Pierre Bérard

Complete Hypersurfaces with Constant Mean Curvature and Finite Total Curvature

The main result of this paper states that the traceless second fundamental tensor \({A}^{0}\)of an n-dimensional complete hypersurface M, with constant mean curvature H and finite total curvature, \(\int{M}^{|A^0|^n{d\nu}}{M} 0,\) any such surface must be compact.

Eigenvalue and “Twisted” eigenvalue problems, applications to CMC surfaces

Abstract In this paper we investigate an eigenvalue problem which appears naturally when one considers the second variation of a constant mean curvature immersion. In this geometric context, the second variation operator is of the form Δ g + b , where b is a real valued function, and it is viewed as acting on smooth functions with compact support and with mean value zero. The condition on the mean value comes from the fact that the variations under consideration preserve some balance of volume. This kind of eigenvalue problem is interesting in itself. In the case of a compact manifold, possibly with boundary, we compare the eigenvalues of this problem with the eigenvalues of the usual (Dirichlet) problem and we in particular show that the two spectra are interwined (in fact strictly interwined generically). As a by-product of our investigation of the case of a complete manifold with infinite volume we prove, under mild geometric conditions when the dimension is at least 3, that the strong and weak Morse indexes of a constant mean curvature hypersurface coincide.

Dirichlet eigenfunctions of the square membrane: Courant's property, and A. Stern's and Å. Pleijel's analyses

In this paper, we revisit Courants nodal domain theorem for theDirichlet eigenfunctions of a square membrane, and the analyses ofA. Stern and A. Pleijel.

A note on Bochner type theorems for complete manifolds.

In this note we give an extension of Bochners vanishing theorem to complete manifolds; this generalizes earlier results of J. Dodziuk, K.D. Elworthy and S. Rosenberg.

A. Stern’s analysis of the nodal sets of some families of spherical harmonics revisited

In this paper, we revisit the analyses of Stern (Bemerkungen über asymptotisches Verhalten von Eigenwerten und Eigenfunktionen, W. Fr. Kaestner, Göttingen, 1925) and Lewy (Commun Partial Differ Equ 2(12):1233–1244, 1977) devoted to the construction of spherical harmonics with two or three nodal domains. Our method yields sharp quantitative results and a better understanding of the occurrence of bifurcations in the families of nodal sets. This paper is a natural continuation of our critical reading of A. Stern’s results for Dirichlet eigenfunctions in the square, see arXiv:14026054.

Lindelöf's theorem for hyperbolic catenoids

In this paper, we study the maximal stable domains on minimal and constant mean curvature catenoids in hyperbolic space. In particular we investigate whether half-vertical catenoids are maximal stable domains (Lindelofs property). Our motivation comes from Lindelofs 1870 paper on catenoids in Euclidean space.A system for restoring images with undefined pixel values at known locations is described. The threshold value and a neighborhood configuration are defined and are used to restore the image. The neighborhood configuration defines a geometric region, typically a fixed number of pixels, surrounding the target pixel. The threshold value specifies a number of pixels in the neighborhood configuration for which pixel values are known. In our system, for each pixel in one of the unknown regions an analysis is performed over the entire area defined by the neighborhood configuration. If the threshold number of pixels within that region is known, then the value of the unknown pixel is calculated. If the threshold value is not achieved, then analysis proceeds to the next pixel location. By continuing the process and reducing the threshold value when necessary or desirable, the complete image can be restored.

Courant-Sharp Eigenvalues for the Equilateral Torus, and for the Equilateral Triangle

We address the question of determining the eigenvalues

GENERAL CURVATURE ESTIMATES FOR STABLE H-SURFACES IMMERSED INTO A SPACE FORM

{\lambda_{n}}