Volume 2018, Issue 739

We give the first rigorous construction of complete, embedded self-shrinking hypersurfaces under mean curvature flow, since Angenent’s torus in 1989. The surfaces exist for any sufficiently large prescribed genus g , and are non-compact with one end. Each has 4⁢g+4{4g+4} symmetries and comes from desingularizing the intersection of the plane and sphere through a great circle, a configuration with very high symmetry. Each is at infinity asymptotic to the cone in ℝ3{\mathbb{R}^{3}} over a 2⁢π/(g+1){2\pi/(g+1)}-periodic graph on an equator of the unit sphere 𝕊2⊆ℝ3{\mathbb{S}^{2}\subseteq\mathbb{R}^{3}}, with the shape of a periodically “wobbling sheet”. This is a dramatic instability phenomenon, with changes of asymptotics that break much more symmetry than seen in minimal surface constructions. The core of the proof is a detailed understanding of the linearized problem in a setting with severely unbounded geometry, leading to special PDEs of Ornstein–Uhlenbeck type with fast growth on coefficients of the gradient terms. This involves identifying new, adequate weighted Hölder spaces of asymptotically conical functions in which the operators invert, via a Liouville-type result with precise asymptotics.

We prove the Hasse principle and weak approximation for varieties defined over number fields by the nonsingular intersection of pairs of quadratic forms in eight variables. The argument develops work of Colliot-Thélène, Sansuc and Swinnerton-Dyer, and centres on a purely local problem about forms which split off three hyperbolic planes.

We construct and describe a family of groupoids over complex curves which serve as the universal domains of definition for solutions to linear ordinary differential equations with singularities. As a consequence, we obtain a direct, functorial method for resumming formal solutions to such equations.

Ardila and Block used tropical results of Brugallé and Mikhalkin to count nodal curves on a certain family of toric surfaces. Building on a linearity result of the first author, we revisit their work in the context of the Göttsche–Yau–Zaslow formula for counting nodal curves on arbitrary smooth surfaces, addressing several questions they raised by proving stronger versions of their main theorems. In the process, we give new combinatorial formulas for the coefficients arising in the Göttsche–Yau–Zaslow formulas, and give correction terms arising from rational double points in the relevant family of toric surfaces.

The present paper studies the group homology of the groups SL2⁡(k⁢[C]){\operatorname{SL}_{2}(k[C])} and PGL2⁡(k⁢[C]){\operatorname{PGL}_{2}(k[C])}, where C=C¯∖{P1,…,Ps}{C=\overline{C}\setminus\{P_{1},\dots,P_{s}\}} is a smooth affine curve over an algebraically closed field k . It is well known that these groups act on a product of trees and the quotients can be described in terms of certain equivalence classes of rank two vector bundles on the curve C¯{\overline{C}}. There is a natural subcomplex consisting of cells with suitably non-trivial isotropy group. The paper provides explicit formulas for the equivariant homology of this “parabolic subcomplex”. These formulas also describe group homology of SL2⁡(k⁢[C]){\operatorname{SL}_{2}(k[C])} above degree s , generalizing a result of Suslin in the case s=1{s=1}.

We introduce an invariant, called mean rank, for any module ℳ{\mathcal{M}} of the integral group ring of a discrete amenable group Γ, as an analogue of the rank of an abelian group. It is shown that the mean dimension of the induced Γ-action on the Pontryagin dual of ℳ{\mathcal{M}}, the mean rank of ℳ{\mathcal{M}}, and the von Neumann–Lück rank of ℳ{\mathcal{M}} all coincide. As applications, we establish an addition formula for mean dimension of algebraic actions, prove the analogue of the Pontryagin–Schnirelmann theorem for algebraic actions, and show that for elementary amenable groups with an upper bound on the orders of finite subgroups, algebraic actions with zero mean dimension are inverse limits of finite entropy actions.

We show that the pluriclosed flow preserves generalized Kähler structures with the extra condition [J+,J-]=0[J_{+},J_{-}]=0, a condition referred to as “split tangent bundle.” Moreover, we show that in this case the flow reduces to a nonconvex fully nonlinear parabolic flow of a scalar potential function. We prove a number of a priori estimates for this equation, including a general estimate in dimension n=2n=2 of Evans–Krylov type requiring a new argument due to the nonconvexity of the equation. The main result is a long-time existence theorem for the flow in dimension n=2n=2, covering most cases. We also show that the pluriclosed flow represents the parabolic analogue to an elliptic problem which is a very natural generalization of the Calabi conjecture to the setting of generalized Kähler geometry with split tangent bundle.

We prove that if F is a tangent to the identity diffeomorphism at 0∈ℂ20\in\mathbb{C}^{2} and Γ is a formal invariant curve of F which is not contained in the set of fixed points then there exists a parabolic curve (attracting or repelling) of F asymptotic to Γ.

We prove that the genus g , the relative irregularity qfq_{f} and the Clifford index cfc_{f} of a non-isotrivial fibration f satisfy the inequality qf≤g-cfq_{f}\leq g-c_{f}. This gives in particular a proof of Xiao’s conjecture for fibrations whose general fibres have maximal Clifford index.