The spectral theory of the Yano rough Laplacian with some of its applications. (English) Zbl 1317.53058
Summary: J. H. Sampson has defined the Laplacian \(\triangle_{\mathrm{sym}}\) acting on the space of symmetric covariant tensors on Riemannian manifolds. This operator is an analogue of the well-known Hodge-de Rham Laplacian \(\triangle\) which acts on the space of skew-symmetric covariant tensors on Riemannian manifolds. In the present paper, we perform properties analysis of Sampson’s operator which acts on one-forms. We show that the Sampson operator is the Yano rough Laplacian. We also find the biggest lower bounds of spectra of the Yano and Hodge-de Rham operators and obtain estimates of their multiplicities for the space of one-forms on compact Riemannian manifolds with negative and positive Ricci curvatures, respectively.
MSC:
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 53C24 | Rigidity results |
| 58J40 | Pseudodifferential and Fourier integral operators on manifolds |
| 58A14 | Hodge theory in global analysis |
Keywords:
Riemannian manifold; second-order elliptic differential operator on 1-forms; eigenvalues and eigen-formsReferences:
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