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$p$-adic shtukas and the theory of global and local Shimura varieties Camb. J. Math. (IF 1.6) Pub Date : 2024-01-30 Georgios Pappas, Michael Rapoport
We establish basic results on p-adic shtukas and apply them to the theory of local and global Shimura varieties, and on their interrelation. We construct canonical integral models for (local, and global) Shimura varieties of Hodge type with parahoric level structure.
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Scattering rigidity for analytic metrics Camb. J. Math. (IF 1.6) Pub Date : 2024-01-30 Yannick Guedes-Bonthonneau, Colin Guillarmou, Malo Jézéquel
For analytic negatively curved Riemannian manifolds with analytic strictly convex boundary, we show that the scattering map for the geodesic flow determines the manifold up to isometry. In particular, one recovers both the topology and the metric. More generally our result holds in the analytic category under the no conjugate point and hyperbolic trapped set assumptions.
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Metric SYZ conjecture for certain toric Fano hypersurfaces Camb. J. Math. (IF 1.6) Pub Date : 2024-01-30 Yang Li
We prove the metric version of the SYZ conjecture for a class of Calabi–Yau hypersurfaces inside toric Fano manifolds, by solving a variational problem whose minimizer may be interpreted as a global solution of the real Monge–Ampère equation on certain polytopes. This does not rely on discrete symmetry.
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Existence of flips for generalized $\operatorname{lc}$ pairs Camb. J. Math. (IF 1.6) Pub Date : 2023-09-29 Christopher D. Hacon, Jihao Liu
$\def\lc{\operatorname{lc}} \def\Q{\mathbb{Q}}$We prove the existence of flips for $\Q$-factorial NQC generalized lc pairs, and the cone and contraction theorems for NQC generalized $\lc$ pairs. This answers a conjecture of Han–Li–Birkar. As an immediate application, we show that we can run the minimal model program for $\Q$-factorial NQC generalized $\lc$ pairs. In particular, we complete the minimal
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Stratification in tensor triangular geometry with applications to spectral Mackey functors Camb. J. Math. (IF 1.6) Pub Date : 2023-09-29 Tobias Barthel, Drew Heard, Beren Sanders
We systematically develop a theory of stratification in the context of tensor triangular geometry and apply it to classify the localizing tensor-ideals of certain categories of spectral $G$-Mackey functors for all finite groups $G$. Our theory of stratification is based on the approach of Stevenson which uses the Balmer–Favi notion of big support for tensor-triangulated categories whose Balmer spectrum
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Length orthospectrum of convex bodies on flat tori Camb. J. Math. (IF 1.6) Pub Date : 2023-09-29 Nguyen Viet Dang, Matthieu Léautaud, Gabriel Rivière
In analogy with the study of Pollicott–Ruelle resonances on negatively curved manifolds, we define anisotropic Sobolev spaces that are well-adapted to the analysis of the geodesic vector field associated with any translation invariant Finsler metric on the torus $\mathbb{T}^d$. Among several applications of this functional point of view, we study properties of geodesics that are orthogonal to two convex
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Classification of noncollapsed translators in $\mathbb{R}^4$ Camb. J. Math. (IF 1.6) Pub Date : 2023-08-08 Kyeongsu Choi, Robert Haslhofer, or Hershkovits
In this paper, we classify all noncollapsed singularity models for the mean curvature flow of $3$-dimensional hypersurfaces in $\mathbb{R}^4$ or more generally in $4$-manifolds. Specifically, we prove that every noncollapsed translating hypersurface in $\mathbb{R}^4$ is either $\mathbb{R} \times 2\textrm{d}$-bowl, or a $3\textrm{d}$ round bowl, or belongs to the one-parameter family of $3\textrm{d}$
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Canonical currents and heights for K3 surfaces Camb. J. Math. (IF 1.6) Pub Date : 2023-08-08 Simion Filip, Valentino Tosatti
We construct canonical positive currents and heights on the boundary of the ample cone of a K3 surface. These are equivariant for the automorphism group and fit together into a continuous family, defined over an enlarged boundary of the ample cone. Along the way, we construct preferred representatives for certain height functions and currents on elliptically fibered surfaces.
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Generalizing the Linearized Doubling approach, I: General theory and new minimal surfaces and self-shrinkers Camb. J. Math. (IF 1.6) Pub Date : 2023-06-06 Nikolaos Kapouleas, Peter McGrath
In Part I of this article we generalize the Linearized Doubling (LD) approach, introduced in earlier work by NK, by proving a general theorem stating that if $\Sigma$ is a closed minimal surface embedded in a Riemannian three-manifold $(N,g)$ and its Jacobi operator has trivial kernel, then given a suitable family of LD solutions on $\Sigma$, a minimal surface $\breve{M}$ resembling two copies of $\Sigma$
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On loop Deligne–Lusztig varieties of Coxeter-type for inner forms of $\mathrm{GL}_n$ Camb. J. Math. (IF 1.6) Pub Date : 2023-06-06 Charlotte Chan, Alexander B. Ivanov
For a reductive group $G$ over a local non-archimedean field $K$ one can mimic the construction from classical Deligne–Lusztig theory by using the loop space functor. We study this construction in the special case that $G$ is an inner form of $\mathrm{GL}_n$ and the loop Deligne–Lusztig variety is of Coxeter type. After simplifying the proof of its representability, our main result is that its $\ell$-adic
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Prismatic $F$-crystals and crystalline Galois representations Camb. J. Math. (IF 1.6) Pub Date : 2023-06-06 Bhargav Bhatt, Peter Scholze
Let $K$ be a complete discretely valued field of mixed characteristic $(0, p)$ with perfect residue field. We prove that the category of prismatic $F$-crystals on $\mathcal{O}_K$ is equivalent to the category of lattices in crystalline $G_K$-representations.
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Modularity of $\mathrm{GL}_2 (\mathbb{F}_p)$-representations over CM fields Camb. J. Math. (IF 1.6) Pub Date : 2023-06-05 Patrick B. Allen, Chandrashekhar Khare, Jack A. Thorne
We prove that many representations $\overline{\rho} : \mathrm {Gal}(\overline{K}/K) \to \mathrm {GL}_2 (\mathbb{F}_3)$, where $K$ is a CM field, arise from modular elliptic curves. We prove similar results when the prime $p=3$ is replaced by $p=2$ or $p=5$. As a consequence, we prove that a positive proportion of elliptic curves over any CM field not containing a 5th root of unity are modular.
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Crossed modular categories and the Verlinde formula for twisted conformal blocks Camb. J. Math. (IF 1.6) Pub Date : 2023-06-05 Tanmay Deshpande, Swarnava Mukhopadhyay
In this paper we give a Verlinde formula for computing the ranks of the bundles of twisted conformal blocks associated with a simple Lie algebra equipped with an action of a finite group $\Gamma$ and a positive integral level $\ell$ under the assumption “$\Gamma$ preserves a Borel”. For $\Gamma = \mathbb{Z} / 2$ and double covers of $\mathbb{P}^1$, this formula was conjectured by Birke–Fuchs–Schweigert
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Algebra structure of multiple zeta values in positive characteristic Camb. J. Math. (IF 1.6) Pub Date : 2022-10-21 Chieh-Yu Chang, Yen-Tsung Chen, Yoshinori Mishiba
This paper is a culmination of [CM21] on the study of multiple zeta values (MZV’s) over function fields in positive characteristic. For any finite place $v$ of the rational function field $k$ over a finite field, we prove that the $v$‑adic MZV’s satisfy the same $\overline{k}$‑algebraic relations that their corresponding $\infty$‑adic MZV’s satisfy. Equivalently, we show that the $v$‑adic MZV’s form
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Topological uniqueness for self-expanders of small entropy Camb. J. Math. (IF 1.6) Pub Date : 2022-10-21 Jacob Bernstein, Lu Wang
For a fixed regular cone in Euclidean space with small entropy we show that all smooth self-expanding solutions of the mean curvature flow that are asymptotic to the cone are in the same isotopy class.
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Free boundary minimal surfaces with connected boundary and arbitrary genus Camb. J. Math. (IF 1.6) Pub Date : 2022-10-21 Alessandro Carlotto, Giada Franz, Mario B. Schulz
We employ min-max techniques to show that the unit ball in $\mathbb{R}^3$ contains embedded free boundary minimal surfaces with connected boundary and arbitrary genus.
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Generalizations of the Eierlegende–Wollmilchsau Camb. J. Math. (IF 1.6) Pub Date : 2022-10-21 Paul Apisa, Alex Wright
We classify a natural collection of $GL(2,\mathbb{R})$-invariant subvarieties, which includes loci of double covers, the orbits of the Eierlegende–Wollmilchsau, Ornithorynque, and Matheus–Yoccoz surfaces, and loci appearing naturally in the study of the complex geometry of Teichmüller space. This classification is the key input in subsequent work of the authors that classifies “high rank” invariant
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Split Milnor–Witt motives and its applications to fiber bundles Camb. J. Math. (IF 1.6) Pub Date : 2022-10-21 Nanjun Yang
We study the Milnor–Witt motives which are a finite direct sum of $\mathbb{Z}(q)[p]$ and $\mathbb{Z}/\eta(q)[p]$. We show that for MW‑motives of this type, we can determine an MW‑motivic cohomology class in terms of a motivic cohomology class and a Witt cohomology class. We define the motivic Bockstein cohomology and show that it corresponds to subgroups of Witt cohomology, if the MW‑motive splits
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Cohomologie des courbes analytiques $p$-adiques Camb. J. Math. (IF 1.6) Pub Date : 2022-07-22 Pierre Colmez, Gabriel Dospinescu, Wiesława Nizioł
The cohomology of affinoids does not behave well; often, this is remedied by making affinoids overconvergent. In this paper, we focus on dimension $1$ and compute, using analogs of pants decompositions of Riemann surfaces, various cohomologies of affinoids. To give a meaning to these decompositions we modify slightly the notion of $p$-adic formal scheme, which gives rise to the adoc (an interpolation
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Horizontal Delaunay surfaces with constant mean curvature in $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$ Camb. J. Math. (IF 1.6) Pub Date : 2022-07-22 José M. Manzano, Francisco Torralbo
We obtain a $1$-parameter family of horizontal Delaunay surfaces with positive constant mean curvature in $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$, being the mean curvature larger than $1$ $2$ in the latter case. These surfaces are not equivariant but singly periodic, and they lie at bounded distance from a horizontal geodesic. We study in detail the geometry of the whole
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A Paley–Wiener theorem for Harish–Chandra modules Camb. J. Math. (IF 1.6) Pub Date : 2022-07-22 Heiko Gimperlein, Bernhard Krötz, Job Kuit, Henrik Schlichtkrull
We formulate and prove a Paley–Wiener theorem for Harish–Chandra modules for a real reductive group. As a corollary we obtain a new and elementary proof of the Helgason conjecture.
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On the $\operatorname{mod}p$ cohomology for $\mathrm{GL}_2$: the non-semisimple case Camb. J. Math. (IF 1.6) Pub Date : 2022-06-23 Yongquan Hu, Haoran Wang
Let $F$ be a totally real field unramified at all places above $p$ and $D$ be a quaternion algebra which splits at either none, or exactly one, of the infinite places. Let $\overline{r} : \mathrm{Gal}(\overline{F} / F) \to \mathrm{GL}_2 (\overline{\mathbb{F}}_p)$ be a continuous irreducible representation which, when restricted to a fixed place $v \vert p$, is non-semisimple and sufficiently generic
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Irreducible components of affine Deligne–Lusztig varieties Camb. J. Math. (IF 1.6) Pub Date : 2022-06-23 Sian Nie
We determine the top-dimensional irreducible components (and their stabilizers in the Frobenius twisted centralizer group) of affine Deligne–Lusztig varieties in the affine Grassmannian of a reductive group, by constructing a natural map from the set of irreducible components to the set of Mirković–Vilonen cycles. This in particular verifies a conjecture by Miaofen Chen and Xinwen Zhu.
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Existence of static vacuum extensions with prescribed Bartnik boundary data Camb. J. Math. (IF 1.6) Pub Date : 2022-04-21 Zhongshan An, Lan-Hsuan Huang
We prove the existence and local uniqueness of asymptotically flat, static vacuum metrics with arbitrarily prescribed Bartnik boundary data that are close to the induced boundary data on any star-shaped hypersurface or a general family of perturbed hypersurfaces in the Euclidean space. It confirms the existence part of the Bartnik static extension conjecture for large classes of boundary data, and
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Trialities of $\mathcal{W}$-algebras Camb. J. Math. (IF 1.6) Pub Date : 2022-04-21 Thomas Creutzig, Andrew R. Linshaw
We prove the conjecture of Gaiotto and Rapčák that the $Y$‑algebras $Y_{L,M,N} [\psi]$ with one of the parameters $L,M,N$ zero, are simple one-parameter quotients of the universal two-parameter $\mathcal{W}_{1+\infty}$‑algebra, and satisfy a symmetry known as triality. These $Y$‑algebras are defined as the cosets of certain non-principal $\mathcal{W}$‑algebras and $\mathcal{W}$‑superalgebras by their
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On the operator norm of non-commutative polynomials in deterministic matrices and iid GUE matrices Camb. J. Math. (IF 1.6) Pub Date : 2022-04-21 Benoît Collins, Alice Guionnet, Félix Parraud
Let $X^N = (X^N_1 , \dotsc , X^N_d)$ be a d‑tuple of $N \times N$ independent GUE random matrices and $Z^{NM}$ be any family of deterministic matrices in $\mathbb{M}_N \mathbb{C}) \otimes \mathbb{M}_M (\mathbb{C})$. Let $P$ be a self-adjoint non-commutative polynomial. A seminal work of Voiculescu shows that the empirical measure of the eigenvalues of $P(X^N)$ converges towards a deterministic measure
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Equivariant Grothendieck–Riemann–Roch theorem via formal deformation theory Camb. J. Math. (IF 1.6) Pub Date : 2022-03-22 Grigory Kondyrev, Artem Prikhodko
We use the formalism of traces in higher categories to prove a common generalization of the holomorphic Atiyah–Bott fixed point formula and the Grothendieck–Riemann–Roch theorem. The proof is quite different from the original one proposed by Grothendieck et al: it relies on the interplay between self dualities of quasi- and ind-coherent sheaves on $X$ and formal deformation theory of Gaitsgory–Rozenblyum
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Higher order obstructions to the desingularization of Einstein metrics Camb. J. Math. (IF 1.6) Pub Date : 2022-03-22 Tristan Ozuch
We find new obstructions to the desingularization of compact Einstein orbifolds by smooth Einstein metrics. These new obstructions, specific to the compact situation, raise the question of whether a compact Einstein $4$-orbifold which is limit of Einstein metrics bubbling out Eguchi–Hanson metrics has to be Kähler. We then test these obstructions to discuss if it is possible to produce a Ricci-flat
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Metrics with Positive constant curvature and modular differential equations Camb. J. Math. (IF 1.6) Pub Date : 2022-03-22 Jia-Wei Guo, Chang-Shou Lin, Yifan Yang
Let $\mathbb{H}^\ast = \mathbb{H} \cup \mathbb{Q} \cup \{ \infty \}$, where $\mathbb{H}$ is the complex upper half-plane, and $Q(z)$ be a meromorphic modular form of weight $4$ on $\mathrm{SL}(2, \mathbb{Z})$ such that the differential equation $\mathcal{L}:y''(z) = Q(z)y(z)$ is Fuchsian on $\mathbb{H}^\ast$. In this paper, we consider the problem when $\mathcal{L}$ is apparent on $\mathbb{H}$, i.e
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On the stability of self-similar blow-up for $C^{1,\alpha}$ solutions to the incompressible Euler equations on $\mathbb{R}^3$ Camb. J. Math. (IF 1.6) Pub Date : 2022-03-22 Tarek M. Elgindi, Tej-Eddine Ghoul, Nader Masmoudi
We study the stability of recently constructed self-similar blowup solutions to the incompressible Euler equation. A consequence of our work is the existence of finite-energy $C^{1,\alpha}$ solutions that become singular in finite time in a locally self-similar manner. As a corollary, we also observe that the Beale–Kato–Majda criterion cannot be improved in the class of $C^{1,\alpha}$ solutions.
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On the analogy between real reductive groups and Cartan motion groups: the Mackey–Higson bijection Camb. J. Math. (IF 1.6) Pub Date : 2021-12-07 Alexandre Afgoustidis
George Mackey suggested in 1975 that there should be analogies between the irreducible unitary representations of a noncompact reductive Lie group $G$ and those of its Cartan motion group $G_0$ — the semidirect product of a maximal compact subgroup of $G$ and a vector space. He conjectured the existence of a natural one-to-one correspondence between “most” irreducible (tempered) representations of
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Hypergeometric sheaves and finite symplectic and unitary groups Camb. J. Math. (IF 1.6) Pub Date : 2021-12-07 Nicholas M. Katz, Pham Huu Tiep
We construct hypergeometric sheaves whose geometric monodromy groups are the finite symplectic groups $\mathrm{Sp}_{2n} (q)$ for any odd $n \geq 3$, for $q$ any power of an odd prime $p$. We construct other hypergeometric sheaves whose geometric monodromy groups are the finite unitary groups $\mathrm{GU}_n (q)$, for any even $n \geq 2$, for $q$ any power of any prime $p$. Suitable Kummer pullbacks
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Geometric flows for the Type IIA string Camb. J. Math. (IF 1.6) Pub Date : 2021-12-07 Teng Fei, Duong H. Phong, Sebastien Picard, Xiangwen Zhang
A geometric flow on $6$-dimensional symplectic manifolds is introduced which is motivated by supersymmetric compactifications of the Type IIA string. The underlying structure turns out to be $\mathrm{SU}(3)$ holonomy, but with respect to the projected Levi–Civita connection of an almost-Hermitian structure. The short-time existence is established, and new identities for the Nijenhuis tensor are found
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Well-posedness of free boundary hard phase fluids in Minkowski background and their Newtonian limit Camb. J. Math. (IF 1.6) Pub Date : 2021-10-07 Shuang Miao, Sohrab Shahshahani, Sijue Wu
The hard phase model describes a relativistic barotropic irrotational fluid with sound speed equal to the speed of light. In this paper, we prove the local well-posedness for this model in the Minkowski background with free boundary. Moreover, we show that as the speed of light tends to infinity, the solution of this model converges to the solution of the corresponding Newtonian free boundary problem
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Congruences of algebraic automorphic forms and supercuspidal representations Camb. J. Math. (IF 1.6) Pub Date : 2021-10-07 Jessica Fintzen, Sug Woo Shin, Raphaël Beuzart-Plessis, Vytautas Paškūnas
Congruences between automorphic forms have been an essential tool in number theory since Ramanujan’s discovery of congruences for the $\tau$‑function, for instance in Iwasawa theory and the Langlands program. Over time, several approaches to congruences have been developed via Fourier coefficients, geometry of Shimura varieties, Hida theory, eigenvarieties, cohomology theories, trace formula, and automorphy
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Motivic infinite loop spaces Camb. J. Math. (IF 1.6) Pub Date : 2021-10-07 Elden Elmanto, Marc Hoyois, Adeel A. Khan, Vladimir Sosnilo, Maria Yakerson
We prove a recognition principle for motivic infinite $\mathsf{P}^1$‑loop spaces over a perfect field. This is achieved by developing a theory of framed motivic spaces, which is a motivic analogue of the theory of $\mathcal{E}_\infty$‑spaces. A framed motivic space is a motivic space equipped with transfers along finite syntomic morphisms with trivialized cotangent complex in K‑theory. Our main result
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Fourier–Jacobi cycles and arithmetic relative trace formula (with an appendix by Chao Li and Yihang Zhu) Camb. J. Math. (IF 1.6) Pub Date : 2021-01-01 Yifeng Liu, Chao Li, Yihang Zhu
In this article, we develop an arithmetic analogue of Fourier–Jacobi period integrals for a pair of unitary groups of equal rank. We construct the so-called Fourier–Jacobi cycles, which are algebraic cycles on the product of unitary Shimura varieties and abelian varieties.We propose the arithmetic Gan–Gross–Prasad conjecture for these cycles, which is related to the central derivatives of certain Rankin–Selberg
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Uniqueness of the minimizer of the normalized volume function Camb. J. Math. (IF 1.6) Pub Date : 2021-01-01 Chenyang Xu, Ziquan Zhuang
We confirm a conjecture of Chi Li which says that the minimizer of the normalized volume function for a $\mathrm{klt}$ singularity is unique up to rescaling. This is achieved by defining stability thresholds for valuations in the local setting, and then showing that a valuation is a minimizer if and only if it is $\mathrm{K}$-semistable, and that $\mathrm{K}$-semistable valuation is unique up to rescaling
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Concavity properties of solutions to Robin problems Camb. J. Math. (IF 1.6) Pub Date : 2021-01-01 Graziano Crasta, Ilaria Fragalà
We prove that the Robin ground state and the Robin torsion function are respectively $\operatorname{log}$-concave and $\frac{1}{2}$ -concave on an uniformly convex domain $\Omega \subset \mathbb{R}^N$ of class $\mathcal{C}^m$, with $[m - \frac{N}{2}] \geq 4$, provided the Robin parameter exceeds a critical threshold. Such threshold depends on $N, m$, and on the geometry of $\Omega$, precisely on the
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On the structure of some $p$-adic period domains Camb. J. Math. (IF 1.6) Pub Date : 2021-01-01 Miaofen Chen, Laurent Fargues, Xu Shen
We study the geometry of the $p$‑adic analogues of the complex analytic period spaces first introduced by Griffiths. More precisely, we prove the Fargues–Rapoport conjecture for $p$‑adic period domains: for a reductive group $G$ over a $p$‑adic field and a minuscule cocharacter $\mu$ of $G$, the weakly admissible locus coincides with the admissible one if and only if the Kottwitz set $B(G, \mu)$ is
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Arithmetic subspaces of moduli spaces of rank one local systems Camb. J. Math. (IF 1.6) Pub Date : 2020-10-02 Hélène Esnault, Moritz Kerz
We show that closed subsets of the character variety of a complex variety with negatively weighted first homology, which are $p$-adically integral and stabilized by an arithmetic Galois action, are motivic.
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One-sided curvature estimates for $H$-disks Camb. J. Math. (IF 1.6) Pub Date : 2020-10-02 William H. Meeks, Giuseppe Tinaglia
In this paper we prove an extrinsic one-sided curvature estimate for disks embedded in $\mathbb{R}^3$ with constant mean curvature, which is independent of the value of the constant mean curvature. We apply this extrinsic one-sided curvature estimate in [26] to prove a weak chord arc result for these disks. In Section 4 we apply this weak chord arc result to obtain an intrinsic version of the one-sided
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Hermitian $K$-theory, Dedekind $\zeta$-functions, and quadratic forms over rings of integers in number fields Camb. J. Math. (IF 1.6) Pub Date : 2020-10-02 Jonas Irgens Kylling, Röndigs Oliver, Paul Arne Østvær
We employ the slice spectral sequence, the motivic Steenrod algebra, and Voevodsky’s solutions of the Milnor and Bloch–Kato conjectures to calculate the hermitian $K$-groups of rings of integers in number fields. Moreover, we relate the orders of these groups to special values of Dedekind $\zeta$-functions for totally real abelian number fields. Our methods apply more readily to the examples of algebraic
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Displacement convexity of Boltzmann’s entropy characterizes the strong energy condition from general relativity Camb. J. Math. (IF 1.6) Pub Date : 2020-10-02 Robert J. McCann
On a Riemannian manifold, lower Ricci curvature bounds are known to be characterized by geodesic convexity properties of various entropies with respect to the Kantorovich–Rubinstein–Wasserstein square distance from optimal transportation. These notions also make sense in a (nonsmooth) metric measure setting, where they have found powerful applications. This article initiates the development of an analogous
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Non-concavity of the Robin ground state Camb. J. Math. (IF 1.6) Pub Date : 2020-04-21 Ben Andrews, Julie Clutterbuck, Daniel Hauer
On a convex bounded Euclidean domain, the ground state for the Laplacian with Neumann boundary conditions is a constant, while the Dirichlet ground state is log-concave. The Robin eigenvalue problem can be considered as interpolating between the Dirichlet and Neumann cases, so it seems natural that the Robin ground state should have similar concavity properties. The aim of this paper is to show that
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Existence of hypersurfaces with prescribed mean curvature I – generic min-max Camb. J. Math. (IF 1.6) Pub Date : 2020-04-21 Xin Zhou, Jonathan J. Zhu
We prove that, for a generic set of smooth prescription functions $h$ on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature $h$. The solution is either an embedded minimal hypersurface with integer multiplicity, or a non-minimal almost embedded hypersurface of multiplicity one. More precisely, we show that our previous min-max theory
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The index and nullity of the Lawson surfaces $\xi_{g,1}$ Camb. J. Math. (IF 1.6) Pub Date : 2020-04-21 Nikolaos Kapouleas, David Wiygul
We prove that the Lawson surface $\xi_{g,1}$ in Lawson’s original notation, which has genus $g$ and can be viewed as a desingularization of two orthogonal great two-spheres in the round three-sphere $\mathbb{S}^3$, has index $2g + 3$ and nullity $6$ for any genus $g \geq 2$. In particular $\xi_{g,1}$ has no exceptional Jacobi fields, which means that it cannot “flap its wings” at the linearized level
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$(1,1)$ forms with specified Lagrangian phase: a priori estimates and algebraic obstructions Camb. J. Math. (IF 1.6) Pub Date : 2020-04-21 Tristan C. Collins, Adam Jacob, Shing-Tung Yau
Let $(X, \alpha)$ be a Kähler manifold of dimension $n$, and let $[\omega] \in H^{1,1} (X, \mathbb{R})$. We study the problem of specifying the Lagrangian phase of $\omega$ with respect to $\alpha$, which is described by the nonlinear elliptic equation\[\sum^{n}_{i=1} \arctan (\lambda_i) = h(x)\]where $\lambda_i$ are the eigenvalues of $\omega$ with respect to $\alpha$. When $h(x)$ is a topological
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$(1,1)$ forms with specified Lagrangian phase: a priori estimates and algebraic obstructions Camb. J. Math. (IF 1.6) Pub Date : 2020-01-01 Tristan C. Collins,Adam Jacob,Shing-Tung Yau
Let $(X,\alpha)$ be a K\"ahler manifold of dimension n, and let $[\omega] \in H^{1,1}(X,\mathbb{R})$. We study the problem of specifying the Lagrangian phase of $\omega$ with respect to $\alpha$, which is described by the nonlinear elliptic equation \[ \sum_{i=1}^{n} \arctan(\lambda_i)= h(x) \] where $\lambda_i$ are the eigenvalues of $\omega$ with respect to $\alpha$. When $h(x)$ is a topological
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The index and nullity of the Lawson surfaces $\xi_{g,1}$ Camb. J. Math. (IF 1.6) Pub Date : 2020-01-01 Nikolaos Kapouleas,David Wiygul
We prove that the Lawson surface $\xi_{g,1}$ in Lawson's original notation, which has genus $g$ and can be viewed as a desingularization of two orthogonal great two-spheres in the round three-sphere ${\mathbb{S}}^3$, has index $2g+3$ and nullity $6$ for any genus $g\ge2$. In particular $\xi_{g,1}$ has no exceptional Jacobi fields, which means that it cannot `flap its wings' at the linearized level
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Non-concavity of the Robin ground state Camb. J. Math. (IF 1.6) Pub Date : 2020-01-01 Ben Andrews,Julie Clutterbuck,Daniel Hauer
On a convex bounded Euclidean domain, the ground state for the Laplacian with Neumann boundary conditions is a constant, while the Dirichlet ground state is log-concave. The Robin eigenvalue problem can be considered as interpolating between the Dirichlet and Neumann cases, so it seems natural that the Robin ground state should have similar concavity properties. In this paper we show that this is false
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An infinite-dimensional phenomenon in finite-dimensional metric topology Camb. J. Math. (IF 1.6) Pub Date : 2020-01-01 Alexander N. Dranishnikov,Steven C. Ferry,Shmuel Weinberger
We show that there are homotopy equivalences $h:N\to M$ between closed manifolds which are induced by cell-like maps $p:N\to X$ and $q:M\to X$ but which are not homotopic to homeomorphisms. The phenomenon is based on construction of cell-like maps that kill certain $\mathbb L$-classes. The image space in these constructions is necessarily infinite-dimensional. In dimension $>6$ we classify all such
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Homotopy invariant presheaves with framed transfers Camb. J. Math. (IF 1.6) Pub Date : 2020-01-01 Grigory Garkusha,Ivan Panin
The category of framed correspondences $Fr_*(k)$, framed presheaves and framed sheaves were invented by Voevodsky in his unpublished notes [12]. Based on the theory, framed motives are introduced and studied in [7]. The main aim of this paper is to prove that for any $\mathbb A^1$-invariant quasi-stable radditive framed presheaf of Abelian groups $\mathcal F$, the associated Nisnevich sheaf $\mathcal
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Existence of hypersurfaces with prescribed mean curvature I – generic min-max Camb. J. Math. (IF 1.6) Pub Date : 2020-01-01 Xin Zhou,Jonathan J. Zhu
We prove that, for a generic set of smooth prescription functions $h$ on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature $h$. The solution is either an embedded minimal hypersurface with integer multiplicity, or a non-minimal almost embedded hypersurface of multiplicity one. More precisely, we show that our previous min-max theory
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Twisted orbital integrals and irreducible components of affine Deligne–Lusztig varieties Camb. J. Math. (IF 1.6) Pub Date : 2020-01-01 Rong Zhou,Yihang Zhu
We analyze the asymptotic behavior of certain twisted orbital integrals arising from the study of affine Deligne-Lusztig varieties. The main tools include the Base Change Fundamental Lemma and $q$-analogues of the Kostant partition functions. As an application we prove a conjecture of Miaofen Chen and Xinwen Zhu, relating the set of irreducible components of an affine Deligne-Lusztig variety modulo
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Higher-order estimates for collapsing Calabi–Yau metrics Camb. J. Math. (IF 1.6) Pub Date : 2020-01-01 Hans-Joachim Hein, Valentino Tosatti
We prove a uniform $C^\alpha$ estimate for collapsing Calabi–Yau metrics on the total space of a proper holomorphic submersion over the unit ball in $\mathbb{C}^m$. The usual methods of Calabi, Evans–Krylov, Caffarelli, et al. do not apply to this setting because the background geometry degenerates. We instead rely on blowup arguments and on linear and nonlinear Liouville theorems on cylinders. In
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Higher $\mathcal{L}$-invariants for $\mathrm{GL}_3 (\mathbb{Q}_p)$ and local-global compatibility Camb. J. Math. (IF 1.6) Pub Date : 2020-01-01 Christophe Breuil, Yiwen Ding
Let $\rho_p$ be a 3-dimensional semi-stable representation of $\operatorname{Gal} (\overline{\mathbb{Q}_p} / \mathbb{Q}_p)$ with Hodge–Tate weights $(0, 1, 2)$ (up to shift) and such that $N^2 \neq 0$ on $D_\mathrm{st} (\rho_p)$. When $\rho_p$ comes from an automorphic representation $\pi$ of $G(\mathbb{A}_{F^+})$ (for a unitary group $G$ over a totally real field $F^+$ which is compact at infinite
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Global well-posedness and scattering for nonlinear Schrödinger equations with algebraic nonlinearity when $d = 2,3$ and $u_0$ is radial Camb. J. Math. (IF 1.6) Pub Date : 2019-01-01 Benjamin Dodson
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Improved Fourier restriction estimates in higher dimensions Camb. J. Math. (IF 1.6) Pub Date : 2019-01-01 Jonathan Hickman,Keith M. Rogers
We consider Guth's approach to the Fourier restriction problem via polynomial partitioning. By writing out his induction argument as a recursive algorithm and introducing new geometric information, known as the polynomial Wolff axioms, we obtain improved bounds for the restriction conjecture, particularly in high dimensions. Consequences for the Kakeya conjecture are also considered.
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A Riemann–Hilbert correspondence in positive characteristic Camb. J. Math. (IF 1.6) Pub Date : 2019-01-01 Bhargav Bhatt,Jacob Lurie
We explain a version of the Riemann-Hilbert correspondence for $p$-torsion \'etale sheaves on an arbitrary $\mathbf{F}_p$-scheme.