Expanding Ricci Solitons Research Articles

Ricci solitons and Einstein-scalar field theory

B List has recently studied a geometric flow whose fixed points correspond to static Ricci flat spacetimes. It is now known that this flow is in fact Ricci flow modulo pullback by a certain diffeomorphism. We use this observation to associate with each static Ricci flat spacetime a local Ricci soliton in one higher dimension. Also, solutions of Euclidean-signature Einstein gravity coupled to a free massless scalar field with nonzero cosmological constant are associated with shrinking or expanding Ricci solitons. We exhibit examples, including an explicit family of complete expanding solitons. These solitons can also be thought of as a Ricci flow for a complete Lorentzian metric. The possible generalization to Ricci-flat stationary metrics leads us to consider an alternative to Ricci flow.

Linear stability of homogeneous Ricci solitons

As a step toward understanding the analytic behavior of Type-III Ricci ‡ow singularities, i.e. immortal solutions that exhibit jRm j C=t curvature decay, we examine the linearization of an equivalent ‡ow at xed points discovered recently by Baird–Danielo and Lott: nongradient homogeneous expanding Ricci solitons on nilpotent or solvable Lie groups. For all explicitly known nonproduct examples, we demonstrate linear stability of the ‡ow at these xed points and prove that the linearizations generate C0 semigroups.

Analysis of weighted Laplacian and applications to Ricci solitons

We study both function theoretic and spectral properties of the weighted Laplacian ∆f on complete smooth metric measure space (M, g, e dv) with its Bakry-Emery curvature Ricf bounded from below by a constant. In particular, we establish a gradient estimate for positive f−harmonic functions and a sharp upper bound of the bottom spectrum of ∆f in terms of the lower bound of Ricf and the linear growth rate of f. We also address the rigidity issue when the bottom spectrum achieves its optimal upper bound under a slightly stronger assumption that the gradient of f is bounded. Applications to the study of the geometry and topology of gradient Ricci solitons are also considered. Among other things, it is shown that the volume of a noncompact shrinking Ricci soliton must be of at least linear growth. It is also shown that a nontrivial expanding Ricci soliton must be connected at infinity provided its scalar curvature satisfies a suitable lower bound.