Applications of Eigenvalue Techniques to Geometry

  • Peter Li
  • Andrejs Treibergs
Part of the The University Series in Mathematics book series (USMA)


Analysis has always been a powerful tool in the study of the geometry of manifolds. This fact has been reconfirmed by the recent developments in connection with the application of geometric analysis to various fields such as differential topology, algebraic geometry, mathematical physics, number theory, etc. Crucial to the application of partial differential equations to geometric problems are techniques developed in studying the Laplace operator and its eigenvalues. This is because in most cases when a partial differential equation occurs in geometry, the linearized equation has a principal term given by the Laplacian. The question of inverting the linearized operator naturally leads to the studying of the spectrum of the Laplacian. Among all the eigenvalues, the first nonzero eigenvalue plays the most important role. This is due to the fact that it occurs in the Poincaré inequality (see Section 1.4), which is one of the most powerful inequalities in the theory of nonlinear analysis.


Minimal Surface Compact Manifold Ricci Curvature Isoperimetric Inequality Compact Riemannian Manifold 
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Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • Peter Li
    • 1
  • Andrejs Treibergs
    • 2
  1. 1.Department of MathematicsUniversity of ArizonaTucsonUSA
  2. 2.Department of MathematicsUniversity of UtahSalt Lake CityUSA

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