Large spectral gaps for Steklov eigenvalues under volume constraints and under localized conformal deformations

Abstract

In this paper we construct compact manifolds with fixed boundary geometry which admit Riemannian metrics of unit volume with arbitrarily large Steklov spectral gap. We also study the effect of localized conformal deformations that fix the boundary geometry. For instance, we prove that it is possible to make the spectral gap arbitrarily large using conformal deformations which are localized on domains of small measure, as long as the support of the deformations contains and connects each component of the boundary. 1 arXiv:1801.07836v1 [math.DG]

From Steklov to Neumann via homogenisation

“…To this end, assume without loss of generality that f (r ) = 1. Following a strategy that was used in [12] and in [9], consider the two following situations. Let t ∈ (r, R), to be fixed later.…”

“…As earlier, write a function as Using the notation for the tangential gradient, the Dirichlet energy of u is expressed as On the other hand, the denominator in ( 45 ) is given by Combining these last two expressions in ( 45 ) and defining the density , we see it is enough to prove that Indeed, working term by term, we prove that any smooth function satisfies To this end, assume without loss of generality that . Following a strategy that was used in [ 12 ] and in [ 9 ], consider the two following situations.…”

Hypersurfaces with Prescribed Boundary and Small Steklov Eigenvalues

“…One of our main interests in recent years has been to understand the particular role that the boundary Σ plays with respect to Steklov eigenvalues. Some papers studying this question are [6,15,4,2,17,11,7,5,16]. In particular, we have considered the effect of various geometric constraints on individual eigenvalues σ k .…”

“…For manifolds M of dimension n ≥ 3 one can also obtain arbitrarily large eigenvalues, but in general not using a perturbation that is localized near the boundary of M . See [4,2]. In [5] a more restrictive constraint was imposed by requiring the manifold M to be a submanifold of R m with prescribed boundary Σ = ∂M ⊂ R m .…”

“…The boundary of S f is the same for each f . We will use the function defined byũ(x, y, z) = x and its restriction u =ũ S f as a trial function in inequality (2). Because ∇ũ = (1, 0, 0), it follows from Lemma 8 that the Dirichlet energy of u := S fũ : S f → R is given by…”