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1. Rigidity and compactness with constant mean curvature in warped product manifolds

   Maggi, Francesco; Santilli, Mario (March 2024, arxiv)

   We prove the rigidity of rectifiable boundaries with constant distributional mean curvature in the Brendle class of warped product manifolds (which includes important models in General Relativity, like the deSitter--Schwarzschild and Reissner--Nordstrom manifolds). As a corollary we characterize limits of rectifiable boundaries whose mean curvatures converge, as distributions, to a constant. The latter result is new, and requires the full strength of distributional CMC-rigidity, even when one considers smooth boundaries whose mean curvature oscillations vanish in arbitrarily strong C^k-norms. 

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2. Bergman metrics with constant holomorphic sectional curvatures

   https://doi.org/10.1515/crelle-2025-0013  

   Huang, Xiaojun; Li, Song-Ying; Treuer, John N (March 2025, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract The paper studies complex manifolds whose Bergman metrics are incomplete but have constant holomorphic sectional curvature.We will construct a real analytic unbounded domain in ${\mathbb{C}}^2$\mathbb{C}^{2}whose Bergman metric is well-defined and has a positive constant holomorphic sectional curvature, which appears to be the first example of this kind.We will answer a long standing folklore conjecture that a Stein manifold has a negative constant holomorphic sectional curvature if and only if it is biholomorphic to a ball with a pluripolar set removed.Together with the uniqueness of a moment problem in the appendix of the paper provided by John Treuer, we will show that, under natural assumptions, there is no complex manifold whose Bergman metric is flat. 

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3. The Neumann problem on the domain in 𝕊 ³ bounded by the Clifford torus

   https://doi.org/10.1515/ans-2022-0072  

   Case, Jeffrey S; Chen, Eric; Wang, Yi; Yang, Paul; Yung, Po-Lam (January 2023, Advanced Nonlinear Studies)

   Abstract In this study, the solution of the Neumann problem associated with the CR Yamabe operator on a subset $\mathrm{\Omega }$\Omegaof the CR manifold ${\mathbb{S}}^3${{\mathbb{S}}}^{3}bounded by the Clifford torus $\mathrm{\Sigma }$\Sigmais discussed. The Yamabe-type problem of finding a contact form on $\mathrm{\Omega }$\Omegawhich has zero Tanaka-Webster scalar curvature and for which $\mathrm{\Sigma }$\Sigmahas a constant $p$p-mean curvature is also discussed. 

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4. Local Well-Posedness of Skew Mean Curvature Flow for Small Data in $$d\ge 4$$ Dimensions

   https://doi.org/10.1007/s00220-021-04303-8  

   Huang, Jiaxi; Tataru, Daniel (January 2022, Communications in Mathematical Physics)

   Abstract The skew mean curvature flow is an evolution equation forddimensional manifolds embedded in$${{\mathbb {R}}}^{d+2}$$ $R^{d+2}$(or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension$$d\ge 4$$ $d\ge 4$. 

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5. ADM mass for C 0 C^{0} metrics and distortion under Ricci–DeTurck flow

   https://doi.org/10.1515/crelle-2023-0085  

   Burkhardt-Guim, Paula (December 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We show that there exists a quantity, depending only on $C^0$C^{0}data of a Riemannian metric, that agrees with the usual ADM mass at infinity whenever the ADM mass exists, but has a well-defined limit at infinity for any continuous Riemannian metric that is asymptotically flat in the $C^0$C^{0}sense and has nonnegative scalar curvature in the sense of Ricci flow.Moreover, the $C^0$C^{0}mass at infinity is independent of choice of $C^0$C^{0}-asymptotically flat coordinate chart, and the $C^0$C^{0}local mass has controlled distortion under Ricci–DeTurck flow when coupled with a suitably evolving test function. 

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