2026-09-25T12:58:54Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/50697 2023-07-11 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Dunfield, Nathan M. Laugesen, Richard S. Dunfield, Nathan M. Alexander, Stephanie B. Leininger, Christopher J. Benson, Brian 2014-09-16T17:25:22Z 2014-09-16T17:25:22Z 2014-08 2014-09-16 2014-08 Buser’s inequality gives an upper bound on the first non-zero eigenvalue of the Laplacian of a closed manifold M in terms of the Cheeger constant h(M). Agol later gave a quantitative improvement of Buser’s inequality. Agol’s result is less transparent since it is given implicitly by a set of equations, one of which is a differential equation Agol could not solve except when M is three-dimensional. We show that a substitution transforms Agol’s differential equation into the Riemann differential equation. Then, we give a proof of Agol’s result and also generalize it using Sturm-Liouville theory. Under the same assumptions on M, we are able to give upper bounds on the higher eigenvalues of M , λ_k(M), in terms of the eigenvalues of a Sturm-Liouville problem which depends on h(M). We then compare the Weyl asymptotic of λ_k(M) given by the works of Cheng, Gromov, and Berard-Besson-Gallot to the asymptotics of our Sturm-Liouville problems given by Atkinson-Mingarelli. Item withdrawn by Laura Spradlin (lspradl2@illinois.edu) on 2014-06-12T18:53:11Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 9 PL.pdf: 3553 bytes, checksum: 23bc646e1ac102315e4983fe4300851c (MD5) nchange.pdf: 4719 bytes, checksum: d5e921f2515cb5d091050f65fc9a9ebd (MD5) mean3.pdf: 3458 bytes, checksum: fb3c7d71883cf49d3f4dfa28d99c3883 (MD5) mean2.pdf: 3474 bytes, checksum: 4c00c6c94b4a82eb07fa66c1ccbd6092 (MD5) mean1.pdf: 3416 bytes, checksum: f09653e9ef62d9a8e5dcbe6c5f96dc10 (MD5) HigherEigen.pdf: 4587 bytes, checksum: 3dccdfe63755218e3ddcfa7cc9693a75 (MD5) drawing5.pdf: 1357 bytes, checksum: e2e99ace1d11ec2b7decd56b78cab69a (MD5) Benson_Brian.tex: 104696 bytes, checksum: 4cf1158395ea69c51540a30538478cd8 (MD5) Benson_Brian.pdf: 459534 bytes, checksum: 76daad65ede05f57d37c805374a770a9 (MD5) Made available in DSpace on 2014-09-16T17:25:22Z (GMT). No. of bitstreams: 10 Brian_Benson.pdf: 459550 bytes, checksum: e505553f7ca56a7b1f9287ade16d734e (MD5) Benson_Brian.tex: 104697 bytes, checksum: 57c417679c15f986316ffadf316484a5 (MD5) PL.pdf: 3553 bytes, checksum: 23bc646e1ac102315e4983fe4300851c (MD5) nchange.pdf: 4719 bytes, checksum: d5e921f2515cb5d091050f65fc9a9ebd (MD5) mean3.pdf: 3458 bytes, checksum: fb3c7d71883cf49d3f4dfa28d99c3883 (MD5) mean2.pdf: 3474 bytes, checksum: 4c00c6c94b4a82eb07fa66c1ccbd6092 (MD5) mean1.pdf: 3416 bytes, checksum: f09653e9ef62d9a8e5dcbe6c5f96dc10 (MD5) HigherEigen.pdf: 4587 bytes, checksum: 3dccdfe63755218e3ddcfa7cc9693a75 (MD5) drawing5.pdf: 1357 bytes, checksum: e2e99ace1d11ec2b7decd56b78cab69a (MD5) license.txt: 4061 bytes, checksum: 0e7bba987897e474b18906ce2dc0f755 (MD5) http://hdl.handle.net/2142/50697 en Copyright 2014 Brian Benson Cheeger constant spectrum of Laplacian eigenvalues of closed Riemannian manifolds Buser's inequality Sturm-Liouville estimates for the spectrum and Cheeger constant text Mathematics 1257 Mathematics 0439 University of Illinois at Urbana-Champaign Dissertation Ph.D. PHD:Mathematics -UIUC 10KS0439PHD