Stability of isoperimetric inequalities for Laplace eigenvalues on surfaces

Abstract

We prove stability estimates for the isoperimetric inequalities for the first and the second nonzero Laplace eigenvalues on surfaces, both globally and in a fixed conformal class. We employ the notion of eigenvalues of measures and show that if a normalized eigenvalue is close to its maximal value, the corresponding measure must be close in the Sobolev space W −1,2 to the set of maximizing measures. In particular, this implies a qualitative stability result: metrics almost maximizing the normalized eigenvalue must be W −1,2 -close to a maximal metric. Following this approach, we prove sharp quantitative stability of the celebrated Hersch's inequality for the first eigenvalue on the sphere, as well as of its counterpart for the second eigenvalue. Similar results are also obtained for the precise isoperimetric eigenvalue inequalities on the projective plane, torus, and Klein bottle. The square of the W −1,2 distance to a maximizing measure in these stability estimates is controlled by the difference between the normalized eigenvalue and its maximal value, indicating that the maxima are in a sense nondegenerate. We construct examples showing that the power of the distance can not be improved, and that the choice of the Sobolev space W −1,2 is optimal.

From Steklov to Laplace: free boundary minimal surfaces with many boundary components

“…The proof follows closely that of [KNPS,Theorem 1.17], with Lemma 5.2 replacing [KNPS,Lemma 2.1] at the final step. As discussed in [KNPS,Section 6], the minimal immersions u : M → S n that induce the λ1 -maximizing metrics on M = RP 2 , T 2 , and K all have maximal Morse index as critical points of the area functional, in the sense that ind A (u) = n + 1 + dim(M 0 (M )), where M 0 (M ) = Met can (M )/Diff 0 (M ) denotes the Teichmüller space of conformal structures on M . In particular, these minimal immersions satisfy the hypotheses of [KNPS,Lemma 6.5].…”

“…To obtain analogous estimates in the cases where M = RP 2 , T 2 , or the Klein bottle K, we combine Lemma 5.2 with the techniques of [KNPS,Section 6]. The case of M = RP 2 -which carries only one conformal structure-is in principle simpler, but we group it with the others for convenience.…”

“…As discussed in [KNPS,Section 6], the minimal immersions u : M → S n that induce the λ1 -maximizing metrics on M = RP 2 , T 2 , and K all have maximal Morse index as critical points of the area functional, in the sense that ind A (u) = n + 1 + dim(M 0 (M )), where M 0 (M ) = Met can (M )/Diff 0 (M ) denotes the Teichmüller space of conformal structures on M . In particular, these minimal immersions satisfy the hypotheses of [KNPS,Lemma 6.5].…”

“…Following the proof of [KNPS,Theorem 6.1], let C max ⊂ M 0 (M ) denote the set of (equivalence classes of) conformal structures g achieving the Λ 1 (M, [g]) = Λ 1 (M ). By [KNPS,Lemma 6.5], there exists a neighborhood U of C max in M 0 (M ) and a family of maps…”

“…Remark 5.5. Using the techniques of [KNPS,Section 2.3] in place of [KNPS,Section 6] 5.2. Structure of nearly-σ 1 -maximizing metrics with many boundary components.…”