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1. Quadrilateral mesh generation II: Meromorphic quartic differentials and Abel–Jacobi condition

   https://doi.org/112980  

   Lei, Na; Zheng, Xiaopeng; Luo, Zhongxuan; Luo, Feng; Gu, Xianfeng (July 2020, Computer methods in applied mechanics and engineering)

   null (Ed.)

   This work discovers the equivalence relation between quadrilateral meshes and meromorphic quartic differentials. Each quad-mesh induces a conformal structure of the surface, and a meromorphic quartic differential, where the configuration of singular vertices corresponds to the configurations of the poles and zeros (divisor) of the meromorphic differential. Due to Riemann surface theory, the configuration of singularities of a quad-mesh satisfies the Abel–Jacobi condition. Inversely, if a divisor satisfies the Abel–Jacobi condition, then there exists a meromorphic quartic differential whose divisor equals the given one. Furthermore, if the meromorphic quartic differential is with finite trajectories, then it also induces a quad-mesh, the poles and zeros of the meromorphic differential correspond to the singular vertices of the quad-mesh. Besides the theoretic proofs, the computational algorithm for verification of Abel–Jacobi condition is also explained in detail. Furthermore, constructive algorithm of meromorphic quartic differential on genus zero surfaces is proposed, which is based on the global algebraic representation of meromorphic differentials. Our experimental results demonstrate the efficiency and efficacy of the algorithm. This opens up a novel direction for quad-mesh generation using algebraic geometric approach. 

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2. Spherical conical metrics and harmonic maps to spheres

   https://doi.org/10.1090/tran/8578  

   Karpukhin, Mikhail; Zhu, Xuwen (February 2022, Transactions of the American Mathematical Society)

   A spherical conical metric g g on a surface Σ \Sigma is a metric of constant curvature 1 1 with finitely many isolated conical singularities. The uniformization problem for such metrics remains largely open when at least one of the cone angles exceeds 2 π 2\pi . The eigenfunctions of the Friedrichs Laplacian Δ g \Delta _g with eigenvalue λ = 2 \lambda =2 play a special role in this problem, as they represent local obstructions to deformations of the metric g g in the class of spherical conical metrics. In the present paper we apply the theory of multivalued harmonic maps to spheres to the question of existence of such eigenfunctions. In the first part we establish a new criterion for the existence of 2 2 -eigenfunctions, given in terms of a certain meromorphic data on Σ \Sigma . As an application we give a description of all 2 2 -eigenfunctions for metrics on the sphere with at most three conical singularities. The second part is an algebraic construction of metrics with large number of 2 2 -eigenfunctions via the deformation of multivalued harmonic maps. We provide new explicit examples of metrics with many 2 2 -eigenfunctions via both approaches, and describe the general algorithm to find metrics with arbitrarily large number of 2 2 -eigenfunctions. 

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3. Schauder estimates for equations with cone metrics, II

   https://doi.org/10.2140/apde.2024.17.757  

   Guo, Bin; Song, Jian (January 2024, Analysis & PDE)

   We continue our work on the linear theory for equations with conical singularities. We derive interior Schauder estimates for linear elliptic and parabolic equations with a background Kähler metric of conical singularities along a divisor of simple normal crossings. As an application, we prove the short-time existence of the conical Kähler–Ricci flow with conical singularities along a divisor with simple normal crossings. 

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4. Classification of asymptotically conical Calabi–Yau manifolds

   https://doi.org/10.1215/00127094-2023-0030  

   Conlon, Ronan J.; Hein, Hans-Joachim (April 2024, Duke Mathematical Journal)

   A Riemannian cone (C,gC) is by definition a warped product C=R+×L with metric gC=dr2⊕r2gL, where (L,gL) is a compact Riemannian manifold without boundary. We say that C is a Calabi-Yau cone if gC is a Ricci-flat Kähler metric and if C admits a gC-parallel holomorphic volume form; this is equivalent to the cross-section (L,gL) being a Sasaki-Einstein manifold. In this paper, we give a complete classification of all smooth complete Calabi-Yau manifolds asymptotic to some given Calabi-Yau cone at a polynomial rate at infinity. As a special case, this includes a proof of Kronheimer's classification of ALE hyper-Kähler 4-manifolds without twistor theory. 

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5. Flat line bundles and the Cappell-Miller torsion in Arakelov geometry

   https://doi.org/10.24033/asens.2409  

   Freixas i Montplet, Gerard; Wentworth, Richard A. (January 2019, Annales scientifiques de lÉcole normale supérieure)

   null (Ed.)

   In this paper, we extend Deligne’s functorial Riemann-Roch isomorphism for Hermitian holomorphic line bundles on Riemann surfaces to the case of flat, not necessarily unitary connections. The Quillen metric and ?-product of Gillet-Soulé are replaced with complex valued logarithms. On the determinant of cohomology side, we show that the Cappell-Miller torsion is the appropriate counterpart of the Quillen metric. On the Deligne pairing side, the logarithm is a refinement of the intersection connections considered in a previous work. The construction naturally leads to an Arakelov theory for flat line bundles on arithmetic surfaces and produces arithmetic intersection numbers valued in C/πi Z. In this context we prove an arithmetic Riemann-Roch theorem. This realizes a program proposed by Cappell-Miller to show that their holomorphic torsion exhibits properties similar to those of the Quillen metric proved by Bismut, Gillet and Soulé. Finally, we give examples that clarify the kind of invariants that the formalism captures; namely, periods of differential forms. 

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