Acknowledgement

In this thesis, we give curvature estimates for strongly stable constant mean curvature surfaces in a complete three dimensional manifold. We use a key observation of Colding and Minicozzi to obtain area and small total curvature estimates of constant mean curvature surfaces. Then following Choi and Schoen we show that small total curvatures yield curvature estimates. By giv-ing a much shorter proof, this thesis extends the work of Bérard and Hauswirth, where they gave curvature estimates for constant mean curvature surfaces in a space form