A bayesian method of parameter identification and prediction by Pavlov N. D.

By Pavlov N. D.

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This implies that the canonical neighborhood around q0 can not be a compact manifold (without boundary) with positive curvature operator. Note that γ∞ is shortest since it is the limit of a sequence of shortest geodesics. Without loss of generality, we may assume ε is suitably small. These imply that as q0 sufficiently close to y∞ , the canonical neighborhood around q0 can not be a 2ε-cap. Thus we conclude that each q0 ∈ γ∞ , which is sufficiently close to y∞ , is the center of a 2ε-neck. 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 .

Note again that the number of compact 58 components is finite. Let us throw away all the compact components lying Ω\Ωσ or with positive curvature operator, and then consider the all components Ωj , 1 ≤ j ≤ k, of Ω which contains points of Ωσ . (We will consider the components of Ω\Ωσ consisting of capped ε-horns and double ε-horns later). We could perform Hamilton’s surgerical procedure in Section D of [19] at every horn of Ωj , 1 ≤ j ≤ k, so that the positive isotropic curvature condition and the pinching assumption is preserved.

4 Assume that the curvature operator of the limit (M∞ , gij (·, t)) at the time slice t = 0 is positive everywhere. Suppose there exists a sequence of 4 points pj ∈ M∞ such that their scalar curvatures R∞ (pj , 0) → +∞ as j → +∞. By the local curvature estimate in Step 1 and the assertion of the above Step 2 (for the marked points pj ) as well as the κ-noncollapsed assumption, a (∞) 4 subsequence of the rescaled and marked manifolds (M∞ , R∞ (pj , 0)gij (·), pj ) 49 ∞ converges in Cloc topology to a smooth nonflat limit Y .

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