Metrics on a closed surface of genus two which maximize the first eigenvalue of the Laplacian

Nayatani, Shin; Shoda, Toshihiro

doi:10.1016/j.crma.2018.11.008

Cited by 25 publications

(23 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Indeed, in [MS17], Matthiesen and Siffert proved that for any closed surface Σ there exists a metric g of area one, smooth except for possibly finitely many points which correspond to conical singularities, that maximizes λ 1 among all other unit-area metrics on Σ. These maximal metrics are induced from branched minimal immersions into a round sphere by first eigenfunctions and do in general possess conical singularities (see [NS18]). Therefore, it is natural to study Theorem 1.1 in the context of branched minimal immersions.…”

Section: Introductionmentioning

confidence: 99%

“…(i) The proof of Theorem 1.2 uses results of Nadirashvili and Sire [NS15] and Petrides [Pet14] on the maximization of Λ 1 (Σ, g) in a conformal class. (ii) Nayatani and Shoda [NS18] recently proved that Λ 1 is maximized by a metric on the Bolza surface with constant curvature one and six conical singularities (this metric was proposed to be maximal in [JLN + 05]). Thus, Theorem 1.2 is optimal in regards to the regularity of a maximal metric.…”

Section: Introductionmentioning

confidence: 99%

“…Finally, the maximal metric is known for Σ 2 , the orientable surface of genus 2. Nayatani and Shoda proved in [NS18] that the metric on the Bolza surface proposed in [JLN + 05] is maximal. As a result, Λ 1 (Σ 2 ) = 16π.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1.4 allows us to construct an example where a maximal metric forλ 1 cannot be induced by a branched minimal immersion whose components form a basis in the λ 1 -eigenspace. Indeed, in [NS18] the authors showed that on a surface of genus 2 there exists a family of maximal metrics forλ 1 induced from a branched minimal immersion to S 2 . Moreover, there are metrics in the family such that the multiplicity of the first eigenvalue is equal to 5.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On branched minimal immersions of surfaces by first eigenfunctions

2019

View full text Add to dashboard Cite

It was proved by Montiel and Ros that for each conformal structure on a compact surface there is at most one metric which admits a minimal immersion into some unit sphere by first eigenfunctions. We generalize this theorem to the setting of metrics with conical singularities induced from branched minimal immersions by first eigenfunctions into spheres. Our primary motivation is the fact that metrics realizing maxima of the first non-zero Laplace eigenvalue are induced by minimal branched immersions into spheres. In particular, we show that the properties of such metrics induced from S 2 differ significantly from the properties of those induced from S m with m > 2. This feature appears to be novel and needs to be taken into account in the existing proofs of the sharp upper bounds for the first non-zero eigenvalue of the Laplacian on the 2-torus and the Klein bottle. In the present paper we address this issue and give a detailed overview of the complete proofs of these upper bounds following the works of Nadirashvili, Jakobson-Nadirashvili-Polterovich,

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On branched minimal immersions of surfaces by first eigenfunctions

2019

View full text Add to dashboard Cite

show abstract

“…The same conformal class contains an optimal metric for a related first eigenvalue problem; see[28]. This optimal metric similarly has finitely many conical singularities.…”

mentioning

confidence: 96%

Systolically extremal nonpositively curved surfaces are flat with finitely many singularities

Katz

Sabourau

2019

J. Topol. Anal.

View full text Add to dashboard Cite

The regularity of systolically extremal surfaces is a notoriously difficult problem already discussed by M. Gromov in 1983, who proposed an argument toward the existence of L 2 -extremizers exploiting the theory of r-regularity developed by P. A. White and others by the 1950s. We propose to study the problem of systolically extremal metrics in the context of generalized metrics of nonpositive curvature. A natural approach would be to work in the class of Alexandrov surfaces of finite total curvature, where one can exploit the tools of the completion provided in the context of Radon measures as studied by Reshetnyak and others. However the generalized metrics in this sense still don't have enough regularity. Instead, we develop a more hands-on approach and show that, for each genus, every systolically extremal nonpositively curved surface is piecewise flat with finitely many conical singularities. This result exploits a decomposition of the surface into flat systolic bands and nonsystolic polygonal regions, as well as the combinatorial/topological estimates of Malestein-Rivin-Theran, Przytycki, Aougab-Biringer-Gaster and Greene on the number of curves meeting at most once, combined with a kite excision move. The move merges pairs of conical singularities on a surface of genus g and leads to an asymptotic upper bound g 4+ǫ on the number of singularities.

show abstract

Extremal Eigenvalue Problems and Free Boundary Minimal Surfaces in the Ball

Fraser

2020

Geometric Analysis

View full text Add to dashboard Cite

Metrics on a closed surface of genus two which maximize the first eigenvalue of the Laplacian

Abstract: In this paper, we settle in the affirmative the Jakobson-Levitin-Nadirashvili-Nigam-Polterovich conjecture, stating that a certain singular metric on the Bolza surface, with area normalized, should maximize the first eigenvalue of the Laplacian.

Cited by 25 publications

References 11 publications

On branched minimal immersions of surfaces by first eigenfunctions

On branched minimal immersions of surfaces by first eigenfunctions

Systolically extremal nonpositively curved surfaces are flat with finitely many singularities

Extremal Eigenvalue Problems and Free Boundary Minimal Surfaces in the Ball

Contact Info

Product

Resources

About