Verify absence of decaying TT zero modes

Determine whether the Riemannian Schwarzschild–Tangherlini metric admits any transverse-traceless solution of the Lichnerowicz equation [?]—specifically, a solution w of [?] [?]—in the weighted dual space required by the Fredholm reduction, thereby establishing that the obstruction space for solving the first Fredholm step is trivial.

Background

The paper explains that the explicit construction of a trace-free tensor H satisfying P(H)=k could potentially be replaced by a Fredholm argument in weighted spaces. After a York decomposition, the first step reduces to solving the Lichnerowicz equation on transverse-traceless tensors. The relevant cokernel consists of decaying TT solutions w of the homogeneous equation Δ_L w=0 in a specified weighted space.

The author identifies the absence of such static TT zero modes with a linearized version of uniqueness for static black holes. This absence is needed to eliminate the Fredholm obstruction and obtain a solution h; the analogous issue arises again in the second step, although its obstruction space is expected to be smaller.

References

The obstruction space consists of TT solutions w of \Delta_Lw=0 with w\in 0_{2-N-\delta}, i.e. decaying faster than r{-N-a_k}. Such a w would be an L2 static TT zero mode of \Delta_L; its absence is expected (it is a linearized version of the uniqueness of static black holes), but it has to be checked.

— Compactly supported instability of Riemannian Schwarzschild--Tangherlini metrics  (2610.13507 - Delay, 9 Oct 2026) in Section 9.1, subsection “Existence of the preimages by the Fredholm alternative,” paragraph “The two steps”