Advances in Mathematics最新文献

In this paper, we introduce a notion, called generalized Reifenberg condition, under which we prove a smooth fibration theorem for collapsed manifolds with Ricci curvature bounded below, which gives a unified proof of smooth fibration theorems in many previous works (including the ones proved by Fukaya and Yamaguchi respectively). A key tool in the proof of this fibration theorem is the transformation technique for almost splitting maps, which originates from Cheeger-Naber ([16]) and Cheeger-Jiang-Naber ([14]). More precisely, we show that a transformation theorem of Cheeger-Jiang-Naber (see Proposition 7.7 in [14]) holds for possibly collapsed manifolds. Some other applications of the transformation theorems are given in this paper.

We study the semiclassical limit $\kappa \to \infty$ of the generalized quantum Langlands kernel associated to a Lie algebra $g$ and an integer level p. This vertex algebra acquires a big centre, containing the ring of functions over the space of $g$-connections. We conjecture that the fibre over the zero connection is the Feigin-Tipunin vertex algebra, whose category of representations should be equivalent to the small quantum group, and that the other fibres are precisely its twisted modules, and that the entire category of representations is related to the quantum group with a big centre. In this sense we present a generalized Kazhdan-Lusztig conjecture, involving deformations by any $g$-connection. We prove our conjectures in small cases $(g,1)$ and $({\mathrm{sl}}_2,2)$ by explicitly computing all vertex algebras and categories involved.

We study ergodic-theoretic properties of coded shift spaces. A coded shift space is defined as a closure of all bi-infinite concatenations of words from a fixed countable generating set. We derive sufficient conditions for the uniqueness of measures of maximal entropy and equilibrium states of Hölder continuous potentials based on the partition of the coded shift into its concatenation set (sequences that are concatenations of generating words) and its residual set (sequences added under the closure). In this case we provide a simple explicit description of the measure of maximal entropy. We also obtain flexibility results for the entropy on the concatenation and residual sets. Finally, we prove a local structure theorem for intrinsically ergodic coded shift spaces which shows that our results apply to a larger class of coded shift spaces compared to previous works by Climenhaga [9], Climenhaga and Thompson [10], [11], and Pavlov [25].

We construct a finite-range potential on a bidimensional full shift on a finite alphabet that exhibits a zero-temperature chaotic behavior as introduced by van Enter and Ruszel. This is the phenomenon where there exists a sequence of temperatures that converges to zero for which the whole set of equilibrium measures at these given temperatures oscillates between two sets of ground states. Brémont's work shows that the phenomenon of non-convergence does not exist for finite-range potentials in dimension one for finite alphabets; Leplaideur obtained a different proof for the same fact. Chazottes and Hochman provided the first example of non-convergence in higher dimensions $d\ge 3$; we extend their result for $d=2$ and highlight the importance of two estimates of recursive nature that are crucial for this proof: the relative complexity and the reconstruction function of an extension.

We note that a different proof of this result was found by Chazottes and Shinoda, at around the same time that this article was initially submitted and that a strong generalization has been found by Gayral, Sablik and Taati.

We study the boundary behavior of solutions to fractional Laplacian. As the first result, the isolation of the first eigenvalue of the fractional Lane-Emden equation is proved in the bounded open sets with Wiener regular boundary. Then, a generalized Hopf's lemma and a global boundary Harnack inequality are proved for the fractional Laplacian.

Recent works [22], [23], [3], [33] have shed light on the topological behavior of geodesic planes in the convex core of a geometrically finite hyperbolic 3-manifold M of infinite volume. In this paper, we focus on the remaining case of geodesic planes outside the convex core of M, giving a complete classification of their closures in M.

In particular, we show that the behavior is different depending on whether exotic roofs exist or not. Here an exotic roof is a geodesic plane contained in an end E of M, which limits on the convex core boundary ∂E, but cannot be separated from the core by a support plane of ∂E.

A necessary condition for the existence of exotic roofs is the existence of exotic rays for the bending lamination. Here an exotic ray is a geodesic ray that has a finite intersection number with a measured lamination $L$ but is not asymptotic to any leaf nor eventually disjoint from $L$. We establish that exotic rays exist if and only if $L$ is not a multicurve. The proof is constructive, and the ideas involved are important in the construction of exotic roofs.

We also show that the existence of geodesic rays satisfying a stronger condition than being exotic, phrased only in terms of the hyperbolic surface ∂E and the bending lamination, is sufficient for the existence of exotic roofs. As a result, we show that geometrically finite ends with exotic roofs exist in every genus. Moreover, in genus 1, when the end is homotopic to a punctured torus, a generic one (in the sense of Baire category) contains uncountably many exotic roofs.

The low-frequency $L^1$ assumption has been extensively applied to the large-time asymptotics of solutions to the compressible Navier-Stokes equations and incompressible Navier-Stokes equations since the classical efforts due to Kawashima, Matsumura, Nishida, Ponce, Schonbek and Wiegner. In this paper, we establish a sharp decay characterization for the compressible Navier-Stokes equations in the critical $L^p$ framework. Precisely, it is proved that the Besov space ${\dot{B}}_{2,\infty }^{{\sigma }_1}$-boundedness condition (with $\frac{d}{2}-\frac{2d}{p}\le {\sigma }_1<\frac{d}{2}-1$) of the low-frequency part of initial perturbation is not only sufficient, but also necessary to achieve those upper bounds of time-decay estimates. Furthermore, we show that the upper and lower bounds of time-decay estimates hold if and only if the low-frequency part of initial perturbation belongs to a nontrivial subset of ${\dot{B}}_{2,\infty }^{{\sigma }_1}$. To the best of our knowledge, our work is the first one addressing the inverse problem for the large-time asymptotics of compressible viscous fluids.

Given a surface Σ in $R^3$ diffeomorphic to $S^2$, Struwe [38] proved that for almost every H below the mean curvature of the smallest sphere enclosing Σ, there exists a branched immersed disk which has constant mean curvature H and boundary meeting Σ orthogonally. We reproduce this result using a different approach and improve it under additional convexity assumptions on Σ. Specifically, when Σ itself is convex and has mean curvature bounded below by $H_0$, we obtain existence for all $H\in (0,H_0)$. Instead of the heat flow in [38], we use a Sacks-Uhlenbeck type perturbation. As in previous joint work with Zhou [7], a key ingredient for extending existence across the measure zero set of H's is a Morse index upper bound.