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1. Uniqueness and stability of Ricci flow through singularities

   https://doi.org/DOI: https://dx.doi.org/10.4310/ACTA.2022.v228.n1.a1  

   Bamler, Richard; Kleiner, Bruce (July 2022, Acta mathematica)

   We verify a conjecture of Perelman, which states that there exists a canonical Ricci flow through singularities starting from an arbitrary compact Riemannian 3‑manifold. Our main result is a uniqueness theorem for such flows, which, together with an earlier existence theorem of Lott and the second named author, implies Perelman’s conjecture. We also show that this flow through singularities depends continuously on its initial condition and that it may be obtained as a limit of Ricci flows with surgery. Our results have applications to the study of diffeomorphism groups of 3‑manifolds—in particular to the generalized Smale conjecture—which will appear in a subsequent paper. 

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2. Blow-up sets of Ricci curvatures of complete conformal metrics

   https://doi.org/10.1515/crelle-2025-0020  

   Han, Qing; Shen, Weiming; Wang, Yue (March 2025, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract A version of the singular Yamabe problem in smooth domains in a closed manifoldyields complete conformal metrics with negative constant scalar curvatures.In this paper, we study the blow-up phenomena of Ricci curvatures of these metrics on domains whose boundary is close to a certain limit set of a lower dimension.We will characterize the blow-up set according to the Yamabe invariant of the underlying manifold.In particular, we will prove that all points in the lower dimension part of the limit set belong to the blow-up set on manifolds not conformally equivalent to the standard sphere and that all but one point in the lower dimension part of the limit set belong to the blow-up set on manifolds conformally equivalent to the standard sphere.In certain cases, the blow-up set can be the entire manifold.We will demonstrate by examples that these results are optimal. 

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3. The Splitting Theorem and topology of noncompact spaces with nonnegative N-Bakry Émery Ricci curvature

   https://doi.org/10.1090/proc/15240  

   Lim, Alice (August 2021, Proceedings of the American Mathematical Society)

   In this paper, we generalize topological results known for noncompact manifolds with nonnegative Ricci curvature to spaces with nonnegative N N -Bakry Émery Ricci curvature. We study the Splitting Theorem and a property called the geodesic loops to infinity property in relation to spaces with nonnegative N N -Bakry Émery Ricci curvature. In addition, we show that if M n M^n is a complete, noncompact Riemannian manifold with nonnegative N N -Bakry Émery Ricci curvature where N > n N>n , then H n − 1 ( M , Z ) H_{n-1}(M,\mathbb {Z}) is 0 0 . 

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4. Nonnegative Ricci curvature, splitting at infinity, and first Betti number rigidity

   https://doi.org/10.1016/j.aim.2024.110028  

   Pan, Jiayin; Ye, Zhu (December 2024, Advances in Mathematics)

   We study the rigidity problems for open (complete and noncompact) $$n$$-manifolds with nonnegative Ricci curvature. We prove that if an asymptotic cone of $$M$$ properly contains a Euclidean $$\mathbb{R}^{k-1}$$, then the first Betti number of $$M$$ is at most $n-k$; moreover, if equality holds, then $$M$$ is flat. Next, we study the geometry of the orbit $$\Gamma\tilde{p}$$, where $$\Gamma=\pi_1(M,p)$$ acts on the universal cover $$(\widetilde{M},\tilde{p})$$. Under a similar asymptotic condition, we prove a geometric rigidity in terms of the growth order of $$\Gamma\tilde{p}$$. We also give the first example of a manifold $$M$$ of $$\mathrm{Ric}>0$$ and $$\pi_1(M)=\mathbb{Z}$$ but with a varying orbit growth order. 

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5. Large Deviation Estimates of Selberg’s Central Limit Theorem and Applications

   https://doi.org/10.1093/imrn/rnad176  

   Arguin, Louis-Pierre; Bailey, Emma (July 2023, International Mathematics Research Notices)

   Abstract For $$V\sim \alpha \log \log T$$ with $$0<\alpha <2$$, we prove $$\begin{align*} & \frac{1}{T}\textrm{meas}\{t\in [T,2T]: \log|\zeta(1/2+ \textrm{i} t)|>V\}\ll \frac{1}{\sqrt{\log\log T}} e^{-V^{2}/\log\log T}. \end{align*}$$This improves prior results of Soundararajan and of Harper on the large deviations of Selberg’s Central Limit Theorem in that range, without the use of the Riemann hypothesis. The result implies the sharp upper bound for the fractional moments of the Riemann zeta function proved by Heap, Radziwiłł, and Soundararajan. It also shows a new upper bound for the maximum of the zeta function on short intervals of length $$(\log T)^{\theta }$$, $$0<\theta <3$$, that is expected to be sharp for $$\theta> 0$$. Finally, it yields a sharp upper bound (to order one) for the moments on short intervals, below and above the freezing transition. The proof is an adaptation of the recursive scheme introduced by Bourgade, Radziwiłł, and one of the authors to prove fine asymptotics for the maximum on intervals of length $$1$$. 

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