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1. Brownian Motions and Heat Kernel Lower Bounds on Kähler and Quaternion Kähler Manifolds

   https://doi.org/10.1093/imrn/rnaa199  

   Baudoin, Fabrice; Yang, Guang (August 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract We study the radial parts of the Brownian motions on Kähler and quaternion Kähler manifolds. Thanks to sharp Laplacian comparison theorems, we deduce as a consequence a sharp Cheeger–Yau-type lower bound for the heat kernels of such manifolds and also sharp Cheng’s type estimates for the Dirichlet eigenvalues of metric balls. 

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2. Logarithmic Sobolev Inequalities on Homogeneous Spaces

   https://doi.org/10.1093/imrn/rnae205  

   Gordina, Maria; Luo, Liangbing (September 2024, International Mathematics Research Notices)

   Abstract We consider sub-Riemannian manifolds which are homogeneous spaces equipped with a sub-Riemannian structure induced by a transitive action by a Lie group. Then the corresponding sub-Laplacian is not an elliptic but a hypoelliptic operator. We study logarithmic Sobolev inequalities with respect to the hypoelliptic heat kernel measure on such spaces. We show that the logarithmic Sobolev constant can be chosen to depend only on the Lie group acting transitively on such a space but the constant is independent of the action of its isotropy group. This approach allows us to track the dependence of the logarithmic Sobolev constant on the geometry of the underlying space, in particular we show that the constant is independent of the dimension of the underlying spaces in several examples. 

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3. 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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4. Weighted Poincaré inequality and the Poisson Equation

   https://doi.org/10.1090/tran/8291  

   Munteanu, Ovidiu; Sung, Chiung-Jue; Wang, Jiaping (March 2021, Transactions of the American Mathematical Society)

   We develop Green’s function estimates for manifolds satisfying a weighted Poincaré inequality together with a compatible lower bound on the Ricci curvature. This estimate is then applied to establish existence and sharp estimates of solutions to the Poisson equation on such manifolds. As an application, a Liouville property for finite energy holomorphic functions is proven on a class of complete Kähler manifolds. Consequently, such Kähler manifolds must be connected at infinity. 

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5. Fractional Yamabe Problem on Locally Flat Conformal Infinities of Poincaré-Einstein Manifolds

   https://doi.org/10.1093/imrn/rnad195  

   Mayer, Martin; Ndiaye, Cheikh_Birahim (August 2023, International Mathematics Research Notices)

   Abstract We study the fractional Yamabe problem first considered by Gonzalez-Qing [36] on the conformal infinity $$(M^{n}, \;[h])$$ of a Poincaré-Einstein manifold $$(X^{n+1}, \;g^{+})$$ with either $n=2$ or $$n\geq 3$$ and $$(M^{n}, \;[h])$$ locally flat, namely $(M, h),$ is locally conformally flat. However, as for the classical Yamabe problem, because of the involved quantization phenomena, the variational analysis of the fractional one exhibits a local situation and also a global one. The latter global situation includes the case of conformal infinities of Poincaré-Einstein manifolds of dimension either $n=2$ or of dimension $$n\geq 3$$ and which are locally flat, and hence the minimizing technique of Aubin [4] and Schoen [48] in that case clearly requires an analogue of the positive mass theorem of Schoen-Yau [49], which is not known to hold. Using the algebraic topological argument of Bahri-Coron [8], we bypass the latter positive mass issue and show that any conformal infinity of a Poincaré-Einstein manifold of dimension either $n=2$ or of dimension $$n\geq 3$$ and which is locally flat admits a Riemannian metric of constant fractional scalar curvature. 

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