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Additional info for Blow up analysis, existence and qualitative properties of solutions for the two dimensional emden-fowler equation with singular potential

Example text

If x ∈ Ω\Ωσ , then there is either an ε-cap or an εneck, adjacent to the initial ε-neck. In the latter case we can take a point on the boundary of the second ε-neck and continue. This procedure can either terminate when we get into Ωσ or an ε-cap, or go on infinitely, producing 53 an ε-horn. The same procedure can be repeated for the other boundary component of the initial ε-neck. Therefore, we conclude that each ε-neck of (Ω, g ij ) is contained in a subset of Ω of one of the following types: (a) an ε-tube with boundary components in Ωσ , or (b) an ε-cap with boundary in Ωσ , or (c) an ε-horn with boundary in Ωσ , or (d) a capped ε-horn, or (e) a double ε-horn.

This is a contradiction. 4, a suitable neighborhood B (for suitable r) of xk would fall into the category (b) (over B4 ) for sufficiently large k. We also get a contradiction. 4. Therefore we have proved the theorem. # 42 4. 1) with gij (x, 0) = gij (x) on M 4 . Since the initial metric gij (x) has positive scalar curvature, it is easy to see that the maximal time T must be finite and the curvature tensor becomes unbounded as t → T . 1 of √ ), the solution gij (x, t) is κ-noncollapsed on the scale T for all t ∈ [0, T ) for some κ > 0.

Denote by 1 U= q0 ∈γ∞ (∞) B(q0 , 24π(R∞ (q0 ))− 2 ) (⊂ (B∞ , gij )) 1 1 where B(q0 , 24π(R∞ (q0 ))− 2 ) is the ball centered at q0 of radius 24π(R∞ (q0 ))− 2 . 3) that U has (∞) nonnegative curvature operator. Since the metric gij is cylindrical at any point q0 ∈ γ∞ which is sufficiently close to y∞ , we see that the metric space U = U ∪{y∞ } by adding the point y∞ , is locally complete and strictly intrinsic near y∞ . Here strictly intrinsic means that the distance between any two points can be realized by shortest geodesics.