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On the gap between the first eigenvalues of the Laplacian on functions and 1-forms. (English) Zbl 0984.58018

Let \((M,g)\) be a compact smooth connected \(m\) dimensional oriented Riemannian manifold. Let \(\lambda_k^p(M,g)\) be the \(k^{th}\) positive eigenvalue of the Laplacian on smooth \(p\) forms. Since exterior differentiation \(d\) embeds the positive spectrum of \(\Delta_0\) in the positive spectrum of \(\Delta_1\), \(\lambda_1^1(M,g)\leq\lambda_1^0(M,g)\); note that if \(m=2\), then equality holds.
The author studies when \((M,g)\) has a ‘gap’, i.e. when \(\lambda_1^1(M,g)<\lambda_1^0(M,g)\) by showing:
Theorem 1.1. If \(m\geq 3\), then there exists a metric \(g\) on \(M\) so \(\lambda_1^1(M,g)=\lambda_1^0(M,g)\).
Theorem 1.2. If \(m\geq 3\) and if the first Betti-number of \(M\) vanishes, then there exists a metric \(g\) on \(M\) so \(\lambda_1^1(M,g)<\lambda_1^0(M,g)\).
The author also studies geometric properties of Riemannian manifolds with gaps and positive Ricci curvature:
Theorem 1.3. Let \((M,g)\) be Einstein manifold with positive Ricci curvature and isometry group \(G\). Assume \(\lambda_1^1(M,g)<\lambda_1^0(M,g)\). If \(\dim G=0\), then the identity map is strongly stable as a harmonic map. If \(\dim G\geq 1\), then the identity map is weakly stable as a harmonic map.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35P15 Estimates of eigenvalues in context of PDEs
53C43 Differential geometric aspects of harmonic maps
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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