Bounds of eigenfunctions of the Laplacian on compact Riemannian manifolds. (English) Zbl 0991.58006
Let \( \varphi \) be an eigenfunction of the Laplacian with eigenvalue \( \lambda \not= 0 \).
The author proves that \[ \|\varphi \|_{\infty} \leq c _1 \cdot \lambda ^{\frac{n-1}{4}} \|\varphi \|_2 \] where \( n \) is the dimension of \( M \) and \( c _1 \) depends only upon a bound for the absolute value of the sectional curvature of \( M \) and a lower bound for the injectivity radius of \( M \).
The author proves that \[ \|\varphi \|_{\infty} \leq c _1 \cdot \lambda ^{\frac{n-1}{4}} \|\varphi \|_2 \] where \( n \) is the dimension of \( M \) and \( c _1 \) depends only upon a bound for the absolute value of the sectional curvature of \( M \) and a lower bound for the injectivity radius of \( M \).
Reviewer: Mircea Puta (Timişoara)
MSC:
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
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