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We give a complete analysis of mode solutions for the linearized Einsteinequations and the $1-$form wave operator on the Kerr metric in the large$\mathfrak{a}$ case. By mode solutions we mean solutions of the form$e^{-it_*\sigma}\tilde{h}(r,\theta,\varphi)$ where $t_*$ is a suitable timevariable. The corresponding Fourier transformed $1-$form wave operator andlinearized Einstein operator are shown to be Fredholm between suitable functionspaces and $\tilde{h}$ has to lie in the domain of these operators. Thesespaces are constructed following the general framework of Vasy. No modesolutions exist for ${\Im}\, \sigma\ge 0,\, \sigma\neq 0$. For $\sigma=0$ modesolutions are Coulomb solutions for the $1-$form wave operator and linearizedKerr solutions plus pure gauge terms in the case of the linearized Einsteinequations. If we fix a De Turck/wave map gauge, then the zero mode solutionsfor the linearized Einstein equations lie in a fixed $7-$dimensional space. Theproof relies on the absence of modes for the Teukolsky equation shown by thethird author and a complete classification of the gauge invariants oflinearized gravity on the Kerr spacetime due to Aksteiner et al.

Mode analysis for the linearized Einstein equations on the Kerr metric : the large $\mathfrak{a}$ case