Tightness and Curvature

 Note that $${1\over 2\pi} \int_M |K|\,dx \ge \left|{1\over 2\pi} \int_M K\,dx\right|$$ is a strict inequality only when $K$ takes on both positive and negative values. Theorem: a surface in a CES is tight if, and only if, its curvature does not change sign.