Volume 2025, Issue 826

By Perelman’s ℒ-geodesic theory, we study the blow-down solutions on a noncompact 𝜅-noncollapsed steady gradient Ricci soliton ( M n , g ) (M^{n},g) ( n ≥ 4 n\geq 4 ) with nonnegative curvature operator and positive Ricci curvature away from a compact set of 𝑀. We prove that any compact split ancient solution of codimension one from the blow-down of ( M , g ) (M,g) is of type I. The result is a generalization of our previous work from n = 4 n=4 to any dimension.

In his groundbreaking work from 2002, Perelman introduced two fundamental monotonic quantities: the reduced volume and the entropy. While the reduced volume was motivated by the Bishop–Gromov volume comparison applied to a suitably constructed 𝑁-space, which becomes Ricci-flat as N → ∞ N\rightarrow\infty , Perelman did not provide a corresponding explanation for the origin of the entropy. In this article, we demonstrate that Perelman’s entropy emerges as the limit of Colding’s monotonic volume for harmonic functions on Ricci-flat manifolds, when appropriately applied to Perelman’s 𝑁-space.

Uniqueness results for asymptotically locally flat and asymptotically flat S 1 S^{1} -symmetric gravitational instantons are proved using a divergence identity of the type used in uniqueness proofs for static black holes, combined with results derived from the 𝐺-signature theorem. Our results include a proof of the S 1 S^{1} -symmetric version of the Euclidean Black Hole Uniqueness Conjecture, a uniqueness result for the Taub-bolt family of instantons, as well as a proof that an ALF S 1 S^{1} -symmetric instanton with the topology of the Chen–Teo family of instantons is Hermitian.

For a geometrically finite Kleinian group Γ, the Bowen–Margulis–Sullivan measure is finite and is the unique measure of maximal entropy for the geodesic flow, as shown by Sullivan and Otal–Peigné respectively. Moreover, it is strongly mixing by a result of Babillot. We obtain a higher-rank analogue of this theorem. Given a relatively Anosov subgroup Γ of a semisimple real algebraic group, there is a family of flow spaces parameterized by linear forms tangent to the growth indicator. We construct a reparameterization of each flow space by the geodesic flow on the Groves–Manning space of Γ which exhibits exponential expansion along unstable foliations. Using this reparameterization, we prove that the Bowen–Margulis–Sullivan measure of each flow space is finite and is the unique measure of maximal entropy. Moreover, it is strongly mixing.

In this work, we study the supersingular locus of the Shimura variety associated to the unitary group GU ⁡ ( 2 , 4 ) \operatorname{GU}(2,4) over a ramified prime. We show that the associated Rapoport–Zink space is flat, and we give an explicit description of the irreducible components of the reduction modulo 𝑝 of the basic locus. In particular, we show that these are universally homeomorphic to either a generalized Deligne–Lusztig variety for a symplectic group or to the closure of a vector bundle over a classical Deligne–Lusztig variety for an orthogonal group. Our results are confirmed in the group-theoretical setting by the reduction method à la Deligne and Lusztig and the study of the admissible set.

We provide a generalization of Jouanolou duality that is applicable to a plethora of situations. The environment where this generalized duality takes place is a new class of rings, that we introduce and call weakly Gorenstein rings. As a consequence, we obtain a new general framework to investigate blowup algebras. We use our results to study and determine the defining equations of the Rees algebra for certain families of ideals.

Let ( Z , p ) (Z,p) be a pointed Gromov–Hausdorff limit of non-collapsing Kähler–Einstein metrics with uniformly bounded Ricci curvature. We show that the singular Kähler–Einstein metric on 𝑍 is conical at 𝑝 if and only if C = W \mathcal{C}=W in Donaldson–Sun’s two-step degeneration theory, assuming curvature grows at most quadratically near 𝑝. Let ( X , p ) (X,p) be a germ of an isolated log terminal algebraic singularity. Following Hein–Sun’s approach, we show that if C = W \mathcal{C}=W in the two-step stable degeneration of ( X , p ) (X,p) and 𝒞 has a smooth link, then every singular Kähler–Einstein metric on 𝑋 with non-positive Ricci curvature and bounded potential is conical at 𝑝.

We consider the long-time existence of the anomaly flow on a compact complex 3-fold with general slope parameter α ′ \alpha^{\prime} . In particular, we obtain integral Shi-type estimates for the flow by adapting an integration-by-parts type argument instead of the usual maximum principle techniques. Following this, we prescribe a sufficient smallness condition on α ′ \alpha^{\prime} in order to extend the flow on [ 0 , τ ) [0,\tau) to [ 0 , τ + ϵ ) [0,\tau+\epsilon) .