In this paper we consider new geometric flow equations, called D-flow, which describe the variation of space-time geometries under the change of the number of dimensions. The D-flow is originating from the non-trivial dependence of the volume of space-time manifolds on the number of space-time dimensions and it is driven by certain curvature invariants. We will work out specific examples of D-flow equations and their solutions for the case of D-dimensional spheres and Freund-Rubin compactified space-time manifolds. The discussion of the paper is motivated from recent swampland considerations, where the number D of space-time dimensions is treated as a new swampland parameter.
Following the recent observation that the Ricci flow and the infinite distance swampland conjecture are closely related to each other, we will investigate in this paper geometric flow equations for AdS space‐time geometries. First, we consider the so called Yamabe and Ricci‐Bourguignon flows and we show that these two flows ‐ in contrast to the Ricci flow ‐ lead to infinite distance fixed points for product spaces like , where denotes d‐dimensional AdS space and corresponds to a p‐dimensional sphere. Second, we consider black hole geometries in AdS space time geometries and their behaviour under the Yamabe and Ricci‐Bourguignon flows. Specifically we will examine if and how the AdS black holes will undergo a Hawking‐Page phase transition under the Ricci flow, the Yamabe flow and under the general Ricci‐Bourguignon flow.
In the following work, an attempt to conciliate the Ricci flow conjecture and the Cobordism conjecture, stated as refinements of the Swampland distance conjecture and of the No global symmetries conjecture respectively, is presented. This is done by starting from a suitable manifold with trivial cobordism class, applying surgery techniques to Ricci flow singularities and trivialising the cobordism class of one of the resulting connected components via the introduction of appropriate defects. The specific example of $$ {\varOmega}_4^{SO} $$ Ω 4 SO is studied in detail. A connection between the process of blowing up a point of a manifold and that of taking the connected sum of such with ℂℙn is explored. Hence, the problem of studying the Ricci flow of a K3 whose cobordism class is trivialised by the addition of 16 copies of ℂℙ2 is tackled by applying both the techniques developed in the previous sections and the classification of singularities in terms of ADE groups.
Ricci flow is a natural gradient flow of the Einstein-Hilbert action. Here we consider the analog for the Einstein-Maxwell action, which gives Ricci flow with a stress tensor contribution coupled to a Yang-Mills flow for the Maxwell field. We argue that this flow is well-posed for static spacetimes with pure electric or magnetic potentials and show it preserves both non-extremal and extremal black hole horizons. In the latter case we find the flow of the near horizon geometry decouples from that of the exterior. The Schwarzschild black hole is an unstable fixed point of Ricci flow for static spacetimes. Here we consider flows of the Reissner-Nordström (RN) fixed point. The magnetic RN solution becomes a stable fixed point of the flow for sufficient charge. However we find that the electric RN black hole is always unstable. Numerically solving the flow starting with a spherically symmetric perturbation of a non-extremal RN solution, we find similar behaviour in the electric case to the Ricci flows of perturbed Schwarzschild, namely the horizon shrinks to a singularity in finite time or expands forever. In the magnetic case, a perturbed unstable RN solution has a similar expanding behaviour, but a perturbation that decreases the horizon size flows to a stable black hole solution rather than a singularity. For extremal RN we solve the near horizon flow for spherical symmetry exactly, and see in the electric case two unstable directions which flow to singularities in finite flow time. However, even turning these off, and fixing the near horizon geometry to be that of RN, we numerically show that the flows appear to become singular in the vicinity of its horizon.
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