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1. Existence of hypersurfaces with prescribed mean curvature I – generic min-max

   Zhou, Xin; Zhu, Jonathan J. (April 2020, Cambridge journal of mathematics)

   We prove that, for a generic set of smooth prescription functions h on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature h. The solution is either an embedded minimal hypersurface with integer multiplicity, or a non-minimal almost embedded hypersurface of multiplicity one. More precisely, we show that our previous min-max theory, developed for constant mean curvature hypersurfaces, can be extended to construct min-max prescribed mean curvature hypersurfaces for certain classes of prescription function, including a generic set of smooth functions, and all nonzero analytic functions. In particular we do not need to assume that h has a sign. 

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2. Ranks of matrix factorizations and sheaf cohomology

   https://doi.org/10.1090/proc/17257  

   Brown, Michael; Walker, Mark (August 2025, Proceedings of the American Mathematical Society)

   Buchweitz-Greuel-Schreyer conjectured in 1987 a lower bound on the ranks of matrix factorizations over certain local hypersurface rings [Invent. Math. 88 (1987), pp. 165–182]. We study a graded version of this conjecture, and we show that it implies a novel conjecture concerning the cohomology of sheaves over non-Fano projective hypersurfaces. 

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3. Shafarevich’s Conjecture for Families of Hypersurfaces over Function Fields

   https://doi.org/10.1093/imrn/rnaf225  

   Engel, Philip; Lin, Alice; Tayou, Salim (August 2025, International Mathematics Research Notices)

   Given a smooth quasi-projective complex algebraic variety $$\mathcal{S}$$, we prove that there are only finitely many Hodge-generic non-isotrivial families of smooth projective hypersurfaces over $$\mathcal{S}$$ of degree $$d$$ in $$\mathbb{P}_{\mathbb C}^{n+1}$$. We prove that the finiteness is uniform in $$\mathcal{S}$$ and give examples where the result is sharp. We also prove similar results for certain complete intersections in $$\mathbb{P}_{\mathbb C}^{n+1}$$ of higher codimension and more generally for algebraic varieties whose moduli space admits a period map that satisfies the infinitesimal Torelli theorem. 

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4. Symmetry of hypersurfaces and the Hopf Lemma

   https://doi.org/10.3934/mine.2023084  

   Li, YanYan (January 2023, Mathematics in Engineering)

   A classical theorem of A. D. Alexandrov says that a connected compact smooth hypersurface in Euclidean space with constant mean curvature must be a sphere. We give exposition to some results on symmetry properties of hypersurfaces with ordered mean curvature and associated variations of the Hopf Lemma. Some open problems will be discussed. 

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5. New asymptotically flat static vacuum metrics with near Euclidean boundary data

   https://doi.org/10.1063/5.0089527  

   An, Zhongshan; Huang, Lan-Hsuan (May 2022, Journal of Mathematical Physics)

   In our prior work toward Bartnik’s static vacuum extension conjecture for near Euclidean boundary data, we establish a sufficient condition, called static regular, and confirm that large classes of boundary hypersurfaces are static regular. In this paper, we further improve some of those prior results. Specifically, we show that any hypersurface in an open and dense subfamily of a certain general smooth one-sided family of hypersurfaces (not necessarily a foliation) is static regular. The proof uses some of our new arguments motivated from studying the conjecture for boundary data near an arbitrary static vacuum metric. 

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