On the first stability eigenvalue of CMC surfaces into warped products with two-dimensional fiber

The purpose of this article is the search of estimates for the first eigenvalue of the stability operator for constant mean curvature compact two-sided surfaces immersed into three-dimensional warped product spaces with two-dimensional fiber satisfying a suitable and well studied convergence condition. We also characterize the cases when the upper bound is reached. As an application, we derive some consequences for those surfaces that are stable, obtaining some classification results.