We present a variational model with local weighted Gaussian curvature as regularizer. We show its convexity for an area-weight function and provide a closed-form solution for this case. The corresponding regularization coefficient has a theoretical bound. Moreover, we prove that the model is convex for a wide range of weight functions and show that it can be efficiently solved using splitting techniques. Finally, we demonstrate several applications of the model in image de-noising, smoothing, texture decomposition, image sharpening, and regularization-coefficient optimization.