Short-time existence for the second order renormalization group flow in general dimensions

Abstract

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