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1. Asymptotic Analysis of Harmonic Maps with Prescribed Singularities

   https://doi.org/10.1007/s40818-026-00247-1  

   Han, Qing; Khuri, Marcus; Weinstein, Gilbert; Xiong, Jingang (December 2026, Annals of PDE)

   Abstract This is the first in a series of two papers to establish the mass-angular momentum inequality for multiple black holes. We study singular harmonic maps from domains of 3-dimensional Euclidean space to the hyperbolic plane having bounded hyperbolic distance to extreme Kerr harmonic maps. We prove that every such harmonic map admits a unique tangent harmonic map at the extreme black hole horizon. The possible tangent maps are classified and shown to be shifted ‘extreme Kerr’ geodesics in the hyperbolic plane that depend on two parameters, one determined by angular momentum and another by conical singularities. In addition, rates of convergence to the tangent map are established. Similarly, expansions in the asymptotically flat end are presented. These results, together with those of Li-Tian [24, 25] and Weinstein [35, 36], provide a complete regularity theory for harmonic maps from$$\mathbb {R}^3\setminus z\text {-axis}$$ $R^3\backslash z\text{-axis}$ to$$\mathbb {H}^2$$ $H^2$ with these prescribed singularities. The analysis is additionally utilized to prove existence of the so called near horizon limit, and to compute the associated near horizon geometries of extreme black holes. 

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2. Riesz energy, $L^2$ discrepancy, and optimal transport of determinantal point processes on the sphere and the flat torus

   https://doi.org/10.1112/mtk.12245  

   Borda, Bence; Grabner, Peter; Matzke, Ryan W (April 2024, Mathematika)

   Abstract Determinantal point processes exhibit an inherent repulsive behavior, thus providing examples of very evenly distributed point sets on manifolds. In this paper, we study the so‐called harmonic ensemble, defined in terms of Laplace eigenfunctions on the sphere and the flat torus , and the so‐called spherical ensemble on , which originates in random matrix theory. We extend results of Beltrán, Marzo, and Ortega‐Cerdà on the Riesz ‐energy of the harmonic ensemble to the nonsingular regime , and as a corollary find the expected value of the spherical cap discrepancy via the Stolarsky invariance principle. We find the expected value of the discrepancy with respect to axis‐parallel boxes and Euclidean balls of the harmonic ensemble on . We also show that the spherical ensemble and the harmonic ensemble on and with points attain the optimal rate in expectation in the Wasserstein metric , in contrast to independent and identically distributed random points, which are known to lose a factor of . 

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3. Experimental simulation of loop quantum gravity on a photonic chip

   https://doi.org/10.1038/s41534-023-00702-y  

   van der Meer, Reinier; Huang, Zichang; Anguita, Malaquias Correa; Qu, Dongxue; Hooijschuur, Peter; Liu, Hongguang; Han, Muxin; Renema, Jelmer J.; Cohen, Lior (December 2023, npj Quantum Information)

   Abstract The unification of general relativity and quantum theory is one of the fascinating problems of modern physics. One leading solution is Loop Quantum Gravity (LQG). Simulating LQG may be important for providing predictions which can then be tested experimentally. However, such complex quantum simulations cannot run efficiently on classical computers, and quantum computers or simulators are needed. Here, we experimentally demonstrate quantum simulations of spinfoam amplitudes of LQG on an integrated photonics quantum processor. We simulate a basic transition of LQG and show that the derived spinfoam vertex amplitude falls within 4% error with respect to the theoretical prediction, despite experimental imperfections. We also discuss how to generalize the simulation for more complex transitions, in realistic experimental conditions, which will eventually lead to a quantum advantage demonstration as well as expand the toolbox to investigate LQG. 

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4. Gaussian curvature on random planar maps and Liouville quantum gravity

   https://doi.org/10.1007/s00440-025-01430-4  

   Hip, Andres_A Contreras; Gwynne, Ewain (April 2026, Probability Theory and Related Fields)

   Abstract We investigate the notion of curvature in the context of Liouville quantum gravity (LQG) surfaces. We define the Gaussian curvature for LQG, which we conjecture is the scaling limit of discrete curvature on random planar maps. Motivated by this, we study asymptotics for the discrete curvature of$$\epsilon $$ $ϵ$ -mated CRT maps. More precisely, we prove that the discrete curvature integrated against a$$C_c^2$$ $C_c^2$ test function is of order$$\epsilon ^{o(1)},$$ ${ϵ}^{o\left(1\right)},$ which is consistent with our scaling limit conjecture. On the other hand, we prove the total discrete curvature on a fixed space-filling SLE segment scaled by$$\epsilon ^{\frac{1}{4}}$$ ${ϵ}^{\frac{1}{4}}$ converges in distribution to an explicit random variable. 

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5. Minimal Ws,ns$$W^{s,\frac{n}{s}}$$‐harmonic maps in homotopy classes

   https://doi.org/10.1112/jlms.12769  

   Mazowiecka, Katarzyna; Schikorra, Armin (August 2023, Journal of the London Mathematical Society)

   Abstract Let a closed ‐dimensional manifold, be a closed manifold, and let for . We extend the monumental work of Sacks and Uhlenbeck by proving that if , then there exists a minimizing ‐harmonic map homotopic to . If , then we prove that there exists a ‐harmonic map from to in a generating set of . Since several techniques, especially Pohozaev‐type arguments, are unknown in the fractional framework (in particular, when , one cannot argue via an extension method), we develop crucial new tools that are interesting on their own: such as a removability result for point singularities and a balanced energy estimate for nonscaling invariant energies. Moreover, we prove the regularity theory for minimizing ‐maps into manifolds. 

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