On Hawking mass and Bartnik mass of CMC surfaces

Given a constant mean curvature surface that bounds a compact manifold with nonnegative scalar curvature, we obtain intrinsic conditions on the surface that guarantee the positivity of its Hawking mass. We also obtain estimates of the Bartnik mass of such surfaces, without assumptions on the integral of the squared mean curvature. If the ambient manifold has negative scalar curvature, our method also applies and yields estimates on the hyperbolic Bartnik mass of these surfaces.