Some properties of the Yamabe soliton and the related nonlinear elliptic equation

2012/08/22 by Hsu, Shu-Yu
#35A01 (Primary) 35B40 #35J70 #58J05 (Secondary) #58J37 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1208.4445

Abstract

We will prove the non-existence of positive radially symmetric solution of the nonlinear elliptic equation (n-1)/(m)Δv^m+αv+βx⋅∇ u=0 in Rⁿ when n≥ 3, 0\fracρn-2>0, the scalar curvature R(r)→ρ as r→∞ if either β>\fracρn-2>0 or ρ=0 and α>0 holds, and lim_r→∞R(r)=0 if ρ<0 and α>0. We give a simple different proof of a result of P.Daskalopoulos and N.Sesum \citeDS2 on the positivity of the sectional curvature of rotational symmetric Yamabe solitons g=v^(4)/(n+2)dx² with v satisfying the above equation with m=(n-2)/(n+2). We will also find the exact value of the sectional curvature of such Yamabe solitons at the origin and at infinity.

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