Type II singularities on complete non-compact Yamabe flow

Beomjun Choi,John King,Panagiota Daskalopoulos

doi:10.1515/crelle-2020-0032

Abstract

Abstract This work concerns with the existence and detailed asymptotic analysis of type II singularities for solutions to complete non-compact conformally flat Yamabe flow with cylindrical behavior at infinity. We provide the specific blow-up rate of the maximum curvature and show that the solution converges, after blowing-up around the curvature maximum points, to a rotationally symmetric steady soliton. It is the first time that the steady soliton is shown to be a finite time singularity model of the Yamabe flow.

Highlights

This work concerns with the existence and detailed asymptotic analysis of Type II singularities for solutions to complete non-compact conformally flat Yamabe flow with cylindrical behavior at infinity
Hamilton [H] himself showed the existence of the normalized Yamabe flow (which is the re-parametrization of (1.1) to keep the volume fixed) for all time; in the case when the scalar curvature of the initial metric is negative, he showed the exponential convergence of the flow to a metric of constant scalar curvature
Schwetlick and Struwe [SS] obtained the convergence of the Yamabe flow on a general compact manifold under a suitable Kazdan-Warner type of condition that rules out the formation of bubbles and that is verified in dimensions 3 ≤ n ≤ 5

Summary

Formal matched asymptotics

We will present the formal construction of our solutions which is based on matched asymptotic analysis. The solutions given by (2.7) describe non-compact surfaces moving in positive η ( positive s) direction, while the corresponding solutions defined on η < 0 describe a symmetric surface just moving on the opposite direction This ansatz, namely setting w(η, τ ) := (n − 1)(n − 2). Assuming that in this region the first term having e−γτ becomes negligible as τ → ∞, the equation is reduced to (n − 1). For a function C(τ ), where w0 is one τ −independent solution of (2.14) By plugging this into (2.13) again, we get an error term e−γτ (1 + γ)w − C (τ ) wξ ≈ 0.

Notation and different scalings

Barrier construction in the outer region

Barrier construction in the inner region

Construction of super and sub-solutions

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