Dissertation > Excellent graduate degree dissertation topics show

Eigenvalue problem of a compact Riemannian manifold research

Author: WangNanZuo
Tutor: XuHongWei
School: Zhejiang University
Course: Basic mathematics
Keywords: Riemannian manifold Eigenvalue estimates Minimal submanifolds First eigenvalue Mean curvature Ricci Dirichlet Harmonic operator Riemannian Bounded Euclidean space The lower bound of the estimated Laplacian Upper bound submanifold Inequality Second fundamental form Isometric Embedding Convex hypersurface The main conclusions
CLC: O186.12
Type: Master's thesis
Year: 2010
Downloads: 22
Quote: 0
Read: Download Dissertation

Abstract


In the first part of this paper, we give a upper bound of the (k+l)-th eigenvalue in terms of the first k eigenvalues of the eigenvalue problem for the Dirichlet biharmonic operator on a connected bounded domain in a complete Riemannian manifold' dissertation">Riemannian manifold, which generalizes the result of paper [4] and [14]. LetΩbe a connected bounded domain in M which is an n-dimensional complete Riemannian manifold, then the eigenvalue problem for the Dirichlet biharmonic operator, or the’Clamped Plate problem’, onΩis: where△is the Laplacian on M and△2 is the biharmonic operator on M. Assuming that are eigenvalues of the above problem, we prove that there exists a constant H02 in terms of the upper bound of the mean curvature, which only depends on M andΩ,such that the following inequality holds: which generalizes the main results of paper [4] and [14]. In order to prove our result, we apply a similar technique as the one proposed by [4], which eliminates the unwanted terms in the deduction perfectly. By making use of Nash’s theorem and introducing k free constants, we derive a universal bound for the (k+l)-th eigenvalue. In fact, the above inequality is one of the sharpest estimates currently for the eigenvalue problems for the Dirichlet biharmonic operators in virtue of the method of [4] which we follow by substituting coordinate components of the position vector of a complete Riemannian manifold for the coordinate functions of a Euclidean space. We also derive the corresponding estimates for the eigenvalues under some specific conditions, namely, M being a minimal submanifold of a convex hypersurface in a Euclidean space, a minimal submanifold of an ellipsoid or a minimal submanifold of a cylinder.In the second part of this paper, we give a new estimate on the lower bound of the first eigenvalue of the Laplacian on a 2-dimensional or 3-dimensional closed Riemannian manifold with positive Ricci curvature in terms of the diameter of the manifold and the lower bound of Ricci curvature, which sharpens one of the results of paper [10]. For an n-dimensional closed Riemannian manifold whose Ricci curvature has a positive lower bound (n-1)K for some constant K> 0, [10] gave both a lower bound of the the first eigenvalueλfor arbitrary n,λ≥π2/d2+0.31(n-1)K, and a better one,λ≥π2/d2+0.375(n-1)K, when n= 2, both of which are the best estimates currently for this eigenvalue problem on compact Riemannian manifold. In our paper, under the low-dimensional condition that n=2,3, we give a better estimate,λ≥π2/d2+0.425(n-1)K, mainly by a similar’Testing Function’ method based on Yau’s ’Gradient Estimation’and a classification discussion as that used in [10], noticing that the classification in [10] can be adjusted for a better estimate.

Related Dissertations

  1. Rigidity of Submanifolds and Problem of Eigenvalue,O186.12
  2. On M(?)bius Form and M(?)bius Isoparametric Hypersurfaces,O186.11
  3. Submanifolds in Riemannian Manifolds of Quasi-constant Curvature,O186.12
  4. Space-like Hypersurfaces in Locally Symmetric Lorentz Spaces,O186.12
  5. Hypersurface of the unit sphere with parallel imitation Blaschke tensor,O186.12
  6. The Geometry and Topology of Submanifolds of the Unit Sphere,O189.31
  7. Submanifolds with Parallel Mean Curvature on a Sphere,O186.12
  8. An Inequality of Submanifolds with Parallel Mean Curvature and the Pinching Problem,O186.12
  9. The Pinching Problem of Submanifolds in Locally Symmetric Space,O186.12
  10. Research on Several Problems of Einstein Manifold,O186.12
  11. Submanifold geometry and topology,O186.12
  12. The rigid geometric theorem submanifold and differential Sphere Theorem,O186.12
  13. Willmore submanifold geometry rigid and eigenvalue problem,O186.12
  14. Quantities of Nonlinear Parabolic Equations on Riemannian Manifolds,O186.12
  15. A Generalized KKM Theorem and Its Applications,O177.91
  16. Study on Vector Equilibrium Problems,O177
  17. The Study of Some Optimization Problems,O224
  18. KKM-Theory and Applications,O177.91
  19. Nonconforming Finite Element Methods for Sobolev Equations and Variational Inequality,O241.82
  20. Applications of Gram Matrix,O151.21

CLC: > Mathematical sciences and chemical > Mathematics > Geometry, topology > Differential geometry,integral geometry > Differential Geometry > Riemannian geometry
© 2012 www.DissertationTopic.Net  Mobile