Contents Chapter 1 Introduction 1 Chapter 2 Eigenvalue comparison theorems 10 2.1 Basic notions and the geometry of spherically symmetric manifolds 11 2.2 A generalized eigenvalue comparison theorem for manifolds with radial Ricci curvature bounded from below 17 2.3 A generalized eigenvalue comparison theorem for manifolds with radial sectional curvature bounded from above 22 2.4 Properties of the model manifolds 27 2.5 Examples 30 2.6 A criterion for the existence of the model spaces 40 2.7 Estimates for the heat kernel 43 2.8 A Cheng-type isoperimetric inequality for the p-Laplace operator and a spectral result for the model spaces 51 2.9 Proof of Theorem 2.34 54 2.10 Estimates for the first eigenvalue of the p-Laplacian 59 Chapter 3 Estimates for lower eigenvalues 65 3.1 Reilly-type inequalities for the first nonzero closed eigenvalue of the p-Laplacian 65 3.2 Isoperimetric bounds for the first eigenvalue of the p-Laplacian 74 3.3 Isoperimetric bounds for the first eigenvalue of the Paneitz operators 77 3.4 Lower bounds for the first eigenvalue of four kinds of eigenvalue problems of the bi-drifting Laplacian 81 Chapter 4 Universal Inequalities 91 4.1 Universal inequalities of the clamped plate problem of the drifting Laplacian 93 4.1.1 Recent developments for universal inequalities of the clamped plate problem 93 4.1.2 The clamped plate problem of the drifting Laplacian 95 4.1.3 Universal inequalities for the clamped plate problem of the drifting Laplacian 104 4.1.4 Universal inequalities on the Gaussian shrinking soliton 115 4.2 Universal inequalities for the buckling problem of the drifting Laplacian 118 4.2.1 Recent developments for universal inequalities of the buckling problem 119 4.2.2 The buckling problem of the drifting Laplacian 120 4.2.3 A general inequality on the Ricci solitons 125 4.2.4 Universal inequalities in a bounded domain on Ricci solitons 138 4.3 Further study 147 4.3.1 Some interesting conjectures 147 4.3.2 Some Problems 151 Chapter 5 Open problems 154 5.1 Spectral problem concerning quantum strips and quantum layers 154 5.2 Connection between curvature flows and eigenvalue problem 160 5.3 Eigenvalue problem in Finsler geometry 163 Appendix A Warped products 166 Bibliography 172
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