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We find out upper bounds for the first eigenvalue of the stability operatorfor compact constant mean curvature orientable surfaces immersed in aRiemannian Killing submersion. As a consequence, the strong stability of suchsurfaces is studied. We also characterize constant mean curvature Hopf tori asthe only ones attaining the bound in certain cases.

First stability eigenvalue characterization of CMC Hopf tori into Riemannian Killing submersions