We analyze the Cauchy problem for symmetric hyperbolic equations with a time singularity of Fuchsian type and establish a global existence theory along with decay estimates for evolutions towards the singular time under a small initial data assumption. We then apply this theory to semilinear wave equations near spatial infinity on Minkowski and Schwarzschild spacetimes, and to the relativistic Euler equations with Gowdy symmetry on Kasner spacetimes.Constants of this type will always be non-negative, non-decreasing, continuous functions of their arguments.Given four vector bundles V , W , Y and Z that sit over Σ, and mapsFor situations, where we want to bound f (t, w, v) by g(t, v) up to an undetermined constant of proportionality, we defineif there exists a R ∈ (0, R) and a map f ∈ C 0 [T 0 , 0), C ∞ (B R (W ) × B R (V ), L(Y, Z))such that
In this work we present solutions to the Ricci flow equations in arbitrary dimensions, particularizing for the 3d and 4d cases. We start by considering the 3d case and note that our solutions belong to the family of maximally symmetric spaces that can be extended to the d ≥ 4 case following an analogue treatment. These solutions can be divided into two scenarios: maximally symmetric spaces with positive curvature i.e. de Sitter spaces, and maximally symmetric spaces with negative curvature i.e. Anti-de Sitter spaces. We show that between both scenarios there is a critical point where the curvature blows up along the flow. Also the solutions for d ≥ 4 satisfy the flow equations with Riemannian or pseudo-Riemannian metrics due to the fact that the considered maximally symmetric spaces do not depend on time neither on the angular coordinates yielding equations that depend only on the radial coordinate and the Ricci flow parameter. Additionally we find an interesting effect of the flow consisting in a change of the signature of the metric when passing the singular point. Besides the signature change, the sign of the curvature also experiences a transition from positive to negative curvature, either with Riemannian or pseudo-Riemannian metrics throughout the singular point.
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