Proof of the transverse-traceless decomposition for bordered surfaces

Prove, or otherwise establish in the existing literature, the decomposition of every smooth symmetric bilinear form on a compact Riemannian surface with boundary into pairwise orthogonal pure-trace, pure-divergence, and transverse-traceless components, with the stated boundary conditions and uniqueness modulo conformal vector fields.

Background

The paper requires a decomposition of symmetric tensors into pure-trace, pure-divergence, and transverse-traceless parts in order to define the two-variable energy index and compare it with the area index. Although the decomposition is classical for closed surfaces, the authors provide a proof for surfaces with boundary because they could not locate one in the literature.

The appendix develops coercivity for the trace-free Lie derivative operator, proves regularity up to the boundary, and derives the decomposition and its uniqueness. The unresolved issue explicitly identified by the authors concerns the availability of such a proof in the literature for the bordered case, rather than the validity of the theorem proved in the appendix.