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1. 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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2. Some local maximum principles along Ricci flows

   https://doi.org/10.4153/s0008414x20000772  

   Lee, Man-Chun; Tam, Luen-Fai (November 2020, Canadian Journal of Mathematics)

   null (Ed.)

   Abstract In this work, we obtain a local maximum principle along the Ricci flow $g(t)$ under the condition that $$\mathrm {Ric}(g(t))\le {\alpha } t^{-1}$$ for $t>0$ for some constant $${\alpha }>0$$ . As an application, we will prove that under this condition, various kinds of curvatures will still be nonnegative for $t>0$ , provided they are non-negative initially. These extend the corresponding known results for Ricci flows on compact manifolds or on complete noncompact manifolds with bounded curvature . By combining the above maximum principle with the Dirichlet heat kernel estimates, we also give a more direct proof of Hochard’s [15] localized version of a maximum principle by Bamler et al. [1] on the lower bound of different kinds of curvatures along the Ricci flows for $t>0$ . 

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3. Cheeger bounds on spin-two fields

   https://doi.org/10.1007/JHEP12(2021)217  

   De Luca, G. Bruno; De Ponti, Nicolò; Mondino, Andrea; Tomasiello, Alessandro (December 2021, Journal of High Energy Physics)

   A bstract We consider gravity compactifications whose internal space consists of small bridges connecting larger manifolds, possibly noncompact. We prove that, under rather general assumptions, this leads to a massive spin-two field with very small mass. The argument involves a recently-noticed relation to Bakry-Émery geometry, a version of the so-called Cheeger constant, and the theory of synthetic Ricci lower bounds. The latter technique allows generalizations to non-smooth spaces such as those with D-brane singularities. For AdS d vacua with a bridge admitting an AdS d +1 interpretation, the holographic dual is a CFT d with two CFT d− 1 boundaries. The ratio of their degrees of freedom gives the graviton mass, generalizing results obtained by Bachas and Lavdas for d = 4. We also prove new bounds on the higher eigenvalues. These are in agreement with the spin-two swampland conjecture in the regime where the background is scale-separated; in the opposite regime we provide examples where they are in naive tension with it. 

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4. Graph Hörmander Systems

   https://doi.org/10.1007/s00023-024-01474-7  

   Li, Haojian; Junge, Marius; LaRacuente, Nicholas (August 2024, Annales Henri Poincaré)

   Abstract This paper extends the Bakry-Émery criterion relating the Ricci curvature and logarithmic Sobolev inequalities to the noncommutative setting. We obtain easily computable complete modified logarithmic Sobolev inequalities of graph Laplacians and Lindblad operators of the corresponding graph Hörmander systems. We develop the anti-transference principle stating that the matrix-valued modified logarithmic Sobolev inequalities of sub-Laplacian operators on a compact Lie group are equivalent to such inequalities of a family of the transferred Lindblad operators with a uniform lower bound. 

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5. Comparison Theorems for 3D Manifolds With Scalar Curvature Bound

   https://doi.org/10.1093/imrn/rnab307  

   Munteanu, Ovidiu; Wang, Jiaping (November 2021, International Mathematics Research Notices)

   Abstract Two sharp comparison results are derived for 3D complete noncompact manifolds with scalar curvature bounded from below. The 1st one concerns the Green’s function. When the scalar curvature is nonnegative, it states that the rate of decay of an energy quantity over the level set is strictly less than that of the Euclidean space unless the manifold itself is isometric to the Euclidean space. The result is in turn converted into a sharp area comparison for the level set of the Green’s function when in addition the Ricci curvature of the manifold is assumed to be asymptotically nonnegative at infinity. The 2nd result provides a sharp upper bound of the bottom spectrum in terms of the scalar curvature lower bound, in contrast to the classical result of Cheng, which involves a Ricci curvature lower bound. 

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