Yamabe Flow on Non-compact Manifolds with Unbounded Initial Curvature
- PDF / 363,377 Bytes
- 15 Pages / 439.37 x 666.142 pts Page_size
- 35 Downloads / 241 Views
Yamabe Flow on Non-compact Manifolds with Unbounded Initial Curvature Mario B. Schulz1 Received: 4 July 2018 © Mathematica Josephina, Inc. 2019
Abstract We prove global existence of Yamabe flows on non-compact manifolds M of dimension m ≥ 3 under the assumption that the initial metric g0 = u 0 g M is conformally equivalent to a complete background metric g M of bounded, non-positive scalar curvature and positive Yamabe invariant with conformal factor u 0 bounded from above and below. We do not require initial curvature bounds. In particular, the scalar curvature of (M, g0 ) can be unbounded from above and below without growth condition. Keywords Yamabe flow · Non-compact · Unbounded scalar curvature · Global existence Mathematics Subject Classification 53C44 · 35K55 · 35A01 Richard Hamilton’s [12] Yamabe flow describes a family of Riemannian metrics g(t) subject to the evolution equation ∂t∂ g = −Rg g, where Rg denotes the scalar curvature corresponding to the metric g. This equation tends to conformally deform a given initial metric towards a metric of vanishing scalar curvature. Hamilton proved existence of Yamabe flows on compact manifolds without boundary. Their asymptotic behaviour was subsequently analysed by Chow [7], Ye [21], Schwetlick and Struwe [17] and Brendle [4,5]. The theory of Yamabe flows on non-compact manifolds is not as developed as in the compact case. Daskalopoulos and Sesum [8] analysed the profiles of self-similar solutions (Yamabe solitons). The question of existence in general was addressed by Ma and An who obtained the following results on complete, noncompact Riemannian manifolds (M, g0 ) satisfying certain curvature assumptions: • If (M, g0 ) has Ricci curvature bounded from below and uniformly bounded, nonpositive scalar curvature, then there exists a global Yamabe flow on M with g0 as initial metric [15].
B 1
Mario B. Schulz [email protected] ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland
123
M.B. Schulz
• If (M, g0 ) is locally conformally flat with Ricci curvature bounded from below and uniformly bounded scalar curvature, then there exists a short-time solution to the Yamabe flow on M with g0 as initial metric [15]. • If (M, g0 ) has non-negative scalar curvature Rg0 (possibly unbounded from above) and if the equation −g0 w = Rg0 has a non-negative solution w in M, then there exists a global Yamabe flow on M with g0 as initial metric [14]. More recently, Bahuaud and Vertman [1,2] constructed Yamabe flows starting from spaces with incomplete edge singularities such that the singular structure is preserved along the flow. Choi, Daskalopoulos and King [6] were able to find solutions to the Yamabe flow on Rm which develop a type II singularity in finite time. In dimension m = 2, where the Yamabe flow coincides with the Ricci flow, Giesen and Topping [10,18,19] introduced the notion of instantaneous completeness and obtained existence and uniqueness of instantaneously complete Ricci flows on arbitrary surfaces. In particular, they do not require any assumptions on
Data Loading...
