Duke Mathematical Journal Articles (Project Euclid)
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The latest articles from Duke Mathematical Journal on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTWed, 01 Jun 2011 09:20 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Crofton measures and Minkowski valuations
http://projecteuclid.org/euclid.dmj/1279140505
<strong>Franz E. Schuster</strong><p><strong>Source: </strong>Duke Math. J., Volume 154, Number 1, 1--30.</p><p><strong>Abstract:</strong><br/>
A description of continuous rigid motion compatible Minkowski valuations is established. As an application, we present a Brunn-Minkowski type inequality for intrinsic volumes of these valuations </p>projecteuclid.org/euclid.dmj/1279140505_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTOn finiteness properties of the Johnson filtrationshttps://projecteuclid.org/euclid.dmj/1525313239<strong>Mikhail Ershov</strong>, <strong>Sue He</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 9, 1713--1759.</p><p><strong>Abstract:</strong><br/>
Let $\Gamma$ be either the automorphism group of the free group of rank $n\geq4$ or the mapping class group of an orientable surface of genus $n\geq12$ with at most $1$ boundary component, and let $G$ be either the subgroup of $\mathrm{IA}$ -automorphisms or the Torelli subgroup of $\Gamma$ . For $N\in\mathbb{N}$ denote by $\gamma_{N}G$ the $N$ th term of the lower central series of $G$ . We prove that
(i) any subgroup of $G$ containing $\gamma_{2}G=[G,G]$ (in particular, the Johnson kernel in the mapping class group case) is finitely generated;
(ii) if $N=2$ or $n\geq8N-4$ and $K$ is any subgroup of $G$ containing $\gamma_{N}G$ (for instance, $K$ can be the $N$ th term of the Johnson filtration of $G$ ), then $G/[K,K]$ is nilpotent and hence the Abelianization of $K$ is finitely generated;
(iii) if $H$ is any finite-index subgroup of $\Gamma$ containing $\gamma_{N}G$ , with $N$ as in (ii), then $H$ has finite Abelianization.
</p>projecteuclid.org/euclid.dmj/1525313239_20180612040040Tue, 12 Jun 2018 04:00 EDTUniversal dynamics for the defocusing logarithmic Schrödinger equationhttps://projecteuclid.org/euclid.dmj/1526436027<strong>Rémi Carles</strong>, <strong>Isabelle Gallagher</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 9, 1761--1801.</p><p><strong>Abstract:</strong><br/>
We consider the Schrödinger equation with a logarithmic nonlinearity in a dispersive regime. We show that the presence of the nonlinearity affects the large time behavior of the solution: the dispersion is faster than usual by a logarithmic factor in time, and the positive Sobolev norms of the solution grow logarithmically in time. Moreover, after rescaling in space by the dispersion rate, the modulus of the solution converges to a universal Gaussian profile. These properties are suggested by explicit computations in the case of Gaussian initial data and remain when an extra power-like nonlinearity is present in the equation. One of the key steps of the proof consists in working in hydrodynamical variables to reduce the equation to a variant of the isothermal compressible Euler equation, whose large time behavior turns out to be governed by a parabolic equation involving a Fokker–Planck operator.
</p>projecteuclid.org/euclid.dmj/1526436027_20180612040040Tue, 12 Jun 2018 04:00 EDTIndependence of $\ell$ for the supports in the decomposition theoremhttps://projecteuclid.org/euclid.dmj/1528855662<strong>Shenghao Sun</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 10, 1803--1823.</p><p><strong>Abstract:</strong><br/>
In this article, we prove a result on the independence of $\ell$ for the supports of irreducible perverse sheaves occurring in the decomposition theorem, as well as for the family of local systems on each support. It generalizes Gabber’s result on the independence of $\ell$ of intersection cohomology to the relative case.
</p>projecteuclid.org/euclid.dmj/1528855662_20180713040712Fri, 13 Jul 2018 04:07 EDTA minimization problem with free boundary related to a cooperative systemhttps://projecteuclid.org/euclid.dmj/1528185619<strong>Luis A. Caffarelli</strong>, <strong>Henrik Shahgholian</strong>, <strong>Karen Yeressian</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 10, 1825--1882.</p><p><strong>Abstract:</strong><br/>
We study the minimum problem for the functional \begin{equation*}\int_{\Omega}(\vert\nabla\mathbf{u}\vert^{2}+Q^{2}\chi_{\{\vert\mathbf{u}\vert\gt 0\}})\,dx\end{equation*} with the constraint $u_{i}\geq0$ for $i=1,\ldots,m$ , where $\Omega\subset\mathbb{R}^{n}$ is a bounded domain and $\mathbf{u}=(u_{1},\ldots,u_{m})\in H^{1}(\Omega;\mathbb{R}^{m})$ . First we derive the Euler equation satisfied by each component. Then we show that the noncoincidence set $\{\vert u\vert\gt 0\}$ is (locally) nontangentially accessible. Having this, we are able to establish sufficient regularity of the force term appearing in the Euler equations and derive the regularity of the free boundary $\Omega\cap\partial\{\vert u\vert\gt 0\}$ .
</p>projecteuclid.org/euclid.dmj/1528185619_20180713040712Fri, 13 Jul 2018 04:07 EDTAnalytic torsion and R-torsion of Witt representations on manifolds with cuspshttps://projecteuclid.org/euclid.dmj/1528358417<strong>Pierre Albin</strong>, <strong>Frédéric Rochon</strong>, <strong>David Sher</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 10, 1883--1950.</p><p><strong>Abstract:</strong><br/>
We establish a Cheeger–Müller theorem for unimodular representations satisfying a Witt condition on a noncompact manifold with cusps. This class of spaces includes all noncompact hyperbolic spaces of finite volume, but we do not assume that the metric has constant curvature nor that the link of the cusp is a torus. We use renormalized traces in the sense of Melrose to define the analytic torsion, and we relate it to the intersection R-torsion of Dar of the natural compactification to a stratified space. Our proof relies on our recent work on the behavior of the Hodge Laplacian spectrum on a closed manifold undergoing degeneration to a manifold with fibered cusps.
</p>projecteuclid.org/euclid.dmj/1528358417_20180713040712Fri, 13 Jul 2018 04:07 EDTThe $p$ -curvature conjecture and monodromy around simple closed loopshttps://projecteuclid.org/euclid.dmj/1529978488<strong>Ananth N. Shankar</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 10, 1951--1980.</p><p><strong>Abstract:</strong><br/>
The Grothendieck–Katz $p$ -curvature conjecture is an analogue of the Hasse principle for differential equations. It states that a set of arithmetic differential equations on a variety has finite monodromy if its $p$ -curvature vanishes modulo $p$ , for almost all primes $p$ . We prove that if the variety is a generic curve, then every simple closed loop on the curve has finite monodromy.
</p>projecteuclid.org/euclid.dmj/1529978488_20180713040712Fri, 13 Jul 2018 04:07 EDTQuadratic Chabauty and rational points, I: $p$ -adic heightshttps://projecteuclid.org/euclid.dmj/1532073621<strong>Jennifer S. Balakrishnan</strong>, <strong>Netan Dogra</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 11, 1981--2038.</p><p><strong>Abstract:</strong><br/>
We give the first explicit examples beyond the Chabauty–Coleman method where Kim’s nonabelian Chabauty program determines the set of rational points of a curve defined over $\mathbb{Q}$ or a quadratic number field. We accomplish this by studying the role of $p$ -adic heights in explicit non-Abelian Chabauty.
</p>projecteuclid.org/euclid.dmj/1532073621_20180808220157Wed, 08 Aug 2018 22:01 EDTExceptional isogenies between reductions of pairs of elliptic curveshttps://projecteuclid.org/euclid.dmj/1530000176<strong>François Charles</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 11, 2039--2072.</p><p><strong>Abstract:</strong><br/>
Let $E$ and $E'$ be two elliptic curves over a number field. We prove that the reductions of $E$ and $E'$ at a finite place $\mathfrak{p}$ are geometrically isogenous for infinitely many $\mathfrak{p}$ , and we draw consequences for the existence of supersingular primes. This result is an analogue for distributions of Frobenius traces of known results on the density of Noether–Lefschetz loci in Hodge theory. The proof relies on dynamical properties of the Hecke correspondences on the modular curve.
</p>projecteuclid.org/euclid.dmj/1530000176_20180808220157Wed, 08 Aug 2018 22:01 EDTEnergy quantization of Willmore surfaces at the boundary of the moduli spacehttps://projecteuclid.org/euclid.dmj/1528358418<strong>Paul Laurain</strong>, <strong>Tristan Rivière</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 11, 2073--2124.</p><p><strong>Abstract:</strong><br/>
We establish an energy quantization result for sequences of Willmore surfaces when the underlying sequence of Riemann surfaces is degenerating in the moduli space. We notably exhibit a new residue which quantifies the possible loss of energy in collar regions. Thanks to this residue, we also establish the compactness (modulo the action of the Möbius group of conformal transformations of ${\mathbb{R}}^{3}\cup\{\infty\}$ ) of the space of Willmore immersions of any arbitrary closed $2$ -dimensional oriented manifold into ${\mathbb{R}}^{3}$ with uniformly bounded conformal class and energy below $12\pi$ .
</p>projecteuclid.org/euclid.dmj/1528358418_20180808220157Wed, 08 Aug 2018 22:01 EDTRigidity of critical circle mapshttps://projecteuclid.org/euclid.dmj/1531900818<strong>Pablo Guarino</strong>, <strong>Marco Martens</strong>, <strong>Welington de Melo</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 11, 2125--2188.</p><p><strong>Abstract:</strong><br/>
We prove that any two $C^{4}$ critical circle maps with the same irrational rotation number and the same odd criticality are conjugate to each other by a $C^{1}$ circle diffeomorphism. The conjugacy is $C^{1+\alpha}$ for a full Lebesgue measure set of rotation numbers.
</p>projecteuclid.org/euclid.dmj/1531900818_20180808220157Wed, 08 Aug 2018 22:01 EDTRecognizing a relatively hyperbolic group by its Dehn fillingshttps://projecteuclid.org/euclid.dmj/1532073620<strong>François Dahmani</strong>, <strong>Vincent Guirardel</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 12, 2189--2241.</p><p><strong>Abstract:</strong><br/>
Dehn fillings for relatively hyperbolic groups generalize the topological Dehn surgery on a noncompact hyperbolic $3$ -manifold such as hyperbolic knot complements. We prove a rigidity result saying that if two nonelementary relatively hyperbolic groups without certain splittings have sufficiently many isomorphic Dehn fillings, then these groups are in fact isomorphic. Our main application is a solution to the isomorphism problem in the class of nonelementary relatively hyperbolic groups with residually finite parabolic groups and with no suitable splittings.
</p>projecteuclid.org/euclid.dmj/1532073620_20180828220249Tue, 28 Aug 2018 22:02 EDTOn the maximum of the C $\beta$ E fieldhttps://projecteuclid.org/euclid.dmj/1533866573<strong>Reda Chhaibi</strong>, <strong>Thomas Madaule</strong>, <strong>Joseph Najnudel</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 12, 2243--2345.</p><p><strong>Abstract:</strong><br/>
In this article, we investigate the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according to the circular beta ensemble (C $\beta$ E). More precisely, assuming that $X_{n}$ is this characteristic polynomial and ${\mathbb{U}}$ is the unit circle, we prove that
\[\sup_{z\in{\mathbb{U}}}\Re\log X_{n}(z)=\sqrt{\frac{2}{\beta}}(\log n-\frac{3}{4}\log\log n+\mathcal{O}(1)),\] as well as an analogous statement for the imaginary part. The notation ${\mathcal{O}}(1)$ means that the corresponding family of random variables, indexed by $n$ , is tight. This answers a conjecture of Fyodorov, Hiary, and Keating, originally formulated for the $\beta=2$ case, which corresponds to the circular unitary ensemble (CUE) field.
</p>projecteuclid.org/euclid.dmj/1533866573_20180828220249Tue, 28 Aug 2018 22:02 EDTCompactification of strata of Abelian differentialshttps://projecteuclid.org/euclid.dmj/1533866574<strong>Matt Bainbridge</strong>, <strong>Dawei Chen</strong>, <strong>Quentin Gendron</strong>, <strong>Samuel Grushevsky</strong>, <strong>Martin Möller</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 12, 2347--2416.</p><p><strong>Abstract:</strong><br/>
We describe the closure of the strata of Abelian differentials with prescribed type of zeros and poles, in the projectivized Hodge bundle over the Deligne–Mumford moduli space of stable curves with marked points. We provide an explicit characterization of pointed stable differentials in the boundary of the closure, both a complex analytic proof and a flat geometric proof for smoothing the boundary differentials, and numerous examples. The main new ingredient in our description is a global residue condition arising from a full order on the dual graph of a stable curve.
</p>projecteuclid.org/euclid.dmj/1533866574_20180828220249Tue, 28 Aug 2018 22:02 EDTThe Gaussian core model in high dimensionshttps://projecteuclid.org/euclid.dmj/1534233620<strong>Henry Cohn</strong>, <strong>Matthew de Courcy-Ireland</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 13, 2417--2455.</p><p><strong>Abstract:</strong><br/>
We prove lower bounds for energy in the Gaussian core model, in which point particles interact via a Gaussian potential. Under the potential function $t\mapsto e^{-\alpha t^{2}}$ with $0\lt \alpha \lt 4\pi/e$ , we show that no point configuration in $\mathbb{R}^{n}$ of density $\rho$ can have energy less than $(\rho+o(1))(\pi/\alpha)^{n/2}$ as $n\to \infty$ with $\alpha$ and $\rho$ fixed. This lower bound asymptotically matches the upper bound of $\rho(\pi/\alpha)^{n/2}$ obtained as the expectation in the Siegel mean value theorem, and it is attained by random lattices. The proof is based on the linear programming bound, and it uses an interpolation construction analogous to those used for the Beurling–Selberg extremal problem in analytic number theory. In the other direction, we prove that the upper bound of $\rho (\pi/\alpha)^{n/2}$ is no longer asymptotically sharp when $\alpha \gt \pi e$ . As a consequence of our results, we obtain bounds in $\mathbb{R}^{n}$ for the minimal energy under inverse power laws $t\mapsto1/t^{n+s}$ with $s\gt 0$ , and these bounds are sharp to within a constant factor as $n\to \infty$ with $s$ fixed.
</p>projecteuclid.org/euclid.dmj/1534233620_20180914220702Fri, 14 Sep 2018 22:07 EDTStochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion processhttps://projecteuclid.org/euclid.dmj/1535356817<strong>Guillaume Barraquand</strong>, <strong>Alexei Borodin</strong>, <strong>Ivan Corwin</strong>, <strong>Michael Wheeler</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 13, 2457--2529.</p><p><strong>Abstract:</strong><br/>
We consider the asymmetric simple exclusion process (ASEP) on the positive integers with an open boundary condition. We show that, when starting devoid of particles and for a certain boundary condition, the height function at the origin fluctuates asymptotically (in large time $\tau$ ) according to the Tracy–Widom Gaussian orthogonal ensemble distribution on the $\tau^{1/3}$ -scale. This is the first example of Kardar–Parisi–Zhang asymptotics for a half-space system outside the class of free-fermionic/determinantal/Pfaffian models.
Our main tool in this analysis is a new class of probability measures on Young diagrams that we call half-space Macdonald processes , as well as two surprising relations. The first relates a special (Hall–Littlewood) case of these measures to the half-space stochastic six-vertex model (which further limits to the ASEP) using a Yang–Baxter graphical argument. The second relates certain averages under these measures to their half-space (or Pfaffian) Schur process analogues via a refined Littlewood identity.
</p>projecteuclid.org/euclid.dmj/1535356817_20180914220702Fri, 14 Sep 2018 22:07 EDTGevrey stability of Prandtl expansions for $2$ -dimensional Navier–Stokes flowshttps://projecteuclid.org/euclid.dmj/1534838557<strong>David Gérard-Varet</strong>, <strong>Yasunori Maekawa</strong>, <strong>Nader Masmoudi</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 13, 2531--2631.</p><p><strong>Abstract:</strong><br/>
We investigate the stability of boundary layer solutions of the $2$ -dimensional incompressible Navier–Stokes equations. We consider shear flow solutions of Prandtl type: $u^{\nu}(t,x,y)=(U^{E}(t,y)+U^{\mathrm{BL}}(t,\frac{y}{\sqrt{\nu}}),0)$ , $0\lt \nu\ll1.$ We show that if $U^{\mathrm{BL}}$ is monotonic and concave in $Y=y/\sqrt{\nu}$ , then $u^{\nu}$ is stable over some time interval $(0,T)$ , $T$ independent of $\nu$ , under perturbations with Gevrey regularity in $x$ and Sobolev regularity in $y$ . We improve in this way the classical stability results of Sammartino and Caflisch in analytic class (both in $x$ and $y$ ). Moreover, in the case where $U^{\mathrm{BL}}$ is steady and strictly concave, our Gevrey exponent for stability is optimal. The proof relies on new and sharp resolvent estimates for the linearized Orr–Sommerfeld operator.
</p>projecteuclid.org/euclid.dmj/1534838557_20180914220702Fri, 14 Sep 2018 22:07 EDTAlmost periodicity in time of solutions of the KdV equationhttps://projecteuclid.org/euclid.dmj/1538121917<strong>Ilia Binder</strong>, <strong>David Damanik</strong>, <strong>Michael Goldstein</strong>, <strong>Milivoje Lukic</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 14, 2633--2678.</p><p><strong>Abstract:</strong><br/>
We study the Cauchy problem for the KdV equation $\partial_{t}u-6u\partial_{x}u+\partial_{x}^{3}u=0$ with almost periodic initial data $u(x,0)=V(x)$ . We consider initial data $V$ , for which the associated Schrödinger operator is absolutely continuous and has a spectrum that is not too thin in a sense we specify, and we show the existence, uniqueness, and almost periodicity in time of solutions. This establishes a conjecture of Deift for this class of initial data. The result is shown to apply to all small analytic quasiperiodic initial data with Diophantine frequency vector.
</p>projecteuclid.org/euclid.dmj/1538121917_20180928040600Fri, 28 Sep 2018 04:06 EDTThe Markoff group of transformations in prime and composite modulihttps://projecteuclid.org/euclid.dmj/1538121918<strong>Chen Meiri</strong>, <strong>Doron Puder</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 14, 2679--2720.</p><p><strong>Abstract:</strong><br/>
The Markoff group of transformations is a group $\Gamma$ of affine integral morphisms, which is known to act transitively on the set of all positive integer solutions to the equation $x^{2}+y^{2}+z^{2}=xyz$ . The fundamental strong approximation conjecture for the Markoff equation states that for every prime $p$ , the group $\Gamma$ acts transitively on the set $X^{*}(p)$ of nonzero solutions to the same equation over ${\mathbb{Z}}/{p\mathbb{Z}}$ . Recently, Bourgain, Gamburd, and Sarnak proved this conjecture for all primes outside a small exceptional set. Here, we study a group of permutations obtained by the action of $\Gamma$ on $X^{*}(p)$ , and show that for most primes, it is the full symmetric or alternating group. We use this result to deduce that $\Gamma$ acts transitively also on the set of nonzero solutions in a big class of composite moduli. Finally, our result also translates to a parallel in the case $r=2$ of a well-known theorem of Gilman and Evans regarding “ $T_{r}$ -systems” of $\operatorname{PSL}(2,p)$ .
</p>projecteuclid.org/euclid.dmj/1538121918_20180928040600Fri, 28 Sep 2018 04:06 EDTCounting points of schemes over finite rings and counting representations of arithmetic latticeshttps://projecteuclid.org/euclid.dmj/1538121919<strong>Avraham Aizenbud</strong>, <strong>Nir Avni</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 14, 2721--2743.</p><p><strong>Abstract:</strong><br/>
We relate the singularities of a scheme $X$ to the asymptotics of the number of points of $X$ over finite rings. This gives a partial answer to a question of Mustata. We use this result to count representations of arithmetic lattices. More precisely, if $\Gamma$ is an arithmetic lattice whose $\mathbb{Q}$ -rank is greater than $1$ , then let $r_{n}(\Gamma)$ be the number of irreducible $n$ -dimensional representations of $\Gamma$ up to isomorphism. We prove that there is a constant $C$ (in fact, any $C\gt 40$ suffices) such that $r_{n}(\Gamma)=O(n^{C})$ for every such $\Gamma$ . This answers a question of Larsen and Lubotzky.
</p>projecteuclid.org/euclid.dmj/1538121919_20180928040600Fri, 28 Sep 2018 04:06 EDTMöbius disjointness for homogeneous dynamicshttps://projecteuclid.org/euclid.dmj/1538121920<strong>Ryan Peckner</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 14, 2745--2792.</p><p><strong>Abstract:</strong><br/>
We prove Sarnak’s Möbius disjointness conjecture for all unipotent translations on homogeneous spaces of real connected Lie groups. Namely, we show that if $G$ is any such group, $\Gamma\subset G$ a lattice, and $u\in G$ an Ad-unipotent element, then for every $x\in\Gamma\backslash G$ and every function $f$ continuous on the $1$ -point compactification of $\Gamma\backslash G$ , the sequence $f(xu^{n})$ cannot correlate with the Möbius function on average.
</p>projecteuclid.org/euclid.dmj/1538121920_20180928040600Fri, 28 Sep 2018 04:06 EDTEffective finiteness of irreducible Heegaard splittings of non-Haken $3$ -manifoldshttps://projecteuclid.org/euclid.dmj/1538532049<strong>Tobias Holck Colding</strong>, <strong>David Gabai</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 15, 2793--2832.</p><p><strong>Abstract:</strong><br/>
The main result is a short effective proof of Tao Li’s theorem that a closed non-Haken hyperbolic $3$ -manifold $N$ has at most finitely many irreducible Heegaard splittings. Along the way we show that $N$ has finitely many branched surfaces of pinched negative sectional curvature carrying all closed index- $\le1$ minimal surfaces. This effective result, together with the sequel with Daniel Ketover, solves the classification problem for Heegaard splittings of non-Haken hyperbolic $3$ -manifolds.
</p>projecteuclid.org/euclid.dmj/1538532049_20181018040118Thu, 18 Oct 2018 04:01 EDTOn the classification of Heegaard splittingshttps://projecteuclid.org/euclid.dmj/1538532048<strong>Tobias Holck Colding</strong>, <strong>David Gabai</strong>, <strong>Daniel Ketover</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 15, 2833--2856.</p><p><strong>Abstract:</strong><br/>
The long-standing classification problem in the theory of Heegaard splittings of $3$ -manifolds is to exhibit for each closed $3$ -manifold a complete list, without duplication, of all its irreducible Heegaard surfaces, up to isotopy. We solve this problem for non-Haken hyperbolic $3$ -manifolds.
</p>projecteuclid.org/euclid.dmj/1538532048_20181018040118Thu, 18 Oct 2018 04:01 EDTLarge deviations and the Lukic conjecturehttps://projecteuclid.org/euclid.dmj/1538532052<strong>Jonathan Breuer</strong>, <strong>Barry Simon</strong>, <strong>Ofer Zeitouni</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 15, 2857--2902.</p><p><strong>Abstract:</strong><br/>
We use the large deviation approach to sum rules pioneered by Gamboa, Nagel, and Rouault to prove higher-order sum rules for orthogonal polynomials on the unit circle. In particular, we prove one half of a conjectured sum rule of Lukic in the case of two singular points, one simple and one double. This is important because it is known that the conjecture of Simon fails in exactly this case, so this article provides support for the idea that Lukic’s replacement for Simon’s conjecture might be true.
</p>projecteuclid.org/euclid.dmj/1538532052_20181018040118Thu, 18 Oct 2018 04:01 EDTApproximate latticeshttps://projecteuclid.org/euclid.dmj/1538532050<strong>Michael Björklund</strong>, <strong>Tobias Hartnick</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 15, 2903--2964.</p><p><strong>Abstract:</strong><br/>
In this article we introduce and study uniform and nonuniform approximate lattices in locally compact second countable (lcsc) groups. These are approximate subgroups (in the sense of Tao) which simultaneously generalize lattices in lcsc group and mathematical quasicrystals (Meyer sets) in lcsc Abelian groups. We show that envelopes of strong approximate lattices are unimodular and that approximate lattices in nilpotent groups are uniform. We also establish several results relating properties of approximate lattices and their envelopes. For example, we prove a version of the Milnor–Schwarz lemma for uniform approximate lattices in compactly generated lcsc groups, which we then use to relate the metric amenability of uniform approximate lattices to the amenability of the envelope. Finally we extend a theorem of Kleiner and Leeb to show that the isometry groups of irreducible higher-rank symmetric spaces of noncompact type are quasi-isometrically rigid with respect to finitely generated approximate groups.
</p>projecteuclid.org/euclid.dmj/1538532050_20181018040118Thu, 18 Oct 2018 04:01 EDTOn an invariant bilinear form on the space of automorphic forms via asymptoticshttps://projecteuclid.org/euclid.dmj/1538726432<strong>Jonathan Wang</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 16, 2965--3057.</p><p><strong>Abstract:</strong><br/>
This article concerns the study of a new invariant bilinear form $\mathfrak{B}$ on the space of automorphic forms of a split reductive group $G$ over a function field. We define $\mathfrak{B}$ using the asymptotics maps from recent work of Bezrukavnikov, Kazhdan, Sakellaridis, and Venkatesh, which involve the geometry of the wonderful compactification of $G$ . We show that $\mathfrak{B}$ is naturally related to miraculous duality in the geometric Langlands program through the functions-sheaves dictionary. In the proof, we highlight the connection between the classical non-Archimedean Gindikin–Karpelevich formula and certain factorization algebras acting on geometric Eisenstein series. We then give another definition of $\mathfrak{B}$ using the constant term operator and the inverse of the standard intertwining operator. The form $\mathfrak{B}$ defines an invertible operator $L$ from the space of compactly supported automorphic forms to a new space of pseudocompactly supported automorphic forms. We give a formula for $L^{-1}$ in terms of pseudo-Eisenstein series and constant term operators which suggests that $L^{-1}$ is an analogue of the Aubert–Zelevinsky involution.
</p>projecteuclid.org/euclid.dmj/1538726432_20181029220247Mon, 29 Oct 2018 22:02 EDTComparable upper and lower bounds for boundary values of Neumann eigenfunctions and tight inclusion of eigenvalueshttps://projecteuclid.org/euclid.dmj/1539137164<strong>Alex H. Barnett</strong>, <strong>Andrew Hassell</strong>, <strong>Melissa Tacy</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 16, 3059--3114.</p><p><strong>Abstract:</strong><br/>
For smooth bounded domains in $\mathbb{R}^{n}$ , we prove upper and lower $L^{2}$ bounds on the boundary data of Neumann eigenfunctions, and we prove quasiorthogonality of this boundary data in a spectral window. The bounds are tight in the sense that both are independent of the eigenvalues; this is achieved by working with an appropriate norm for boundary functions, which includes a spectral weight, that is, a function of the boundary Laplacian. This spectral weight is chosen to cancel concentration at the boundary that can happen for whispering gallery -type eigenfunctions. These bounds are closely related to wave equation estimates due to Tataru. Using this, we bound the distance from an arbitrary Helmholtz parameter $E\gt 0$ to the nearest Neumann eigenvalue in terms of boundary normal derivative data of a trial function $u$ solving the Helmholtz equation $(\Delta-E)u=0$ . This inclusion bound improves over previously known bounds by a factor of $E^{5/6}$ , analogously to a recently improved inclusion bound in the Dirichlet case due to the first two authors. Finally, we apply our theory to present an improved numerical implementation of the method of particular solutions for computation of Neumann eigenpairs on smooth planar domains. We show that the new inclusion bound improves the relative accuracy in a computed Neumann eigenvalue (around the $42000$ th) from nine to fourteen digits, with negligible extra numerical effort.
</p>projecteuclid.org/euclid.dmj/1539137164_20181029220247Mon, 29 Oct 2018 22:02 EDTGlobal smoothing of a subanalytic sethttps://projecteuclid.org/euclid.dmj/1538532051<strong>Edward Bierstone</strong>, <strong>Adam Parusiński</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 16, 3115--3128.</p><p><strong>Abstract:</strong><br/>
We give rather simple answers to two long-standing questions in real-analytic geometry, on global smoothing of a subanalytic set, and on transformation of a proper real-analytic mapping to a mapping with equidimensional fibers by global blowings-up of the target. These questions are related: a positive answer to the second can be used to reduce the first to the simpler semianalytic case. We show that the second question has a negative answer, in general, and that the first problem nevertheless has a positive solution.
</p>projecteuclid.org/euclid.dmj/1538532051_20181029220247Mon, 29 Oct 2018 22:02 EDTThe bounded Borel class and $3$ -manifold groupshttps://projecteuclid.org/euclid.dmj/1540454549<strong>Michelle Bucher</strong>, <strong>Marc Burger</strong>, <strong>Alessandra Iozzi</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 17, 3129--3169.</p><p><strong>Abstract:</strong><br/>
If $\Gamma\lt \operatorname{PSL}(2,\mathbb{C})$ is a lattice, we define an invariant of a representation $\Gamma\rightarrow\operatorname{PSL}(n,\allowbreak\mathbb{C})$ using the Borel class $\beta(n)\in\mathrm{H}_{\mathrm{c}}^{3}(\operatorname{PSL}(n,\mathbb{C}),\mathbb{R})$ . We show that this invariant satisfies a Milnor–Wood type inequality and its maximal value is attained precisely by the representations conjugate to the restriction to $\Gamma$ of the irreducible complex $n$ -dimensional representation of $\operatorname{PSL}(2,\mathbb{C})$ or its complex conjugate. Major ingredients of independent interest are the study of our extension to degenerate configurations of flags of a cocycle defined by Goncharov, as well as the identification of $\mathrm{H}_{\mathrm{b}}^{3}(\operatorname{SL}(n,\mathbb{C}),\mathbb{R})$ as a normed space.
</p>projecteuclid.org/euclid.dmj/1540454549_20181108220204Thu, 08 Nov 2018 22:02 ESTIntegrality of Hausel–Letellier–Villegas kernelshttps://projecteuclid.org/euclid.dmj/1539137165<strong>Anton Mellit</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 17, 3171--3205.</p><p><strong>Abstract:</strong><br/>
We prove that the coefficients of the generating function of Hausel, Letellier, and Rodriguez-Villegas and its recent generalization by Carlsson and Rodriguez-Villegas, which according to various conjectures should compute mixed Hodge numbers of character varieties and moduli spaces of Higgs bundles of curves of genus $g$ with $n$ punctures, are polynomials in $q$ and $t$ with integer coefficients for any $g,n\geq 0$ .
</p>projecteuclid.org/euclid.dmj/1539137165_20181108220204Thu, 08 Nov 2018 22:02 ESTConserved energies for the cubic nonlinear Schrödinger equation in one dimensionhttps://projecteuclid.org/euclid.dmj/1540540826<strong>Herbert Koch</strong>, <strong>Daniel Tataru</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 17, 3207--3313.</p><p><strong>Abstract:</strong><br/>
We consider the cubic nonlinear Schrödinger (NLS) equation as well as the modified Korteweg–de Vries (mKdV) equation in one space dimension. We prove that for each $s\gt -\frac{1}{2}$ there exists a conserved energy which is equivalent to the $H^{s}$ -norm of the solution. For the Korteweg–de Vries (KdV) equation, there is a similar conserved energy for every $s\ge -1$ .
</p>projecteuclid.org/euclid.dmj/1540540826_20181108220204Thu, 08 Nov 2018 22:02 ESTHigh-frequency backreaction for the Einstein equations under polarized $\mathbb{U}(1)$ -symmetryhttps://projecteuclid.org/euclid.dmj/1542337536<strong>Cécile Huneau</strong>, <strong>Jonathan Luk</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 18, 3315--3402.</p><p><strong>Abstract:</strong><br/>
Known examples in plane symmetry or Gowdy symmetry show that, given a $1$ -parameter family of solutions to the vacuum Einstein equations, it may have a weak limit which does not satisfy the vacuum equations, but instead has a nontrivial stress-energy-momentum tensor. We consider this phenomenon under polarized $\mathbb{U}(1)$ -symmetry—a much weaker symmetry than most of the known examples—such that the stress-energy-momentum tensor can be identified with that of multiple families of null dust propagating in distinct directions. We prove that any generic local-in-time small-data polarized $\mathbb{U}(1)$ -symmetric solution to the Einstein–multiple null dust system can be achieved as a weak limit of vacuum solutions. Our construction allows the number of families to be arbitrarily large and appears to be the first construction of such examples with more than two families.
</p>projecteuclid.org/euclid.dmj/1542337536_20181128220332Wed, 28 Nov 2018 22:03 ESTHow large is $A_{g}(\mathbb{F}_{q})$ ?https://projecteuclid.org/euclid.dmj/1542250833<strong>Michael Lipnowski</strong>, <strong>Jacob Tsimerman</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 18, 3403--3453.</p><p><strong>Abstract:</strong><br/>
Let $B(g,p)$ denote the number of isomorphism classes of $g$ -dimensional Abelian varieties over the finite field of size $p$ . Let $A(g,p)$ denote the number of isomorphism classes of principally polarized $g$ -dimensional Abelian varieties over the finite field of size $p$ . We derive upper bounds for $B(g,p)$ and lower bounds for $A(g,p)$ for $p$ fixed and $g$ increasing. The extremely large gap between the lower bound for $A(g,p)$ and the upper bound for $B(g,p)$ implies some statistically counterintuitive behavior for Abelian varieties of large dimension over a fixed finite field.
</p>projecteuclid.org/euclid.dmj/1542250833_20181128220332Wed, 28 Nov 2018 22:03 ESTEspaces de Banach–Colmez et faisceaux cohérents sur la courbe de Fargues–Fontainehttps://projecteuclid.org/euclid.dmj/1542337535<strong>Arthur-César Le Bras</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 18, 3455--3532.</p><p><strong>Abstract:</strong><br/>
We give a new definition, simpler but equivalent, of the abelian category of Banach–Colmez spaces introduced by Colmez, and we explain the precise relationship with the category of coherent sheaves on the Fargues–Fontaine curve. One goes from one category to the other by changing the $t$ -structure on the derived category. Along the way we obtain a description of the proétale cohomology of the open disk and the affine space, which is of independent interest.
</p>projecteuclid.org/euclid.dmj/1542337535_20181128220332Wed, 28 Nov 2018 22:03 ESTHypersymplectic 4-manifolds, the $G_{2}$ -Laplacian flow, and extension assuming bounded scalar curvaturehttps://projecteuclid.org/euclid.dmj/1541473293<strong>Joel Fine</strong>, <strong>Chengjian Yao</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 18, 3533--3589.</p><p><strong>Abstract:</strong><br/>
A hypersymplectic structure on a 4-manifold $X$ is a triple $\underline{\omega}$ of symplectic forms which at every point span a maximal positive definite subspace of $\Lambda^{2}$ for the wedge product. This article is motivated by a conjecture by Donaldson: when $X$ is compact, $\underline{\omega}$ can be deformed through cohomologous hypersymplectic structures to a hyper-Kähler triple. We approach this via a link with $G_{2}$ -geometry. A hypersymplectic structure $\underline{\omega}$ on a compact manifold $X$ defines a natural $G_{2}$ -structure $\phi$ on $X\times\mathbb{T}^{3}$ which has vanishing torsion precisely when $\underline{\omega}$ is a hyper-Kähler triple. We study the $G_{2}$ -Laplacian flow starting from $\phi$ , which we interpret as a flow of hypersymplectic structures. Our main result is that the flow extends as long as the scalar curvature of the corresponding $G_{2}$ -structure remains bounded. An application of our result is a lower bound for the maximal existence time of the flow in terms of weak bounds on the initial data (and with no assumption that scalar curvature is bounded along the flow).
</p>projecteuclid.org/euclid.dmj/1541473293_20181128220332Wed, 28 Nov 2018 22:03 ESTMirror symmetry for the Landau–Ginzburg $A$ -model $M=\mathbb{C}^{n}$ , $W=z_{1}\cdots z_{n}$https://projecteuclid.org/euclid.dmj/1545037298<strong>David Nadler</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 1, 1--84.</p><p><strong>Abstract:</strong><br/>
We calculate the category of branes in the Landau–Ginzburg $A$ -model with background $M=\mathbb{C}^{n}$ and superpotential $W=z_{1}\cdots z_{n}$ in the form of microlocal sheaves along a natural Lagrangian skeleton. Our arguments employ the framework of perverse schobers, and our results confirm expectations from mirror symmetry.
</p>projecteuclid.org/euclid.dmj/1545037298_20190103040203Thu, 03 Jan 2019 04:02 ESTCohomologically induced distinguished representations and cohomological test vectorshttps://projecteuclid.org/euclid.dmj/1541646042<strong>Binyong Sun</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 1, 85--126.</p><p><strong>Abstract:</strong><br/>
Let $G$ be a real reductive group, and let $\chi$ be a character of a reductive subgroup $H$ of $G$ . We construct $\chi$ -invariant linear functionals on certain cohomologically induced representations of $G$ , and we show that these linear functionals do not vanish on the bottom layers. Applying this construction, we prove two Archimedean nonvanishing hypotheses which are vital to the arithmetic study of special values of certain $L$ -functions via modular symbols.
</p>projecteuclid.org/euclid.dmj/1541646042_20190103040203Thu, 03 Jan 2019 04:02 ESTArithmetic theta lifts and the arithmetic Gan–Gross–Prasad conjecture for unitary groupshttps://projecteuclid.org/euclid.dmj/1545037299<strong>Hang Xue</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 1, 127--185.</p><p><strong>Abstract:</strong><br/>
We propose a precise formula relating the height of certain diagonal cycles on the product of unitary Shimura varieties and the central derivative of some tensor product $L$ -functions. This can be viewed as a refinement of the arithmetic Gan–Gross–Prasad conjecture. We use the theory of arithmetic theta lifts to prove some endoscopic cases of it for $\operatorname{U}(2)\times\operatorname{U}(3)$ .
</p>projecteuclid.org/euclid.dmj/1545037299_20190103040203Thu, 03 Jan 2019 04:02 ESTOn the rationality problem for quadric bundleshttps://projecteuclid.org/euclid.dmj/1541646041<strong>Stefan Schreieder</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 2, 187--223.</p><p><strong>Abstract:</strong><br/>
We classify all positive integers $n$ and $r$ such that (stably) nonrational complex $r$ -fold quadric bundles over rational $n$ -folds exist. We show in particular that, for any $n$ and $r$ , a wide class of smooth $r$ -fold quadric bundles over $\mathbb{P}^{n}_{\mathbb{C}}$ are not stably rational if $r\in[2^{n-1}-1,2^{n}-2]$ . In our proofs we introduce a generalization of the specialization method of Voisin and of Colliot-Thélène and Pirutka which avoids universally $\mathrm{CH}_{0}$ -trivial resolutions of singularities.
</p>projecteuclid.org/euclid.dmj/1541646041_20190122220240Tue, 22 Jan 2019 22:02 ESTLegendrian fronts for affine varietieshttps://projecteuclid.org/euclid.dmj/1547110820<strong>Roger Casals</strong>, <strong>Emmy Murphy</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 2, 225--323.</p><p><strong>Abstract:</strong><br/>
In this article we study Weinstein structures endowed with a Lefschetz fibration in terms of the Legendrian front projection. First, we provide a systematic recipe for translating from a Weinstein Lefschetz bifibration to a Legendrian handlebody. Then we present several new applications of this technique to symplectic topology. This includes the detection of flexibility and rigidity for several families of Weinstein manifolds and the existence of closed, exact Lagrangian submanifolds. In particular, we prove that the Koras–Russell cubic is Stein deformation-equivalent to $\mathbb{C}^{3}$ , and we verify the affine parts of the algebraic mirrors of two Weinstein $4$ -folds.
</p>projecteuclid.org/euclid.dmj/1547110820_20190122220240Tue, 22 Jan 2019 22:02 ESTAsymptotics of Chebyshev polynomials, II: DCT subsets of ${\mathbb{R}}$https://projecteuclid.org/euclid.dmj/1547024421<strong>Jacob S. Christiansen</strong>, <strong>Barry Simon</strong>, <strong>Peter Yuditskii</strong>, <strong>Maxim Zinchenko</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 2, 325--349.</p><p><strong>Abstract:</strong><br/>
We prove Szegő–Widom asymptotics for the Chebyshev polynomials of a compact subset of $\mathbb{R}$ which is regular for potential theory and obeys the Parreau–Widom and DCT conditions.
</p>projecteuclid.org/euclid.dmj/1547024421_20190122220240Tue, 22 Jan 2019 22:02 ESTNice triples and the Grothendieck–Serre conjecture concerning principal G-bundles over reductive group schemeshttps://projecteuclid.org/euclid.dmj/1546938026<strong>Ivan Panin</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 2, 351--375.</p><p><strong>Abstract:</strong><br/>
The main result of this article is to reduce a proof of the conjecture to a statement about principal bundles on affine line over a regular local scheme. This reduction is obtained via a theory of nice triples, which goes back to the ideas of Voevodsky. As an application, an unpublished result due to Gabber is proved.
</p>projecteuclid.org/euclid.dmj/1546938026_20190122220240Tue, 22 Jan 2019 22:02 ESTNonabelian Cohen–Lenstra momentshttps://projecteuclid.org/euclid.dmj/1548730815<strong>Melanie Matchett Wood</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 3, 377--427.</p><p><strong>Abstract:</strong><br/>
In this article, we give a conjecture for the average number of unramified $G$ -extensions of a quadratic field for any finite group $G$ . The Cohen–Lenstra heuristics are the specialization of our conjecture to the case in which $G$ is abelian of odd order. We prove a theorem toward the function field analogue of our conjecture and give additional motivations for the conjecture, including the construction of a lifting invariant for the unramified $G$ -extensions that takes the same number of values as the predicted average and an argument using the Malle–Bhargava principle. We note that, for even $|G|$ , corrections for the roots of unity in ${\mathbb{Q}}$ are required, which cannot be seen when $G$ is abelian.
</p>projecteuclid.org/euclid.dmj/1548730815_20190207040155Thu, 07 Feb 2019 04:01 ESTThe class of Eisenbud–Khimshiashvili–Levine is the local $\mathbf{A}^{1}$ -Brouwer degreehttps://projecteuclid.org/euclid.dmj/1547607998<strong>Jesse Leo Kass</strong>, <strong>Kirsten Wickelgren</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 3, 429--469.</p><p><strong>Abstract:</strong><br/>
Given a polynomial function with an isolated zero at the origin, we prove that the local $\mathbf{A}^{1}$ -Brouwer degree equals the Eisenbud–Khimshiashvili–Levine class. This answers a question posed by David Eisenbud in 1978. We give an application to counting nodes, together with associated arithmetic information, by enriching Milnor’s equality between the local degree of the gradient and the number of nodes into which a hypersurface singularity bifurcates to an equality in the Grothendieck–Witt group.
</p>projecteuclid.org/euclid.dmj/1547607998_20190207040155Thu, 07 Feb 2019 04:01 ESTOn the $\ell ^{p}$ -norm of the discrete Hilbert transformhttps://projecteuclid.org/euclid.dmj/1548666102<strong>Rodrigo Bañuelos</strong>, <strong>Mateusz Kwaśnicki</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 3, 471--504.</p><p><strong>Abstract:</strong><br/>
Using a representation of the discrete Hilbert transform in terms of martingales arising from Doob $h$ -processes, we prove that its $\ell ^{p}$ -norm, $1\lt p\lt \infty $ , is bounded above by the $L^{p}$ -norm of the continuous Hilbert transform. Together with the already known lower bound, this resolves the long-standing conjecture that the norms of these operators are equal.
</p>projecteuclid.org/euclid.dmj/1548666102_20190207040155Thu, 07 Feb 2019 04:01 ESTThree combinatorial formulas for type $A$ quiver polynomials and $K$ -polynomialshttps://projecteuclid.org/euclid.dmj/1549270814<strong>Ryan Kinser</strong>, <strong>Allen Knutson</strong>, <strong>Jenna Rajchgot</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 4, 505--551.</p><p><strong>Abstract:</strong><br/>
We provide combinatorial formulas for the multidegree and $K$ -polynomial of an arbitrarily oriented type $A$ quiver locus. These formulas are generalizations of three formulas by Knutson, Miller, and Shimozono from the equioriented setting. In particular, we prove the $K$ -theoretic component formula conjectured by Buch and Rimányi.
</p>projecteuclid.org/euclid.dmj/1549270814_20190221220146Thu, 21 Feb 2019 22:01 ESTMetaplectic covers of Kac–Moody groups and Whittaker functionshttps://projecteuclid.org/euclid.dmj/1549594819<strong>Manish M. Patnaik</strong>, <strong>Anna Puskás</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 4, 553--653.</p><p><strong>Abstract:</strong><br/>
Starting from some linear algebraic data (a Weyl group invariant bilinear form) and some arithmetic data (a bilinear Steinberg symbol), we construct a central extension of a Kac–Moody group generalizing the work of Matsumoto. Specializing our construction over non-Archimedean local fields, for each positive integer $n$ we obtain the notion of $n$ -fold metaplectic covers of Kac–Moody groups. In this setting, we prove a Casselman–Shalika-type formula for Whittaker functions.
</p>projecteuclid.org/euclid.dmj/1549594819_20190221220146Thu, 21 Feb 2019 22:01 ESTNon-LERFness of arithmetic hyperbolic manifold groups and mixed $3$ -manifold groupshttps://projecteuclid.org/euclid.dmj/1549270813<strong>Hongbin Sun</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 4, 655--696.</p><p><strong>Abstract:</strong><br/>
We will show that for any noncompact arithmetic hyperbolic $m$ -manifold with ${m\gt 3}$ , and any compact arithmetic hyperbolic $m$ -manifold with $m\gt 4$ that is not a $7$ -dimensional one defined by octonions, its fundamental group is not locally extended residually finite (LERF). The main ingredient in the proof is a study on abelian amalgamations of hyperbolic $3$ -manifold groups. We will also show that a compact orientable irreducible $3$ -manifold with empty or tori boundary supports a geometric structure if and only if its fundamental group is LERF.
</p>projecteuclid.org/euclid.dmj/1549270813_20190221220146Thu, 21 Feb 2019 22:01 ESTOn topological and measurable dynamics of unipotent frame flows for hyperbolic manifoldshttps://projecteuclid.org/euclid.dmj/1549392546<strong>François Maucourant</strong>, <strong>Barbara Schapira</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 4, 697--747.</p><p><strong>Abstract:</strong><br/>
We study the dynamics of unipotent flows on frame bundles of hyperbolic manifolds of infinite volume. We prove that they are topologically transitive and that the natural invariant measure, the so-called Burger–Roblin measure, is ergodic, as soon as the geodesic flow admits a finite measure of maximal entropy, and this entropy is strictly greater than the codimension of the unipotent flow inside the maximal unipotent flow. The latter result generalizes a theorem of Mohammadi and Oh.
</p>projecteuclid.org/euclid.dmj/1549392546_20190221220146Thu, 21 Feb 2019 22:01 ESTOn the polynomial Szemerédi theorem in finite fieldshttps://projecteuclid.org/euclid.dmj/1548990127<strong>Sarah Peluse</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 5, 749--774.</p><p><strong>Abstract:</strong><br/>
Let $P_{1},\dots,P_{m}\in\mathbb{Z}[y]$ be any linearly independent polynomials with zero constant term. We show that there exists $\gamma\gt 0$ such that any subset of $\mathbb{F}_{q}$ of size at least $q^{1-\gamma}$ contains a nontrivial polynomial progression $x,x+P_{1}(y),\dots,x+P_{m}(y)$ , provided that the characteristic of $\mathbb{F}_{q}$ is large enough.
</p>projecteuclid.org/euclid.dmj/1548990127_20190321040049Thu, 21 Mar 2019 04:00 EDTA gradient estimate for nonlocal minimal graphshttps://projecteuclid.org/euclid.dmj/1551841227<strong>Xavier Cabré</strong>, <strong>Matteo Cozzi</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 5, 775--848.</p><p><strong>Abstract:</strong><br/>
We consider the class of measurable functions defined in all of $\mathbb{R}^{n}$ that give rise to a nonlocal minimal graph over a ball of $\mathbb{R}^{n}$ . We establish that the gradient of any such function is bounded in the interior of the ball by a power of its oscillation. This estimate, together with previously known results, leads to the $C^{\infty}$ regularity of the function in the ball. While the smoothness of nonlocal minimal graphs was known for $n=1,2$ —but without a quantitative bound—in higher dimensions only their continuity had been established. To prove the gradient bound, we show that the normal to a nonlocal minimal graph is a supersolution of a truncated fractional Jacobi operator, for which we prove a weak Harnack inequality. To this end, we establish a new universal fractional Sobolev inequality on nonlocal minimal surfaces. Our estimate provides an extension to the fractional setting of the celebrated gradient bounds of Finn and of Bombieri, De Giorgi, and Miranda for solutions of the classical mean curvature equation.
</p>projecteuclid.org/euclid.dmj/1551841227_20190321040049Thu, 21 Mar 2019 04:00 EDTCongruences of $5$ -secant conics and the rationality of some admissible cubic fourfoldshttps://projecteuclid.org/euclid.dmj/1551754840<strong>Francesco Russo</strong>, <strong>Giovanni Staglianò</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 5, 849--865.</p><p><strong>Abstract:</strong><br/>
The works of Hassett and Kuznetsov identify countably many divisors $C_{d}$ in the open subset of ${\mathbb{P}}^{55}={\mathbb{P}}(H^{0}(\mathcal{O}_{{\mathbb{P}}^{5}}(3)))$ parameterizing all cubic fourfolds and conjecture that the cubics corresponding to these divisors are precisely the rational ones. Rationality has been known classically for the first family $C_{14}$ . We use congruences of $5$ -secant conics to prove rationality for the first three of the families $C_{d}$ , corresponding to $d=14,26,38$ in Hassett’s notation.
</p>projecteuclid.org/euclid.dmj/1551754840_20190321040049Thu, 21 Mar 2019 04:00 EDTSchwarzian derivatives, projective structures, and the Weil–Petersson gradient flow for renormalized volumehttps://projecteuclid.org/euclid.dmj/1551495708<strong>Martin Bridgeman</strong>, <strong>Jeffrey Brock</strong>, <strong>Kenneth Bromberg</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 5, 867--896.</p><p><strong>Abstract:</strong><br/>
To a complex projective structure $\Sigma $ on a surface, Thurston associates a locally convex pleated surface. We derive bounds on the geometry of both in terms of the norms $\|\phi _{\Sigma }\|_{\infty }$ and $\|\phi _{\Sigma }\|_{2}$ of the quadratic differential $\phi _{\Sigma }$ of $\Sigma $ given by the Schwarzian derivative of the associated locally univalent map. We show that these give a unifying approach that generalizes a number of important, well-known results for convex cocompact hyperbolic structures on $3$ -manifolds, including bounds on the Lipschitz constant for the nearest-point retraction and the length of the bending lamination. We then use these bounds to begin a study of the Weil–Petersson gradient flow of renormalized volume on the space $\operatorname{CC}(N)$ of convex cocompact hyperbolic structures on a compact manifold $N$ with incompressible boundary, leading to a proof of the conjecture that the renormalized volume has infimum given by one half the simplicial volume of $\mathit{DN}$ , the double of $N$ .
</p>projecteuclid.org/euclid.dmj/1551495708_20190321040049Thu, 21 Mar 2019 04:00 EDTApproximation theorems for parabolic equations and movement of local hot spotshttps://projecteuclid.org/euclid.dmj/1551495707<strong>Alberto Enciso</strong>, <strong>MªÁngeles García-Ferrero</strong>, <strong>Daniel Peralta-Salas</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 5, 897--939.</p><p><strong>Abstract:</strong><br/>
We prove a global approximation theorem for a general parabolic operator $L$ , which asserts that if $v$ satisfies the equation $Lv=0$ in a space-time region $\Omega\subset\mathbb{R}^{n+1}$ satisfying a certain necessary topological condition, then it can be approximated in a Hölder norm by a global solution $u$ to the equation. If $\Omega$ is compact and $L$ is the usual heat operator, then one can instead approximate the local solution $v$ by the unique solution that falls off at infinity to the Cauchy problem with a suitably chosen smooth, compactly supported initial datum. We then apply these results to prove the existence of global solutions to the equation $Lu=0$ with a local hot spot that moves along a prescribed curve for all time, up to a uniformly small error. Global solutions that exhibit isothermic hypersurfaces of prescribed topologies for all times and applications to the heat equation on the flat torus are also discussed.
</p>projecteuclid.org/euclid.dmj/1551495707_20190321040049Thu, 21 Mar 2019 04:00 EDT