[Paper Review] A remark on the Laplacian operator which acts on symmetric tensors

С. Е. Степанов, Irina Tsyganok+1 more|arXiv (Cornell University)|Nov 7, 2014

Geometric Analysis and Curvature Flows13 references4 citations

TL;DR

This paper establishes that the symmetric tensor Laplacian Δ_sym, defined by J. H. Samson for p=1 and n > 1, coincides with the Yano rough Laplacian on compact Riemannian manifolds. The authors prove spectral properties of Δ_sym, showing it is a second-order elliptic differential operator on 1-forms with well-defined eigenvalues and eigenforms, thereby linking a classical geometric operator to the theory of symmetric tensors.

ABSTRACT

More than forty years ago J. H. Samson has defined the Laplacian $Δ_{sym}$ acting on the space of symmetric covariant $p$-tensors on an $n$-dimensional Riemannian manifold $(M, g)$. This operator is an analogue of the well known Hodge-de Rham Laplacian $Δ$ which acts on the space of exterior differential $p$-forms ($1 \le p \le n$) on $(M, g)$. In the present paper we will prove that for $n > p = 1$ the operator $Δ_{sym}$ is the Yano rough Laplacian and show its spectrum properties on a compact Riemannian manifold.

Motivation & Objective

• To clarify the relationship between the Laplacian Δ_sym acting on symmetric 1-tensors and known geometric operators.
• To investigate the spectral theory of Δ_sym on compact Riemannian manifolds.
• To determine whether Δ_sym is equivalent to the Yano rough Laplacian in the case p=1 and n > 1.
• To analyze the eigenvalues and eigenforms of Δ_sym as a second-order elliptic differential operator.

Proposed method

• The authors analyze the action of Δ_sym on symmetric covariant 1-tensors over an n-dimensional Riemannian manifold (M, g).
• They compare Δ_sym with the Yano rough Laplacian using intrinsic geometric and tensorial identities.
• The proof relies on differential geometric techniques, including curvature terms and symmetric tensor decomposition.
• The spectral analysis is conducted on compact Riemannian manifolds, leveraging the self-adjointness and ellipticity of Δ_sym.
• The authors use the Hodge-de Rham Laplacian as a reference point for comparison due to its well-known spectral properties.
• The analysis is restricted to the case p=1 and n > 1, where the equivalence is established.

Experimental results

Research questions

• RQ1 Is the symmetric tensor Laplacian Δ_sym equivalent to the Yano rough Laplacian when acting on symmetric 1-tensors for n > 1?
• RQ2 What are the spectral properties of Δ_sym on a compact Riemannian manifold?
• RQ3 How does Δ_sym relate to the Hodge-de Rham Laplacian in the context of symmetric tensors?
• RQ4 Does Δ_sym qualify as a second-order elliptic differential operator on 1-forms?
• RQ5 What are the eigenvalues and eigenforms of Δ_sym in the case p=1 and n > 1?

Key findings

• For n > p = 1, the symmetric tensor Laplacian Δ_sym is equivalent to the Yano rough Laplacian on symmetric 1-tensors.
• Δ_sym is a second-order elliptic differential operator acting on 1-forms on compact Riemannian manifolds.
• The operator Δ_sym admits a discrete spectrum consisting of real, non-negative eigenvalues.
• Eigenforms of Δ_sym are smooth sections of the bundle of symmetric 1-tensors, and the spectrum is bounded below by zero.
• The spectral theory of Δ_sym is well-behaved, with eigenfunctions in L² and finite multiplicity.
• The equivalence to the Yano rough Laplacian confirms Δ_sym as a natural geometric operator in the context of symmetric tensor analysis.