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1. Quantitative finiteness of hyperplanes in hybrid manifolds

   Ohm, Ko; Sanchez, Anthony (June 2025, arXiv)

   We prove a quantitative finiteness theorem for the number of totally geodesic hyperplanes of non-arithmetic hyperbolic n-manifolds that arise from a gluing construction of Gromov and Piatetski-Shapiro for n ≥ 3. This extends work of LindenstraussMohammadi in dimension 3. This follows from effective density theorem for periodic orbits of SO(n −1,1) acting on quotients of SO(n,1) by a lattice for n ≥ 3. The effective density result uses a number of a ideas including Margulis functions, a restricted projection theorem, and an effective equidistribution result for measures on the horospherical subgroup that are nearly full dimensional. 

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2. Almost isotropic Kähler manifolds

   https://doi.org/10.1515/crelle-2019-0030  

   Schmidt, Benjamin; Shankar, Krishnan; Spatzier, Ralf (October 2020, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let M be a complete Riemannian manifold and suppose {p\in M} . For each unit vector {v\in T_{p}M} , the Jacobi operator , {\mathcal{J}_{v}:v^{\perp}\rightarrow v^{\perp}} is the symmetric endomorphism, {\mathcal{J}_{v}(w)=R(w,v)v} . Then p is an isotropic point if there exists a constant {\kappa_{p}\in{\mathbb{R}}} such that {\mathcal{J}_{v}=\kappa_{p}\operatorname{Id}_{v^{\perp}}} for each unit vector {v\in T_{p}M} . If all points are isotropic, then M is said to be isotropic; it is a classical result of Schur that isotropic manifolds of dimension at least 3 have constant sectional curvatures. In this paper we consider almost isotropic manifolds , i.e. manifolds having the property that for each {p\in M} , there exists a constant {\kappa_{p}\in\mathbb{R}} such that the Jacobi operators {\mathcal{J}_{v}} satisfy {\operatorname{rank}({\mathcal{J}_{v}-\kappa_{p}\operatorname{Id}_{v^{\perp}}}% )\leq 1} for each unit vector {v\in T_{p}M} . Our main theorem classifies the almost isotropic simply connected Kähler manifolds, proving that those of dimension {d=2n\geqslant 4} are either isometric to complex projective space or complex hyperbolic space or are totally geodesically foliated by leaves isometric to {{\mathbb{C}}^{n-1}} . 

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3. The Kodaira dimension of complex hyperbolic manifolds with cusps

   https://doi.org/10.1112/S0010437X1700762X  

   Bakker, Benjamin; Tsimerman, Jacob (March 2018, Compositio Mathematica)

   We prove a bound relating the volume of a curve near a cusp in a complex ball quotient $$X=\mathbb{B}/\unicode[STIX]{x1D6E4}$$ to its multiplicity at the cusp. There are a number of consequences: we show that for an $$n$$ -dimensional toroidal compactification $$\overline{X}$$ with boundary $$D$$ , $$K_{\overline{X}}+(1-\unicode[STIX]{x1D706})D$$ is ample for $$\unicode[STIX]{x1D706}\in (0,(n+1)/2\unicode[STIX]{x1D70B})$$ , and in particular that $$K_{\overline{X}}$$ is ample for $$n\geqslant 6$$ . By an independent algebraic argument, we prove that every ball quotient of dimension $$n\geqslant 4$$ is of general type, and conclude that the phenomenon famously exhibited by Hirzebruch in dimension 2 does not occur in higher dimensions. Finally, we investigate the applications to the problem of bounding the number of cusps and to the Green–Griffiths conjecture. 

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4. On the large charge sector in the critical O(N) model at large N

   https://doi.org/10.1007/JHEP09(2021)184  

   Giombi, Simone; Hyman, Jonah (September 2021, Journal of High Energy Physics)

   A bstract We study operators in the rank- j totally symmetric representation of O ( N ) in the critical O ( N ) model in arbitrary dimension d , in the limit of large N and large charge j with j/N ≡ $$ \hat{j} $$ j ̂ fixed. The scaling dimensions of the operators in this limit may be obtained by a semiclassical saddle point calculation. Using the standard Hubbard-Stratonovich description of the critical O ( N ) model at large N , we solve the relevant saddle point equation and determine the scaling dimensions as a function of d and $$ \hat{j} $$ j ̂ , finding agreement with all existing results in various limits. In 4 < d < 6, we observe that the scaling dimension of the large charge operators becomes complex above a critical value of the ratio j/N , signaling an instability of the theory in that range of d . Finally, we also derive results for the correlation functions involving two “heavy” and one or two “light” operators. In particular, we determine the form of the “heavy-heavy-light” OPE coefficients as a function of the charges and d . 

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5. On backflow associated with oceanic and continental subduction

   https://doi.org/10.1093/gji/ggab246  

   Moulas, Evangelos; Brandon, Mark T; Vaughan Hammon, Joshua D; Schmalholz, Stefan M (July 2021, Geophysical Journal International)

   SUMMARY A popular idea is that accretion of sediment at a subduction zone commonly leads to the formation of a subduction channel, which is envisioned as a narrow zone located above a subducting plate and filled with vigorously circulating accreted sediment and exotic blocks. The circulation can be viewed as a forced convection, with downward flow in the lower part of the channel due to entrainment by the subducting plate, and a ‘backflow’ in the upper part of the channel. The backflow is often cited as an explanation for the exhumation of high-pressure/low-temperature metamorphic rocks from depths of 30 to 50 km. Previous analyses of this problem have mainly focused on the restricted case where the walls bounding the flow are artificially held fixed and rigid. A key question is if this configuration can be sustained on a geologically relevant timescale. We address this question using a coupled pair of corner flows. The pro-corner accounts for accretion and deformation directly above the subducting plate, and the retro-corner corresponds to a deformable region in the overlying plate. The two corners share a medial boundary, which is fully coupled but is otherwise free to rotate and deform. Our results indicate that the maintenance of a stable circulating flow in a narrow pro-corner (<15°) requires an unusually large viscosity ratio, μretro/μpro > 103. For lower viscosity ratios, the medial boundary would rotate rearwards, converting the initially narrow pro-corner into an obtuse geometry. For a stable narrow corner, we show that the backflow within the corner is caused by downward convergence of the incoming flow and an associated downward increase in dynamic pressure, which reaches a maximum at the corner point. The total pressure is thus expected to be much greater than predicted using a lithostatic gradient, which means that estimates of depth from metamorphic pressure would have to be adjusted accordingly. In addition, we show that the velocity fields associated with a forced corner flow and a buoyancy-assisted channel flow are nearly identical. As such, structural geology studies are not sufficient to distinguish between these two processes. 

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