In this paper, we prove that any Kähler Ricci shrinker surface has bounded sectional curvature. Combining this estimate with earlier work by many authors, we provide a complete classification of all Kähler Ricci shrinker surfaces.
While there may be many Thurston metric geodesics between a pair of points in Teichm\"uller space, we find that by imposing an additional energy minimization constraint on the geodesics, thought of as limits of harmonic map rays, we select a unique Thurston geodesic through those points. Extending the target surface to the Thurston boundary yields, for each point $Y$ in Teichm\"uller space, an "exponential map" of rays from that point $Y$ onto Teichm\"uller space with visual boundary the Thurston boundary of Teichm\"uller space. We first depict harmonic map ray structures on Teichm\"uller space as a geometric transition between Teichm\"uller ray structures and Thurston geodesic ray structures. In particular, by appropriately degenerating the source of a harmonic map between hyperbolic surfaces (along "harmonic map dual rays"), the harmonic map rays through the target converge to a Thurston geodesic; by appropriately degenerating the target of the harmonic map, those harmonic map dual rays through the domain converge to Teichm\"uller geodesics. We then extend this transition to one from Teichm\"uller disks through Hopf differential disks to stretch-earthquake disks. These results apply to surfaces with boundary, resolving a question on stretch maps between such surfaces.
We generalize a classical reciprocity law due to Rédei using our recently developed description of the 2-torsion of class groups of multiquadratic fields. This result is then used to prove a variety of new reflection principles for class groups, one of which involves a symbol similar to the spin symbol as defined in work of Friedlander, Iwaniec, Mazur and Rubin. We combine these reflection principles with Smith's techniques to prove Stevenhagen's conjecture on the solubility of the negative Pell equation.
Consider an area minimizing current modulo p of dimension m in a smooth Riemannian manifold of dimension m+1. We prove that its interior singular set is, up to a relatively closed set of dimension at most m-2, a C^1,α submanifold of dimension m-1 at which, locally, N≤ p regular sheets of the current join transversally, each sheet counted with a positive multiplicity k_i so that ∑_i k_i = p. This completes the analysis of the structure of the singular set of area minimizing hypersurfaces modulo p, initiated by J. Taylor for m=2 and p=3 and extended by the authors to arbitrary m and all odd p in arXiv:2105.08135. We tackle the remaining case of even p by showing that the set of singular points admitting a flat blow-up is of codimension at least two in the current. First, we prove a structural result for the singularities of minimizers in the linearized problem, by combining an epiperimetric inequality with an analysis of homogeneous minimizers to conclude that the corresponding degrees of homogeneity are always integers; second, we refine Almgren's blow-up procedure to prove that all flat singularities of the current persist as singularities of the Dir-minimizing limit. An important ingredient of our analysis is the uniqueness of flat tangent cones at singular points, recently established by Minter and Wickramasekera in arXiv:2111.11202.
