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Register a new userPerelmans Stability Theorem
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Authors
Vitali Kapovitch
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We give a proof of the celebrated stability theorem of Perelman stating that for a noncollapsing sequence $X_i$ of Alexandrov spaces with curvature bounded below Gromov-Hausdorff converging to a compact Alexandrov space $X$, $X_i$ is homeomorphic to $X$ for all large $i$.
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In this expository note, we study the second variation of Perelmans entropy on the space of Kahler metrics at a Kahler-Ricci soliton. We prove that the entropy is stable in the sense of variations. In particular, Perelmans entropy is stable along the Kahler-Ricci flow. The Chinese version of this note has appeared in a volume in honor of professor K.C.Chang (Scientia Sinica Math., 46 (2016), 685-696).
We obtain a Central Limit Theorem for closed Riemannian manifolds, clarifying along the way the geometric meaning of some of the hypotheses in Bhattacharya and Lins Omnibus Central Limit Theorem for Frechet means. We obtain our CLT assuming certain stability hypothesis for the cut locus, which always holds when the manifold is compact but may not be satisfied in the non-compact case.
We prove a product formula for $delta$-invariant and as an application, we show that product of K-(semi, poly)stable Fano varieties is also K-(semi, poly)stable.
The $pi_2$-diffeomorphism finiteness result (cite{FR1,2}, cite{PT}) asserts that the diffeomorphic types of compact $n$-manifolds $M$ with vanishing first and second homotopy groups can be bounded above in terms of $n$, and upper bounds on the absolute value of sectional curvature and diameter of $M$. In this paper, we will generalize this $pi_2$-diffeomorphism finiteness by removing the condition that $pi_1(M)=0$ and asserting the diffeomorphism finiteness on the Riemannian universal cover of $M$.
Mosers theorem (1965) states that the diffeomorphism group of a compact manifold acts transitively on the space of all smooth positive densities with fixed volume. Here we describe the extension of this result to manifolds with corners. In particular we obtain Mosers theorem on simplices. The proof is based on Banyagas paper (1974), where Mosers theorem is proven for manifolds with boundary. A cohomological interpretation of Banyagas operator is given, which allows a proof of Lefschetz duality using differential forms.
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