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Pierre Bérard

Publications and source records attributed to Pierre Bérard.

27 records · Page 2Linked to original sources
arXiv2013

Stability Properties of Rotational Catenoids in the Heisenberg Groups

In this paper, we determine the maximally stable, rotationally invariant domains on the catenoids $\cC_a$ (minimal surfaces invariant by rotations) in the Heisenberg group with a left-invariant metric. We show that these catenoids have Morse index at least 3 and we bound the index from above in terms of the parameter $a$. We also show that the index of $\cC_a$ tends to infinity with $a$. Finally, we study the rotationally symmetric stable domains on the higher dimensional catenoids.

math.DG↗
arXiv2013

Remarks on the boundary set of spectral equipartitions

Given a bounded open set $Ω$ in $\mathbb{R}^n$ (or a compact Riemannian manifold with boundary), and a partition of $Ω$ by $k$ open sets $ω_j$, we consider the quantity $\max_j λ(ω_j)$, where $λ(ω_j)$ is the ground state energy of the Dirichlet realization of the Laplacian in $ω_j$. We denote by $\mathfrak{L}_k(Ω)$ the infimum of $\max_j λ(ω_j)$ over all $k$-partitions. A minimal $k$-partition is a partition which realizes the infimum. The purpose of this paper is to revisit properties of nodal sets and to explore if they are also true for minimal partitions, or more generally for spectral equipartitions. We focus on the length of the boundary set of the partition in the 2-dimensional situation.

math.SP↗
arXiv2013

Spectral positivity and Riemannian coverings

Let $(M,g)$ be a complete non-compact Riemannian manifold. We consider operators of the form $Δ_g + V$, where $Δ_g$ is the non-negative Laplacian associated with the metric $g$, and $V$ a locally integrable function. Let $ρ: (\hat{M},\hat{g}) \to (M,g)$ be a Riemannian covering, with Laplacian $Δ_{\hat{g}}$ and potential $\hat{V} = V \circ ρ$. If the operator $Δ+ V$ is non-negative on $(M,g)$, then the operator $Δ_{\hat{g}} + \hat{V}$ is non-negative on $(\hat{M},\hat{g})$. In this note, we show that the converse statement is true provided that $π_1(\hat{M})$ is a co-amenable subgroup of $π_1(M)$.

math.DG↗
arXiv2010

Eigenvalue estimates for hypersurfaces in $H^m \times R$ and applications

In this paper, we give a lower bound for the spectrum of the Laplacian on minimal hypersurfaces immersed into $H^m \times R$. As an application, in dimension 2, we prove that a complete minimal surface with finite total extrinsic curvature has finite index. On the other hand, for stable, minimal surfaces in $H^3$ or in $H^2 \times \R$, we give an upper bound on the infimum of the spectrum of the Laplacian and on the volume growth.

math.DG↗
arXiv2009

Minimal hypersurfaces in $\HH^n \times \R$, total curvature and index

In this paper, we consider minimal hypersurfaces in the product space $\mathbb{H}^n \times \mathbb{R}$. We begin by studying examples of rotation hypersurfaces and hypersurfaces invariant under hyperbolic translations. We then consider minimal hypersurfaces with finite total curvature. This assumption implies that the corresponding curvature goes to zero uniformly at infinity. We show that surfaces with finite total intrinsic curvature have finite index. The converse statement is not true as shown by our examples which also serve as useful barriers.

math.DG↗
arXiv2009

Lindelöf's theorem for catenoids revisited

In this paper we study the maximal stable domains on minimal catenoids in Euclidean and hyperbolic spaces and in $H^2 \times R$. We in particular investigate whether half-vertical catenoids are maximal stable domains (\emph{Lindelöf's property}). We also consider stable domains on catenoid-cousins in hyperbolic space. Our motivations come from Lindelöf's 1870 paper on catenoids in Euclidean space.

math.DG↗
arXiv2009

Examples of $H$-hypersurfaces in $H^n \times R$ and geometric applications

In this paper we describe all rotation $H$-hypersurfaces in $H^n \times R$ and use them as barriers to prove existence and characterization of certain vertical $H$-graphs and to give symmetry and uniqueness results for compact $H$-hypersurfaces whose boundary is one or two parallel submanifolds in slices. We also describe examples of translation $H$-hypersurfaces in $H^n \times R$. For $n>2$ we obtain a complete embedded translation hypersurface generated by a compact, simple, strictly convex curve. When $0 < H < \frac{n-1}{n}$ we obtain a complete non-entire vertical graph over the non-mean convex domain bounded by an equidistant hypersurface taking infinite boundary value data and infinite asymptotic boundary value data.

math.DG↗
arXiv2000

Index Growth of hypersurfaces with constant mean curvature

In this paper we give the precise index growth for the embedded hypersurfaces of revolution with constant mean curvature (cmc) 1 in $\R^{n}$ (Delaunay unduloids). When $n=3$, using the asymptotics result of Korevaar, Kusner and Solomon, we derive an explicit asymptotic index growth rate for finite topology cmc 1 surfaces with properly embedded ends. Similar results are obtained for hypersurfaces with cmc bigger than 1 in hyperbolic space.

math.DG↗