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1. Aspherical complex surfaces, the Singer conjecture, and Gromov–Lück inequality $$\chi \ge |\sigma |$$

   https://doi.org/10.1017/S0305004125101400  

   ALBANESE, MICHAEL; DI_CERBO, LUCA F; LOMBARDI, LUIGI (July 2025, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract We discuss the Singer conjecture and Gromov–Lück inequality$$\chi\geq |\sigma|$$for aspherical complex surfaces. We give a proof of the Singer conjecture for aspherical complex surfaces with residually finite fundamental group that does not rely on Gromov’s Kähler groups theory. Without the residually finiteness assumption, we observe that this conjecture can be proven for all aspherical complex surfaces except possibly those in Class$$\mathrm{VII}_0^+$$(a positive answer to the global spherical shell conjecture would rule out the existence of aspherical surfaces in this class). We also sharpen the Gromov-Lück inequality for aspherical complex surfaces that are not in Class$$\mathrm{VII}_0^+$$. This is achieved by connecting the circle of ideas of the Singer conjecture with the study of Reid’s conjecture. 

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2. The Riemannian and symplectic geometry of the space of generalized Kähler structures

   https://doi.org/10.4171/CMH/585  

   Apostolov, Vestislav; Streets, Jeffrey; Ustinovskiy, Yury (February 2025, Commentarii Mathematici Helvetici)

   On a compact complex manifold(M, J)endowed with a holomorphic Poisson tensor \pi_{J}and a de Rham class\alpha\in H^{2}(M, \mathbb{R}), we study the space of generalized Kähler (GK) structures defined by a symplectic formF\in \alphaand whose holomorphic Poisson tensor is\pi_{J}. We define a notion of generalized Kähler class of such structures, and use the moment map framework of Boulanger (2019) and Goto (2020) to extend the Calabi program to GK geometry. We obtain generalizations of the Futaki–Mabuchi extremal vector field (1995) and the Calabi–Lichnerowicz–Matsushima result (1982, 1958, 1957) for the Lie algebra of the group of automorphisms of(M, J, \pi_{J}). We define a closed1-form on a GK class, which yields a generalization of the Mabuchi energy and thus a variational characterization of GK structures of constant scalar curvature. Next we introduce a formal Riemannian metric on a given GK class, generalizing the fundamental construction of Mabuchi–Semmes–Donaldson (1987, 1992, 1997) We show that this metric has nonpositive sectional curvature, and that the Mabuchi energy is convex along geodesics, leading to a conditional uniqueness result for constant scalar curvature GK structures. We finally examine the toric case, proving the uniqueness of extremal generalized Kähler structures and showing that their existence is obstructed by the uniform relative K-stability of the corresponding Delzant polytope. Using the resolution of the Yau–Tian–Donaldson conjecture in the toric case by Chen–Cheng (2021) and He (2019), we show in some settings that this condition suffices for existence and thus construct new examples. 

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3. Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems

   https://doi.org/10.1051/cocv/2021033  

   Oudet, Édouard; Kao, Chiu-Yen; Osting, Braxton (January 2021, ESAIM: Control, Optimisation and Calculus of Variations)

   null (Ed.)

   Recently Fraser and Schoen showed that the solution of a certain extremal Steklov eigenvalue problem on a compact surface with boundary can be used to generate a free boundary minimal surface, i.e. , a surface contained in the ball that has (i) zero mean curvature and (ii) meets the boundary of the ball orthogonally (doi: 10.1007/s00222-015-0604-x ). In this paper, we develop numerical methods that use this connection to realize free boundary minimal surfaces. Namely, on a compact surface, Σ, with genus γ and b boundary components, we maximize σ j (Σ, g )  L ( ∂ Σ, g ) over a class of smooth metrics, g , where σ j (Σ, g ) is the j th nonzero Steklov eigenvalue and L ( ∂ Σ, g ) is the length of ∂ Σ. Our numerical method involves (i) using conformal uniformization of multiply connected domains to avoid explicit parameterization for the class of metrics, (ii) accurately solving a boundary-weighted Steklov eigenvalue problem in multi-connected domains, and (iii) developing gradient-based optimization methods for this non-smooth eigenvalue optimization problem. For genus γ = 0 and b = 2, …, 9, 12, 15, 20 boundary components, we numerically solve the extremal Steklov problem for the first eigenvalue. The corresponding eigenfunctions generate a free boundary minimal surface, which we display in striking images. For higher eigenvalues, numerical evidence suggests that the maximizers are degenerate, but we compute local maximizers for the second and third eigenvalues with b = 2 boundary components and for the third and fifth eigenvalues with b = 3 boundary components. 

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4. Substructures in Compact Disks of the Taurus Star-forming Region

   https://doi.org/10.3847/1538-4357/acd334  

   Zhang, Shangjia; Kalscheur, Matt; Long, Feng; Zhang, Ke; Long, Deryl E.; Bergin, Edwin A.; Zhu, Zhaohuan; Trapman, Leon (July 2023, The Astrophysical Journal)

   Abstract Observations of substructure in protoplanetary disks have largely been limited to the brightest and largest disks, excluding the abundant population of compact disks, which are likely sites of planet formation. Here, we reanalyze ∼0.″1, 1.33 mm Atacama Large Millimeter/submillimeter Array (ALMA) continuum observations of 12 compact protoplanetary disks in the Taurus star-forming region. By fitting visibilities directly, we identify substructures in six of the 12 compact disks. We then compare the substructures identified in the full Taurus sample of 24 disks in single-star systems and the ALMA DSHARP survey, differentiating between compact (R_eff,90%< 50 au) and extended (R_eff,90%≥50 au) disk sources. We find that substructures are detected at nearly all radii in both small and large disks. Tentatively, we find fewer wide gaps in intermediate-sized disks withR_eff,90%between 30 and 90 au. We perform a series of planet–disk interaction simulations to constrain the sensitivity of our visibility-fitting approach. Under the assumption of planet–disk interaction, we use the gap widths and common disk parameters to calculate potential planet masses within the Taurus sample. We find that the young planet occurrence rate peaks near Neptune masses, similar to the DSHARP sample. For 0.01M_J/M_⊙≲M_p/M_*≲0.1M_J/M_⊙, the rate is 17.4% ± 8.3%; for 0.1M_J/M_⊙≲M_p/M_*≲1M_J/M_⊙, it is 27.8% ± 8.3%. Both of them are consistent with microlensing surveys. For gas giants more massive than 5M_J, the occurrence rate is 4.2% ± 4.2%, consistent with direct imaging surveys. 

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5. The JWST /MIRI view of the planetary nebula NGC 6302 – I. A UV-irradiated torus and a hot bubble triggering PAH formation

   https://doi.org/10.1093/mnras/staf1194  

   Matsuura, Mikako; Volk, Kevin; Kavanagh, Patrick; Balick, Bruce; Wesson, Roger; Zijlstra, Albert_A; Dinerstein, Harriet_L; Peeters, Els; Sterling, N_C; Cami, Jan; et al (August 2025, Monthly Notices of the Royal Astronomical Society)

   ABSTRACT NGC 6302 is a spectacular bipolar planetary nebula (PN) whose spectrum exhibits fast outflows and highly ionized emission lines, indicating the presence of a very hot central star ($${\sim}$$220 000 K). Its infrared spectrum reveals a mixed oxygen and carbon dust chemistry, displaying both silicate and polycyclic aromatic hydrocarbon (PAH) features. Using the James Webb Space Telescope Mid-Infrared Instrument and Medium Resolution Spectrometer, a mosaic map was obtained over the core of NGC 6302, covering the wavelength range of 5–28 $$\mu$$m and spanning an area of $${\sim}$$18.5 arcsec $$\times$$ 15arcsec. The spatially resolved spectrum reveals $${\sim}$$200 molecular and ionized lines from species requiring ionization potentials of up to 205 eV. The spatial distributions highlight a complex structure at the nebula’s centre. Highly ionized species such as [Mg vii] and [Si vii] show compact structures, while lower ionization species such as H$^+$ extend much farther outwards, forming filament-defined rims that delineate a bubble. Within the bubble, the H$^+$ and H$$_2$$ emission coincide, while the PAH emission appears farther out, indicating an ionization structure distinct from typical photodissociation regions, such as the Orion Bar. This may be the first identification of a PAH formation site in a PN. This PN appears to be shaped not by a steady, continuous outflow, but by a series of dynamic, impulsive bubble ejections, creating local conditions conducive to PAH formation. A dusty torus surrounds the core, primarily composed of large ($$\mu$$m-sized) silicate grains with crystalline components. The long-lived torus contains a substantial mass of material, which could support an equilibrium chemistry and a slow dust-formation process. 

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