Lower bounds for eigenfunctions on Riemannian manifolds. (English) Zbl 0619.58038
Let M be a complete Riemannian manifold with Ricci curvature bounded below by -(n-1)c, \(c>0\). Suppose \(\phi \in L^ 2M\) is an eigenfunction of the Laplacian \(\Delta\) with eigenvalue \(\mu\) lying below the essential spectrum. If \(\mu\) is the infimum of the spectrum, we give an exponential lower bound for \(\phi\), requiring only a lower bound on the Ricci curvature of M. When \(\mu\) lies above the infimum, we give a lower bound with exponential decay for the spherical averages of \(\phi\), imposing several restrictive hypotheses on M. In particular, M must be simply connected and negatively curved with sectional curvatures approaching -1 at infinity.
MSC:
| 58J70 | Invariance and symmetry properties for PDEs on manifolds |
| 53C20 | Global Riemannian geometry, including pinching |
Keywords:
eigenfunction of the Laplacian; essential spectrum; exponential lower bound; Ricci curvatureReferences:
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