Let $K$ be a branched surface whose branch set $S$ is an embedded circle and such that $K\backslash S$ is connected and oriented. We show that $K$ does not admit expanding immersions. Combined with our previous result [3], this implies that among branched surfaces with branch sets single embedded circles, there are only two which admit expanding immersions.

An immersion of a differentiable manifold into an almost Hermitian manifold is called a \textit{general slant immersion} if it has constant Wirtinger angle ([3, 6]). A general slant immersion which is neither holomorphic nor totally real is called a proper slant immersion. In the first part of this article, we prove that every general slant immersion of a compact manifold into the complex Euclidean $m$-space $\mathbf{C}^m$ is totally real. This result generalizes the well-known fact that there exist no compact holomorphic submanifolds in any complex Euclidean space. In the second part, we classify proper slant surfaces in $\mathbf{C}^2$ when they are contained in a hypersphere $S^3$, or contained in a hyperplane $E^3$, or when their Gauss maps have rank $<2$.

We study complete space-like hypersurfaces with constant mean curvature of a Lorentz space form.