Band 2025, Heft 818

We prove an equivariant main conjecture in Iwasawa theory along any rank-one, sign-normalized Drinfeld modular, split-at-∞ Iwasawa tower of a general function field of characteristic 𝑝, for the Iwasawa modules recently considered by Greither and Popescu in their proof of the classical equivariant main conjecture along the (arithmetic) cyclotomic Iwasawa tower. As a consequence, we prove an equivariant main conjecture for a projective limit of certain Ritter–Weiss type modules, along the same Drinfeld modular Iwasawa towers. This generalizes the results of Anglès, Bandini, Bars, Coscelli and Longhi for the split-at-∞ piece of the Iwasawa towers considered in their work, and refines the results by Greither and Popescu.

In his lecture notes on mean curvature flow, Ilmanen conjectured the existence of noncompact self-shrinkers with arbitrary genus. Here, we employ min-max techniques to give a rigorous existence proof for these surfaces. Conjecturally, the self-shrinkers that we obtain have precisely one (asymptotically conical) end. We confirm this for large genus via a precise analysis of the limiting object of sequences of such self-shrinkers for which the genus tends to infinity. Finally, we provide numerical evidence for a further family of noncompact self-shrinkers with odd genus and two asymptotically conical ends.

We introduce a new subclass of Fano varieties (Casagrande–Druel varieties) that are 𝑛-dimensional varieties constructed from Fano double covers of dimension n − 1 n-1 . We conjecture that a Casagrande–Druel variety is K-polystable if the double cover and its base space are K-polystable. We prove this for smoothable Casagrande–Druel threefolds, and for Casagrande–Druel varieties constructed from double covers of P n − 1 \mathbb{P}^{n-1} ramified over smooth hypersurfaces of degree 2 ⁢ d 2d with n > d > n 2 > 1 n>d>\frac{n}{2}>1 . As an application, we describe the connected components of the K-moduli space parametrizing smoothable K-polystable Fano threefolds in the families № 3.9 and № 4.2 in the Mori–Mukai classification.

The main goal of this article is to describe a relation between the asymptotic properties of filtrations on section rings and the geometry at infinity of the space of Kähler potentials. More precisely, for a polarized projective manifold and an ample test configuration, Phong and Sturm associated a geodesic ray of plurisubharmonic metrics on the polarizing line bundle. On the other hand, for the same data, Witt Nyström associated a filtration on the section ring of the polarized manifold. In this article, we establish a folklore conjecture that the pluripotential chordal distance between the geodesic rays associated with two ample test configurations coincides with the spectral distance between the associated filtrations on the section ring. This gives an algebraic description of the boundary at infinity of the space of positive metrics, viewed – as it is usually done for spaces of negative curvature – through geodesic rays.

We solve the equivariant generalized Nash problem for any non-rational normal variety with torus action of complexity one. Namely, we give an explicit combinatorial description of the Nash order on the set of equivariant divisorial valuations on any such variety. Using this description, we positively solve the classical Nash problem in this setting, showing that every essential valuation is a Nash valuation. We also describe terminal valuations and use our results to answer negatively a question of de Fernex and Docampo by constructing examples of Nash valuations which are neither minimal nor terminal, thus illustrating a striking new feature of the class of singularities under consideration.

We develop a min-max theory for the construction of capillary surfaces in 3-manifolds with smooth boundary. In particular, for a generic set of ambient metrics, we prove the existence of nontrivial, smooth, almost properly embedded surfaces with any given constant mean curvature 𝑐, and with smooth boundary contacting at any given constant angle 𝜃. Moreover, if 𝑐 is nonzero and 𝜃 is not π 2 \frac{\pi}{2} , then our min-max solution always has multiplicity one. We also establish a stable Bernstein theorem for minimal hypersurfaces with certain contact angles in higher dimensions.

We prove that the knot Floer complex of a fibered knot detects whether the monodromy of its fibration is right-veering. In particular, this leads to a purely knot Floer-theoretic characterization of tight contact structures, by the work of Honda–Kazez–Matić. Our proof makes use of the relationship between the Heegaard Floer homology of mapping tori and the symplectic Floer homology of area-preserving surface diffeomorphisms. We describe applications of this work to Dehn surgeries and taut foliations.

We prove a sharp upper bound for the fourth moment of the Hurwitz zeta function ζ ⁢ ( s , α ) \zeta(s,\alpha) on the critical line when the shift parameter 𝛼 is irrational and of irrationality exponent strictly less than 3. As a consequence, we determine the order of magnitude of the 2 ⁢ k 2k th moment for all 0 ⩽ k ⩽ 2 0\leqslant k\leqslant 2 in this case. In contrast to the Riemann zeta function and other 𝐿-functions from arithmetic, these grow like T ⁢ ( log ⁡ T ) k T(\log T)^{k} . This suggests, and we conjecture, that the value distribution of ζ ⁢ ( s , α ) \zeta(s,\alpha) on the critical line is Gaussian.