An annulus and a half-helicoid maximize Laplace eigenvalues

2015/10/07 by Sinan Ariturk, Ariturk, Sinan
Mathematics · #35P15 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP) #math.AP #math.DG #math.SP #msc:35P15

paper · pdf · doi:10.48550/arxiv.1510.02030

arxiv created 2015/10/07 · arxiv updated 2015/10/08

Abstract

The Dirichlet eigenvalues of the Laplace-Beltrami operator are larger on an annulus than on any other surface of revolution in ℝ³ with the same boundary. This is established by defining a sequence of shrinking cylinders about the axis of symmetry and proving that flattening a surface outside of each cylinder successively increases the eigenvalues. A similar argument shows that the Dirichlet eigenvalues of the Laplace-Beltrami operator are larger on a half-helicoid than on any other screw surface in ℝ² × \mathbbS¹ with the same boundary.

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