A pinching theorem for the first eigenvalue of the Laplacian on hypersurfaces of the Euclidean space

Abstract

Abstract. In this paper, we give pinching theorems for the first nonzero eigenvalue λ 1 (M) of the Laplacian on the compact hypersurfaces of the Euclidean space. Indeed, we prove that if the volume of M is 1 then, for any ε > 0, there exists a constant C ε depending on the dimension n of M and the L ∞ -norm of the mean curvature H , so that if the, then the Hausdorff-distance between M and a round sphere of radius (n/λ 1 (M)) 1/2 is smaller than ε. Furthermore, we prove that if C is a small enough constant depending on n and the L ∞ -norm of the second fundamental form, then the pinching condition n H 2 2p − C < λ 1 (M) implies that M is diffeomorphic to an n-dimensional sphere. Mathematics Subject Classification (2000). 53A07, 53C21.

Eigenvalue pinching and application to the stability and the almost umbilicity of hypersurfaces

“…However, the above inequality is a kind of pinching on Reilly's inequality, that is a condition of almost equality. Such conditions have been studied for Reilly's inequality in Euclidean space in [7]. In the present paper we will generalize the results of [7] to the inequality (2) for hypersurfaces (i.e.…”

Upper Bounds for the First Eigenvalue of the Laplacian of Hypersurfaces in terms of Anisotropic Mean Curvatures

“…And what do we understand by close ? The following theorems, generalizing results of [2] for r = 1, and [16] for any r, give an answer to this question.…”

“…The proof of the lemma 7 uses a Nirenberg-Moser type of proof (see [2,3]) based on a Sobolev inequality due to Michael-Simon and Hoffman-Spruck (see [7], [8] and [11]). …”

A New Result About Almost Umbilical Hypersurfaces of Real Space Forms

“…First, we recall that the results obtained in [16] and [6] are consequences of pinching results for the first eigenvalue of the Laplacian proved in [5] and [6]. A key tool for these pinching results is the Michael-Simon's extrinsic Sobolev inequality for submanifolds of the Euclidean space [10] and its generalization by Hoffman and Spruck for any ambient manifold [8].…”

“…An immediate consequence of this inequality is that 1 K(n, α)||H|| ∞ V ol(M ) 1/n by taking f = 1. This extrinsic Sobolev inequality is of crucial importance to obtain pinching results for the first eigenvalue of the Laplacian (see [5,6]). An other important fact under these assumptions is that the diameter of the hypersurface is bounded from above in terms of the mean curvature.…”