✦ LIBER ✦

Bounds for the first Dirichlet eigenvalue attained at an infinite family of Riemannian manifolds

✍ Scribed by Ana Lluch; Vicente Miquel

Publisher: Springer
Year: 1996
Tongue: English
Weight: 990 KB
Volume: 61
Category: Article
ISSN: 0046-5755
DOI: 10.1007/bf00149419

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✦ Synopsis

Let M be a compact Riemannian manifold with smooth boundary OM. We get bounds for the first eigenvalue of the Dirichlet eigenvalue problem on M in terms of bounds of the sectional curvature of M and the normal curvatures of OM. We discuss the equality, which is attained precisely on certain model spaces defined by J. H. Eschenburg. We also get analog results for Kfihler manifolds. We show how the same technique gives comparison theorems for the quotient volume(P)/volume(M), M being a compact Riemannian or Kfihler manifold and P being a compact real hypersurface of M.

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An estimate for the first eigenvalue of the Dirac operator on compact Riemannian spin manifold admitting parallel one-form

✍ B. Alexandrov; G. Grantcharov; S. Ivanov 📂 Article 📅 1998 🏛 Elsevier Science 🌐 English ⚖ 492 KB

An estimate for the first eigenvalue of the Dirac operator on compact Riemannian spin manifold of positive scalar curvature admitting a parallel one-form is found. The possible universal covering spaces of the manifolds on which the smallest possible eigenvalue is attained are also listed. Moreover,