We discuss some classification results for Ricci solitons, that is, self similar solutions of the Ricci Flow. New simpler proofs of some known results will be presented. In detail, we will take the equation point of view, trying to avoid the tools provided by considering the dynamic properties of the Ricci flow.

We introduce a new continuity method which, although less natural than flows such as the Kähler-Ricci flow, has the advantage of preserving a lower bound on the Ricci curvature, hence allowing the application of comparison geometry techniques, such as Cheeger-Colding-Tian's compactness theory.

We present here a theory of fractional electricity and magnetism which is capable of describing phenomenon as disparate as the non-locality of the Pippard kernel in superconductivity and anomalous dimensions for conserved currents in holographic dilatonic models. While it is a standard result in field theory that the scaling dimension of conserved currents and their associated gauge fields are determined strictly by dimensional analysis and hence cannot change under any amount of renormalization, it is also the case that the standard conservation laws for currents, dJ = 0, remain unchanged in form if any differential operator that commutes with the total exterior derivative, [d,Ŷ ] = 0, multiplies the current. Such an operator, effectively changing the dimension of the current, increases the allowable gauge transformations in electromagnetism and is at the heart of Nöther's second theorem. However, this observation has not been exploited to generate new electromagnetisms. Here we develop a consistent theory of electromagnetism that exploits this hidden redundancy in which the standard gauge symmetry in electromagnetism is modified by the rotationally invariant operator, the fractional Laplacian. We show that the resultant theories all allow for anomalous (nontraditional) scaling dimensions of the gauge field and the associated current. Using the Caffarelli/Silvestre(Caffarelli and Silvestre, 2007) theorem, its extension(Nave and Phillips, 2019) to p-forms and the membrane paradigm, we show that either the boundary (UV) or horizon (IR) theory of holographic dilatonic models are both described by such fractional electromagnetic theories. We also show that the non-local Pippard kernel introduced to solve the problem of the Meissner effect in elemental superconductors can also be formulated as a special case of fractional electromagnetism. Because the holographic dilatonic models produce boundary theories that are equivalent to those arising from a bulk theory with a massive gauge field along the radial direction, the common thread linking both of these problems is the breaking of U (1) symmetry down to Z 2 . We show that the standard charge quantization rules fail when the gauge field acquires an anomalous dimension. The breakdown of charge quantization is discussed extensively in terms of the experimentally measurable modified Aharonov-Bohm effect in the strange metal phase of the cuprate superconductors.

In this paper we investigate the differential geometric and algebro-geometric properties of the noncollapsing limit in the continuity method that was introduced by the first two named authors in [19].

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