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.)
Full Text: