We consider regular surfaces $M$ that are given as the zeros of a polynomial function $p : \mathbb{R}^3 \to \mathbb{R}$, where the gradient of $p$ vanishes nowhere. We assume that $M$ has non-zero constant mean curvature and prove that there exist only two examples of such surfaces, namely the sphere and the circular cylinder.

We provide an equivariant description / classification of all complete (compact or not) nonnegatively curved manifolds $M$ together with a co-compact action by a reflection group $\mathsf{W}$, and moreover, classify such $\mathsf{W}$. In particular, we show that the building blocks consist of the classical constant curvature models and generalized open books with nonnegatively curved bundle pages, and derive a corresponding splitting theorem for the universal cover.

Let $(M,g, k)$ be an initial data set for the Einstein equations of general relativity.

We prove that there exist solutions of the Plateau problem for marginally outer trapped surfaces (MOTSs) that are stable in the sense of MOTSs. This answers a question of G. Galloway and N. O’Murchadha raised in “Some remarks on the size of bodies and black holes,” [Classical Quantum Gravity 25 (2008), no. 10, 105009, 9. MR 2416045] and is an ingredient in the proof of the spacetime positive mass theorem given by L.-H. Huang, D. Lee, R. Schoen, and the first named author.

We show that a canonical solution of the Jang equation exists in the complement of the union of all weakly future outer trapped regions in the initial data set with respect to a given end, provided that this complement contains no weakly past outer trapped regions. The graph of this solution relates the area of the horizon to the global geometry of the initial data set in a non-trivial way. We prove the existence of a Scherk-type solution of the Jang equation outside the union of all weakly future or past outer trapped regions in the initial data set. This result is a natural exterior analogue for the Jang equation of the classical Jenkins–Serrin theory.

We extend and complement existence theorems for Scherk–type constant mean curvature graphs over polygonal domains in $(M,g)$, where $(M,g)$ is a complete Riemannian surface. We can dispense with the a priori assumptions that a sub solution exists and that $(M,g)$ has particular symmetries. Also, our method generalizes to higher dimensions.

Three explicit families of spacelike Zoll surface admitting a Killing field are provided. It allows to prove the existence of spacelike Zoll surfaces not smoothly conformal to a cover of de Sitter space as well as the existence of Lorentzian Möbius strips of non constant curvature all of whose spacelike geodesics are closed. Further the conformality problem for spacelike Zoll cylinders is studied.

We study a class of localized indices for the Dirac type operators on a complete Riemannian orbifold, where a discrete group acts properly, co-compactly, and isometrically. These localized indices, generalizing the $L^2$-index of Atiyah, are obtained by taking certain traces of the higher index for the Dirac type operators along conjugacy classes of the discrete group, subject to some trace assumption. Applying the local index technique, we also obtain an $L^2$-version of the Lefschetz fixed-point formulas for orbifolds. These cohomological formulas for the localized indices give rise to a class of refined topological invariants for the quotient orbifold.

In this erratum, we give corrections regarding Theorem 4.2 of our paper that appeared in Volume 97, Number 3, 2014, pages 395–425.