Singular Ricci Solitons and Their Stability under the Ricci Flow

Abstract

We introduce certain spherically symmetric singular Ricci solitons and study their stability under the Ricci flow from a dynamical PDE point of view. The solitons in question exist for all dimensions n + 1 ≥ 3, and all have a point singularity where the curvature blows up; their evolution under the Ricci flow is in sharp contrast to the evolution of their smooth counterparts. In particular, the family of diffeomorphisms associated with the Ricci flow "pushes away" from the singularity causing the evolving soliton to open up immediately becoming an incomplete (but non-singular) metric. In the second part of this paper we study the local-in time stability of this dynamical evolution, under spherically symmetric perturbations of the initial soliton metric. We prove a local well-posedness result for the Ricci flow near the singular initial data, which in particular implies that the "opening up" of the singularity persists for the perturbations also.

On the Backward Stability of the Schwarzschild Black Hole Singularity

“…To our surprise the present evolution bears some resemblance at an analytical level with a prior work on the stability of singular Ricci solitons [1]. Although of different nature, hyperbolic/parabolic (respectively), they share a couple of key features such as the opening up rate of the singularity and the "borderline" singularities in the coefficients involved.…”

“…4 Recall that the vector field tangent to the r = const. hypersurfaces (Figure 1) is Killing and we may hence utilize it to shift Σ 0 and (u = 1, v = 1) to whichever point on {uv = 1} we wish; Figure ( Realizing the above plan we thus prove the existence of a class of non-spherically symmetric vacuum spacetimes for which (1) the leading asymptotics of the blow up of curvature and in general of all the geometric quantities (metric, second fundamental form etc.) coincide with their Schwarzschild counterparts, as one approaches the singularity, and (2) the singularity is realized as the limit of uniformly spacelike hypersurfaces, which in the forward direction "pinch off" in finite time at one sphere.…”

“…where by H 1 0 we denote the set of functions in H 1 (Σ 0 ) having zero trace on the boundary of Σ 0 . 18 We fix dg (1) , dK (1) and study Ψ under variations of the variables dg (0) , dK (0) . Whence, we can view Ψ as a map Ψ : (H 4,α+ 3 2…”

“…is non-empty, for every pair (dg (1) , dK (1) ) in a sufficiently small ball of H 4,α+ 3 2 × H 3,α . Remark 6.4.…”

“…Moreover, taking |a| 1, the 'difference' of the corresponding initial data sets from the Schwarzschild one (with the same M > 0), measured in standard Sobolev norms, 1 can be made arbitrarily small. 1 The difference can be defined, for example, component wise for the two pairs of 2-tensors with respect to a common coordinate system and measured in W s,p Sobolev spaces used in the literature [4].…”