“…Following this, Yang and Yau [24] showed (see also [10]) that for a compact orientable surface of genus γ , we have…”

Section: Historical Background and Motivationmentioning

confidence: 99%

Extremal G-invariant eigenvalues of the Laplacian of G-invariant metrics

Colbois

Dryden²,

Soufi³

2007

Math. Z.

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The study of extremal properties of the spectrum often involves restricting the metrics under consideration. Motivated by the work of Abreu and Freitas in the case of the sphere S 2 endowed with S 1 -invariant metrics, we consider the subsequence λ G k of the spectrum of a Riemannian manifold M which corresponds to metrics and functions invariant under the action of a compact Lie group G. If G has dimension at least 1, we show that the functional λ G k admits no extremal metric under volume-preserving G-invariant deformations. If, moreover, M has dimension at least three, then the functional λ G k is unbounded when restricted to any conformal class of G-invariant metrics of fixed volume. As a special case of this, we can consider the standard O(n)-action on S n ; however, if we also require the metric to be induced by an embedding of S n in R n+1 , we get an optimal upper bound on λ G k .

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“…Proof. The proof uses standard arguments (see [11,15]). We consider the map dµ and B 2n+2 is the unit Euclidean ball.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%