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1. The p-widths of a surface

   https://doi.org/10.1007/s10240-023-00141-7  

   Chodosh, Otis; Mantoulidis, Christos (June 2023, Publications mathématiques de l'IHÉS)

   Abstract The $$p$$ p -widths of a closed Riemannian manifold are a nonlinear analogue of the spectrum of its Laplace–Beltrami operator, which corresponds to areas of a certain min-max sequence of possibly singular minimal submanifolds. We show that the $$p$$ p -widths of any closed Riemannian two-manifold correspond to a union of closed immersed geodesics, rather than simply geodesic nets. We then prove optimality of the sweepouts of the round two-sphere constructed from the zero set of homogeneous polynomials, showing that the $$p$$ p -widths of the round sphere are attained by $$\lfloor \sqrt{p}\rfloor $$ ⌊ p ⌋ great circles. As a result, we find the universal constant in the Liokumovich–Marques–Neves–Weyl law for surfaces to be $$\sqrt{\pi }$$ π . En route to calculating the $$p$$ p -widths of the round two-sphere, we prove two additional new results: a bumpy metrics theorem for stationary geodesic nets with fixed edge lengths, and that, generically, stationary geodesic nets with bounded mass and bounded singular set have Lusternik–Schnirelmann category zero. 

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2. The Flex Divisor of a K3 Surface

   https://doi.org/10.1093/imrn/rnac089  

   Alexeev, Valery; Engel, Philip (May 2022, International Mathematics Research Notices)

   Abstract The flex divisor$$R_{\textrm flex}$$ of a primitively polarized K3 surface $(X,L)$ is, generically, the set of all points $$x\in X$$ for which there exists a pencil $$V\subset |L|$$ whose base locus is $$\{x\}$$. We show that if $L^2=2d$ then $$R_{\textrm flex}\in |n_dL|$$ with $$ \begin{align*} &n_d= \frac{(2d)!(2d+1)!}{d!^2(d+1)!^2} =(2d+1)C(d)^2,\end{align*}$$where $C(d)$ is the Catalan number. We also show that there is a well-defined notion of flex divisor over the whole moduli space $$F_{2d}$$ of polarized K3 surfaces. 

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3. The Morse Index of a Minimal Surface

   https://doi.org/10.1090/noti2291  

   Chodosh, Otis; Maximo, Davi (June 2021, Notices of the American Mathematical Society)
4. The degree of a tropical root surface of type A

   https://doi.org/10.2140/om.2024.1.243  

   Ardila-Mantilla, Federico; Cordero_Aguilar, Mont (January 2025, Orbita Mathematicae)

   We prove that the tropical surface of the root system A_{n−1} has degree n(n − 1)(n − 2)/2. 

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5. Shapes of a filament on the surface of a bubble

   https://doi.org/10.1098/rspa.2021.0353  

   Prasath, S. Ganga; Marthelot, Joel; Govindarajan, Rama; Menon, Narayanan (September 2021, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   The shape assumed by a slender elastic structure is a function both of the geometry of the space in which it exists and the forces it experiences. We explore, by experiments and theoretical analysis, the morphological phase space of a filament confined to the surface of a spherical bubble. The morphology is controlled by varying bending stiffness and weight of the filament, and its length relative to the bubble radius. When the dominant considerations are the geometry of confinement and elastic energy, the filament lies along a geodesic and when gravitational energy becomes significant, a bifurcation occurs, with a part of the filament occupying a longitude and the rest along a curve approximated by a latitude. Far beyond the transition, when the filament is much longer than the diameter, it coils around the selected latitudinal region. A simple model with filament shape as a composite of two arcs captures the transition well. For better quantitative agreement with the subcritical nature of bifurcation, we study the morphology by numerical energy minimization. Our analysis of the filament’s morphological space spanned by one geometric parameter, and one parameter that compares elastic energy with body forces, may provide guidance for packing slender structures on complex surfaces. 

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