A complete surface of constant mean curvature 1 (CMC-1) in hyperbolic 3- space with constant curvature $-1$ has two natural notions of ‘‘total curvature’’—one is the total absolute curvature which is the integral over the surface of the absolute value of the Gaussian curvature, and the other is the dual total absolute curvature which is the total absolute curvature of the dual CMC-1 surface. In this paper, we completely classify CMC-1 surfaces with dual total absolute curvature at most $4\pi$. Moreover, we give new examples and partially classify CMC-1 surfaces with dual total absolute curvature at most $8\pi$.

In this paper we characterize the existence of large solutions for a general class of porous medium logistic equations in the presence of a vanishing carrying capacity. The decay rate of the carrying capacity at the boundary of the underlying domain determines the exact blow-up rate of the large solutions. Its explicit knowledge allows us to obtain a general uniqueness result.

We study the singular limit of a degenerate nonlinear diffusion equation which appears in a chemotaxis-growth model. We prove the convergence to the solution of a free boundary problem where the motion equation of the interface involve the gradient of the chemotactic concentration and the critical velocity of a degenerate Fisher equation.