Isoperimetric properties of higher eigenvalues of elliptic operators

We prove, in the setting of a measure energy space (M, μ, (Ɛ,Ƒ)), that if the smallest eigenvalue λ₁(Ω) of the generator of the Dirichlet form Ɛ in any precompact open set Ω ⊂ M admits the estimate λ₁(Ω) ≥ ν(Ω)^-α where ν is a measure absolutely continuous with respect to μ and α > 0 then a similar estimate holds for the kth smallest eigenvalue: λ_k(Ω) ≥ const (k/ν(Ω))^α. As an application, we obtain an upper estimate of the stability index of a minimal surface in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] via the total curvature.