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The principle of convergence stability for geometric flows is the combination of the continuous dependence of the flow on initial conditions, with the stability of fixed points. It implies that if the flow from an initial state $g_0$ exists for all time and converges to a stable fixed point, then the flows of solutions that start near $g_0$ also converge to fixed points. We show this in the case of the Ricci flow, carefully proving the continuous dependence on initial conditions. Symmetry assumptions on initial geometries are often made to simplify geometric flow equations. As an application of our results, we extend known convergence results to open sets of these initial data, which contain geometries with no symmetries., 18 pages

We prove local existence for the second order Renormalization Group flow initial value problem on closed Riemannian manifolds (M, g) in general dimensions, for initial metrics whose sectional curvatures KP satisfy the condition 1 + αKP > 0, at all points p ∈ M and planes P ⊂ TpM . This extends results previously proven for two and three dimensions. The second order approximation of the Renormalization Group flow for the nonlinear sigma model of quantum field theory, which we label the RG-2 flow, is specified by the PDE system (1) ∂ ∂t g = −2 Rc− 2 Rm . Here g is a Riemannian metric, Rc is its Ricci curvature, Rmij = g ggRiklmRjpqn, and α is a positive parameter. We note that for our purposes here, α can assume any real value. For α = 0, this system (1) reduces to the Ricci flow. One can see that the sign of the right hand side, which is roughly 1+α×Curvature, should have an impact on the behavior of the flow, and this has been confirmed in various settings: in particular, the size of the term influences the parabolicity of the flow. Oliynyk has shown in [10] that on a two-dimensional manifold, if the Gaussian curvature K satisfies the condition 1 + αK > 0, then the flow is (weakly) parabolic; while if 1 + αK 0 is satisfied for all sectional curvatures KP . In this note we extend this curvature criterion for short-time existence for RG-2 flow to all dimensions, as first announced in [7]. Our main result is the following: Theorem 1. Let (M, g0) be a closed n-dimensional Riemannian manifold. If 1 + αKP > 0 for all sectional curvatures KP (g0), at all points p ∈M and planes P ⊂ TpM , then there exists a unique solution g(t) of the initial value problem ∂ ∂tg = −2 Rc− α 2 Rm , g(0) = g0, on some time interval [0, T ). Remark 2. In [10], Oliynyk finds open subspaces of the space of smooth metrics that are invariant under the two-dimensional RG-2 flow, and for which the flow remains parabolic (resp. backward parabolic). We are currently investigating this for general dimensions. Proof. To prove the theorem, we calculate the principal symbol of the DeTurck-modified version of RG-2 flow, which is generated by the PDE system (compare with (1) above) (2) ∂ ∂t gij = −2Rij + LWu,ggij − α 2 Rmij . Here Wu,g = −giju−1 jk gg(∇puql− 1 2∇lupq) is the standard vector field usually chosen to modify the Ricci flow into the related (parabolic) DeTurck version of Ricci flow, with u a fixed metric. Letting φt be the one-parameter family of diffeomorphisms generated by the vector field −Wu,g, then φt g is a solution of the RG-2 flow (see also [8]). As in the analogous Ricci flow case, if one can show (for a class of choices of the initial metric) that the PDE system (2) is parabolic, then short-time existence holds for the RG-2 flow (1) as well as for the DeTurck-modified flow (2). Date: January 7, 2014. KG is partially supported by the NSF under grant DGE-1144155. CG is partially supported by the Simons Foundation Collaboration Grant for Mathematicians 283083. JI is partially supported by the NSF under grant PHY-1306441 at the University of Oregon. He also wishes to thank the Mathematical Sciences Research Institute in Berkeley, California, for support under grant 0932078 000. Some of this work was carried out while JI was in residence at MSRI during the fall of 2013. 1 2 KARSTEN GIMRE, CHRISTINE GUENTHER, AND JAMES ISENBERG To calculate the symbol of the system (2), we first linearize the flow. For the first two terms of the right hand side of (2), this linearization effectively produces the Laplacian (see [4], or Theorem 2.1 in [5]). For the remaining term, Rm, it is useful to recall the formula for the variation of the Riemann curvature tensor with respect to the metric (see pg. 74 in [2]): [DRmg(h)] l ijk = [ ∂ ∂e Rm(g + eh) ∣∣∣∣ e=0 ]l

The Ricci flow has been of fundamental importance in mathematics, most famously through its use as a tool for proving the Poincare conjecture and Thurston’s geometrization conjecture. It has a parallel life in physics, arising as the first-order approximation of the renormalization group flow for the nonlinear sigma model of quantum field theory. There recently has been interest in the second-order approximation of this flow, called the RG-2 flow, which mathematically appears as a natural nonlinear deformation of the Ricci flow. A curvature flow arising from quantum field theory seems to us to capture the spirit of Yvonne Choquet-Bruhat’s extensive work in mathematical physics, and so in this commemorative article we give a geometric introduction to the RG-2 flow. A number of new results are presented as part of this narrative: short-time existence and uniqueness results in all dimensions if the sectional curvatures K ij satisfy certain inequalities; the calculation of fixed points for n = 3 dimensions; a reformulation of constant curvature solutions in terms of the Lambert W function; a classification of the solutions that evolve only by homothety; an analogue for RG flow of the 2-dimensional Ricci flow solution known to mathematicians as the cigar soliton, and discussed in the physics literature as Witten’s black hole. We conclude with a list of open problems whose resolutions would substantially increase our understanding of the RG-2 flow both physically and mathematically.

In this article we prove the existence of a fundamental solution for the linear parabolic operator L(u) = (Δ − ∂/∂t − h)u, on a compact n-dimensional manifold M with a time-parameterized family of smooth Riemannian metrics g(t). Δ is the time-dependent Laplacian based on g(t), and h(x, t) is smooth. Uniqueness, positivity, the adjoint property, and the semigroup property hold. We further derive a Harnack inequality for positive solutions of L(u) = 0 on (M, g(t) on a time interval depending on curvature bounds and the dimension of M, and in the special case of Ricci flow, use it to find lower bounds on the fundamental solution of the heat operator in terms of geometric data and an explicit Euclidean type heat kernel.

We study the behavior of the second order Renormalization Group flow on locally homogeneous metrics on closed three-manifolds. In the cases $\mathbb R^3$ and $\text{SO}(3)\times \R$, the flow is qualitatively the same as the Ricci flow. In the cases $\text{H}(3)$ and $\text{H}(2)\times \R$, if the curvature is small, then the flow expands as in the Ricci flow case, while if the curvature is large, then the flow contracts and forms a singularity in finite time. The main focus of the paper is the flow on the $\text{SU}(2)$, $\text{Nil}$, $\text{Sol}$, and $\text{SL}(2,\R)$ 3-geometries, with two of the three principal directions set equal. The configuration spaces for these geometries are two dimensional, and we can consequently apply phase plane techniques to the study. For the $\text{SU}(2)$ case, the flow is everywhere qualitatively the same as Ricci flow. For the $\text{Nil}$, $\text{Sol}$, and $\text{SL}(2,\R)$ cases, we show that the configuration space is partitioned into two regions which are delineated by a solution curve of the flow that depends on the coupling parameter: in one of the regions, the flow develops cigar or pancake singularities characteristic of the Ricci flow, while in the other both directions shrink. In the $\text{Nil}$ case we obtain a characterization of the full 3-dimensional flow.

As a step toward understanding the analytic behavior of Type-III Ricci flow singularities, i.e. immortal solutions that exhibit |Rm, Comment: This is an expanded version that proves linear stability of all explicitly known nonproduct examples of homogeneous expanding Ricci solitons on nilpotent or solvable Lie groups. (To appear in Int. Math. Res. Not.)