Shu-Yu Hsu — Neckpinch

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

Sun, 09 May 2021 00:00:00 GMT

Shu-Yu Hsu — 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)hᵣᵣ(r)=(n-1)h(r)(h(r)−1)+rhᵣ(r)(rhᵣ(r)−λr-(n-1)), h(r)>0, in (0,∞) satisfying h(0)=1, hᵣ(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\.

Another proof of the existence of homothetic solitons of the inverse mean curvature flow

Sat, 18 Jan 2020 00:00:00 GMT

Shu-Yu Hsu — We will give a new proof of the existence of non-compact homothetic solitons of the inverse mean curvature flow (cf.) in ℝⁿ× ℝ, n≥ 2, of the form (r,y(r)) or (r(y),y) where r=|x|, x∈ℝⁿ, is the radially symmetric coordinate and y∈ ℝ. More precisely for any 1/n<λ<1/(n-1) and μ<0, we will give a new proof of the existence of a unique solution r(y)∈ C²(μ,∞)∩ C([μ,∞)) of the equation fracr_yy(y)1+r_y(y)²=(n-1)/(r(y))−(1+r_y(y)²)/(λ(r(y)−yr_y(y))), r(y)>0, in (μ,∞) which satisfies r(μ)=0 and r_y(μ)=lim_(y↘μ)r_y(y)=+∞. We also prove that there exist constants y₂>y₁>0 such that r_y(y)>0 for any μy₁, r_yy(y)<0 for any μ0 for any y>y₂. Moreover lim_(y→ +∞)r(y)=0 and lim_(y→ +∞)yr_y(y)=0.

Another proof of the local curvature estimate for the Ricci flow

Fri, 07 Apr 2017 00:00:00 GMT

Shu-Yu Hsu — By using the De Giorgi iteration method we will give a new simple proof of the recent result of B.Kotschwar, O.Munteanu, J.Wang [KMW] and N.Sesum [S] on the local boundedness of the Riemmanian curvature tensor of solutions of Ricci flow in terms of its inital value on a given ball and a local uniform bound on the Ricci curvature.

Existence and properties of ancient solutions of the Yamabe flow

Thu, 09 Jun 2016 00:00:00 GMT

Shu-Yu Hsu — Let n≥ 3 and m=(n-2)/(n+2). We construct 5-parameters, 4-parameters, 3-parameters ancient solutions of the equation vₜ=(v^m)ₓₓ+v-v^m, v>0, in ℝ× (-∞,T) for some T∈ℝ. This equation arises in the study of Yamabe flow. We obtain various properties of the ancient solutions of this equation including exact decay rate of ancient solutions as |x|→∞. We also prove that both the 3-parameters ancient solution and the 4-parameters ancient solution are singular limit solution of the 5-parameters ancient solutions.