The spectral theory of the Yano rough Laplacian with some of its applications

Abstract

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 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.