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

On the Sampson Laplacian

“…In the third and fourth paragraphs of the paper we consider the properties of the Sampson operator acting on one-forms and symmetric two-tensors. Theorems and corollaries of the present paper complement our results from the papers [3,31,33,39,41,42,44].…”

“…Second, the operator ∆ S −∆ has the order zero and can be defined by symmetric endomorphisms of the bundle S p M. This means that we have the Weitzenböck decomposition formula ∆ S =∆ − Γ p , and Γ p : S p M → S p M is an algebraic symmetric operator that depends linearly in a known way on the curvature tensor R and the Ricci tensor Ricc of the metric (see [38, p. 147]). In particular, for special case of 1-forms, we have ∆ S =∆ − Ricc (see also [32,42,43]).…”

“…The authors of these papers and monograph have determined this operator and obtained its Weitzenböck decomposition but did not quote the source [38]. On the other hand, we were the first and only who began to study the properties of this operator in details (see [3,31,33,39,41,42,44]). To these lists, we can add two papers [24] and [25] in which there are terms "Sampson Laplacian" and "Sampson operator" but there are no new results on the geometry of the Sampson operator.…”

Some differential operators in the symmetric bundle

“…Recently this operator has been investigated in the category of Lee algebroids in [1]. For k = 1 in [2] a similar operator: the Yano rough Laplacian was analyzed in a context of its spectral properties. Some elliptic operators in the bundle of symmetric forms were also investigated in [3] in a context of so-called conformal Killing tensors.…”

“…Recently, such operators were investigated e.g. in [1], by Balcerzak and Pierzchalski in the category of Lie algebroids or, in [2], by Stepanov and Mikes, in the case of one tensors where some spectral properties of the Yano rough Laplacian -the operator similar to the one considered here Sampson Laplacian ∆ s -were analyzed. It is also worth noticing a very recent paper [3], by Heil, Moroianu and Semmelmann where some elliptic operators in the bundle of symmetric forms were investigated in the context of Killing and conformal Killing tensors.…”