A new proof for the existence of rotationally symmetric gradient Ricci solitons

2021/05/09 by Shu-Yu Hsu, Hsu, Shu-Yu
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2105.03805

Abstract

We give a new proof for the existence of rotationally symmetric steady and expanding gradient Ricci solitons in dimension n+1, 2≤ n≤ 4, with metric g=(da²)/(h(a²))+a²d σ for some function h where dσ is the standard metric on the unit sphere Sⁿ in ℝⁿ. More precisely for any λ≥ 0, 2≤ n≤ 4 and μ₁∈ℝ, we prove the existence of unique solution h∈ C²((0,∞))∩ C¹([0,∞)) for the equation 2r²h(r)h_rr(r)=(n-1)h(r)(h(r)-1)+rh_r(r)(rh_r(r)-λr-(n-1)), h(r)>0, in (0,∞) satisfying h(0)=1, h_r(0)=μ₁. We also prove the existence of unique analytic solution of the about equation on [0,∞) for any λ≥ 0, n≥ 2 and μ₁∈ℝ. Moreover we will prove the asymptotic behaviour of the solution h for any n≥ 2, λ≥ 0 and μ₁∈ℝ∖\0\.

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