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We give a condition under which the findings of the paper cited above work well and determine the surfaces that were not considered before. In this paper, we show that a parallel mean curvature surface of a general type in a complex two-dimensional complex space form depends on one real-valued harmonic function on the surface and five real constants if the ambient space is not flat, the mean curvature vector does not vanish, and the Kaehler angle is not constant.
We study the evolution of compact convex curves in two-dimensional space forms. The normal speed is given by the difference of the weighted inverse curvature with the support function, and in the case where the ambient space is the Euclidean plane, i
We prove that under some assumptions on the mean curvature the set of umbilical points of an immersed surface in a $3$-dimensional space form has positive measure. In case of an immersed sphere our result can be seen as a generalization of the celebrated Hopf theorem.
We prove a converse to well-known results by E. Cartan and J. D. Moore. Let $fcolon M^n_ctoQ^{n+p}_{tilde c}$ be an isometric immersion of a Riemannian manifold with constant sectional curvature $c$ into a space form of curvature $tilde c$, and free
The Alexandrov Soap Bubble Theorem asserts that the distance spheres are the only embedded closed connected hypersurfaces in space forms having constant mean curvature. The theorem can be extended to more general functions of the principal curvatures
We study harmonic maps from a 3-manifold with boundary to $mathbb{S}^1$ and prove a special case of dihedral rigidity of three dimensional cubes whose dihedral angles are $pi / 2$. Furthermore we give some applications to mapping torus hyperbolic 3-manifolds.