In this paper we prove a global existence theorem, in the direction of cosmological expansion, for sufficiently small perturbations of a family of n + 1-dimensional, spatially compact spacetimes, which generalizes the k = −1 Friedmann–Lemaˆıtre-Robertson– Walker vacuum spacetime. This work extends the result from [3].The background spacetimes we consider are Lorentz cones over negative Einstein spaces of dimension n  3. We use a variant of the constant mean curvature, spatially harmonic (CMCSH) gauge introduced in [2]. An important difference from the 3+1 dimensional case is that one may have a nontrivial moduli space of negative Einstein geometries. This makes it necessary to introduce a time-dependent background metric, which is used to define the spatially harmonic coordinate system that goes into the gauge. Instead of the Bel-Robinson energy used in [3], we here use an expression analogous to the wave equation type of energy introduced in [2] for the Einstein equations in CMCSH gauge. In order to prove energy estimates, it turns out to be necessary to assume stability of the Einstein geometry. Further, for our analysis it is necessary to have a smooth moduli space. Fortunately, all known examples of negative Einstein geometries satisfy these conditions. We give examples of families of Einstein geometries which have nontrivial moduli spaces. A product construction allows one to generate new families of examples. Our results demonstrate causal geodesic completeness of the perturbed spacetimes, in the expanding direction, and show that the scale-free geometry converges toward an element in the moduli space of Einstein geometries, with a rate of decay depending on the stability properties of the Einstein geometry.

No abstract uploaded!

We study minimal graphic functions on complete Riemannian manifolds $\Sigma$ with non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay. We derive global bounds for the gradients for minimal graphic functions of linear growth only on one side. Then we can obtain a Liouville type theorem with such growth via splitting for tangent cones of $\Sigma$ at infinity. When, in contrast, we do not impose any growth restrictions for minimal graphic functions, we also obtain a Liouville type theorem under a certain non-radial Ricci curvature decay condition on $\Sigma$. In particular, the borderline for the Ricci curvature decay is sharp by our example in the last section.

We consider surfaces with parallel mean curvature vector (pmc surfaces) in complex space forms and introduce a holomorphic differential on such surfaces. When the complex dimension of the ambient space is equal to two we find a second holomorphic differential and then determine those pmc surfaces on which both differentials vanish. We also provide a reduction of codimension theorem and prove a non-existence result for pmc 2-spheres in complex space forms.

In this paper, we illustrate the wave character of the metrics and curvatures of manifolds and introduce a new understanding toolthe hyperbolic geometric flow. This kind of flow is new and very natural to understand certain wave phenomena in the nature as well as the geometry of manifolds. It possesses many interesting properties from both mathematics and physics. Several applications of this method have been found.