Spectrum of hypersurfaces with small extrinsic radius or large λ1 in Euclidean spaces

Abstract

Abstract. In this paper, we prove that Euclidean hypersurfaces with almost extremal extrinsic radius or λ1 have a spectrum that asymptotically contains the spectrum of the extremal sphere in the Reilly or Hasanis-Koutroufiotis Inequalities. We also consider almost extremal hypersurfaces which satisfy a supplementary bound on vM B n α and show that their spectral and topological properties depends on the position of α with respect to the critical value dim M . The study of the metric shape of these extremal hypersurfaces will be done in AG1[3], using estimates of the present paper. IntroductionThroughout the paper, X: M n → R n+1 is a closed, connected, immersed Euclidean hypersurface (with n 2). We set v M its volume, B its second fundamental form, H = 1 n tr B its mean curvature, r M its extrinsic radius (i.e. the least radius of the Euclidean balls containing M ), (λ M i ) i∈N the non-decreasing sequence of its eigenvalues labelled with multiplicities andThe Hasanis-Koutroufiotis inequality asserts that n H 2 2 , once again with equality if and only if M is the sphere S M (we give some short proof of these inequalities in section prel 2). Our aim is to study the spectral properties of the hypersurfaces that are almost extremal for each of this Inequalities. The results and estimates of this paper are used in AG1[3] to study the metric shape of the almost extremal hypersurfaces. We set µ2 the k-th eigenvalue of S M (labelled without multiplicities) and m k its multiplicity. Throughout the paper we shall adopt the notation that τ (ε|n, · · · ) is a positive function which depends on n, · · · and which converges to zero with ε → 0 when n, · · · are fixed.

On the boundary of almost isoperimetric domains

“…R Ω From this we deduce that Note thatρ R Ω was already obtained in (2). On the other hand, we have that…”

“…1 nα + C(n) P (Ω) H n (B x (ρ /2) ∩ ∂Ω) 1 2 δ(Ω) 1 4 Now (6.7) and (6.8) imply that δ(Ω) 1/8n 1/γC 4 and C 4 C(n)/γ which gives ρ 2 ∈ [C(n)δ(Ω) 1/8 R Ω , R Ω ]. So we can apply Theorem 7, and since we have ρ R Ω r(n, p) (see (2) in the proof of the previous lemma), we get…”

“…Now it is clear that for C 1 (n, p) large enough, we haveρ R Ω C(n)δ(Ω) with C 3 (n, p) fixed in (2). Let x ∈ S x Ω (R Ω ).…”

“…Note that a upper bound on H p with p < n − 1 is not sufficient. Indeed, we can refer to examples constructed by the authors in [2,3]: by adding small tubular neighbourhood of well chosen trees to B 0 (1), we get a set almost isoperimetric domains on which H p is uniformly bounded for any p < n − 1 and that is dense for the Hausdorff distance among all the closed set of R n+1 that contain B 0 (1). and so if ∂Ω is connected then we have N = 0 and A δ(Ω) 1/4 ∪ T is connected.…”

1st eigenvalue pinching for convex hypersurfaces in a Riemannian manifold

“…At present quantitative rigidity results with respect to the upper curvature bound δ are rarely known. There are relative more rigidity results for convex domains to be of constant curvature via information along their boundary and interior's lower (Ricci) curvature bound (e.g., [ We refer to [14], [11], [1], [7], and [13] for related results in manifolds and space forms. Let us give the main idea in proving Theorem 1.1.…”