2,307 research outputs found
Range of the first three eigenvalues of the planar Dirichlet Laplacian
We conduct extensive numerical experiments aimed at finding the admissible
range of the ratios of the first three eigenvalues of a planar Dirichlet
Laplacian. The results improve the previously known theoretical estimates of M
Ashbaugh and R Benguria. We also prove some properties of a maximizer of the
ratio .Comment: 21 page; 9 figure
Families of short cycles on Riemannian surfaces
Let be a closed Riemannian surface of genus . We construct a family of
1-cycles on that represents a non-trivial element of the k'th homology
group of the space of cycles and such that the mass of each cycle is bounded
above by . This result is optimal
up to a multiplicative constant.Comment: 16 pages, 3 figures. Exposition improved, to appear in Duke
Mathematical Journa
Integrable Systems and Discrete Geometry
This is an expository article for Elsevier's Encyclopedia of Mathematical
Physics on the subject in the title. Comments/corrections welcome.Comment: 22 pages, 7 figure
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