Acta Mathematica Articles (Project Euclid)
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The latest articles from Acta Mathematica on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2017 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 17 Aug 2017 13:00 EDTThu, 17 Aug 2017 13:00 EDThttps://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Global bifurcation of steady gravity water waves with critical layers
https://projecteuclid.org/euclid.acta/1502989201
<strong>Adrian Constantin</strong>, <strong>Walter Strauss</strong>, <strong>Eugen Vărvărucă</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 217, Number 2, 195--262.</p><p><strong>Abstract:</strong><br/>
We construct families of two-dimensional travelling water waves propagating under the influence of gravity in a flow of constant vorticity over a flat bed, in particular establishing the existence of waves of large amplitude. A Riemann–Hilbert problem approach is used to recast the governing equations as a one-dimensional elliptic pseudodifferential equation with a scalar constraint. The structural properties of this formulation, which arises as the Euler–Lagrange equation of an energy functional, enable us to develop a theory of analytic global bifurcation.
</p>projecteuclid.org/euclid.acta/1502989201_20170817130019Thu, 17 Aug 2017 13:00 EDTMaximum independent sets on random regular graphs
https://projecteuclid.org/euclid.acta/1502989202
<strong>Jian Ding</strong>, <strong>Allan Sly</strong>, <strong>Nike Sun</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 217, Number 2, 263--340.</p><p><strong>Abstract:</strong><br/>
We determine the asymptotics of the independence number of the random d -regular graph for all ${d\geq d_0}$ . It is highly concentrated, with constant-order fluctuations around ${n\alpha_*-c_*\log n}$ for explicit constants ${\alpha_*(d)}$ and ${c_*(d)}$ . Our proof rigorously confirms the one-step replica symmetry breaking heuristics for this problem, and we believe the techniques will be more broadly applicable to the study of other combinatorial properties of random graphs.
</p>projecteuclid.org/euclid.acta/1502989202_20170817130019Thu, 17 Aug 2017 13:00 EDTLocal Hodge theory of Soergel bimodules
https://projecteuclid.org/euclid.acta/1502989203
<strong>Geordie Williamson</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 217, Number 2, 341--404.</p><p><strong>Abstract:</strong><br/>
We prove the local hard Lefschetz theorem and local Hodge–Riemann bilinear relations for Soergel bimodules. Using results of Soergel and Kübel, one may deduce an algebraic proof of the Jantzen conjectures. We observe that the Jantzen filtration may depend on the choice of non-dominant regular deformation direction.
</p>projecteuclid.org/euclid.acta/1502989203_20170817130019Thu, 17 Aug 2017 13:00 EDTTits geometry and positive curvaturehttps://projecteuclid.org/euclid.acta/1505401886<strong>Fuquan Fang</strong>, <strong>Karsten Grove</strong>, <strong>Gudlaugur Thorbergsson</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 218, Number 1, 1--53.</p>projecteuclid.org/euclid.acta/1505401886_20170914111134Thu, 14 Sep 2017 11:11 EDTA hyper-Kähler compactification of the intermediate Jacobian fibration associated with a cubic 4-foldhttps://projecteuclid.org/euclid.acta/1505401887<strong>Radu Laza</strong>, <strong>Giulia Saccà</strong>, <strong>Claire Voisin</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 218, Number 1, 55--135.</p>projecteuclid.org/euclid.acta/1505401887_20170914111134Thu, 14 Sep 2017 11:11 EDTEquivariant Dirac operators and differentiable geometric invariant theoryhttps://projecteuclid.org/euclid.acta/1505401888<strong>Paul-Emile Paradan</strong>, <strong>Michèle Vergne</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 218, Number 1, 137--199.</p>projecteuclid.org/euclid.acta/1505401888_20170914111134Thu, 14 Sep 2017 11:11 EDTHitchin characters and geodesic laminationshttps://projecteuclid.org/euclid.acta/1517426684<strong>Francis Bonahon</strong>, <strong>Guillaume Dreyer</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 218, Number 2, 201--295.</p><p><strong>Abstract:</strong><br/>
For a closed surface $S$, the Hitchin component $\mathrm{Hit}_n (S)$ is a preferred component of the character variety consisting of group homomorphisms from the fundamental group $\pi_1(S)$ to the Lie group $\mathrm{PSL}_n (\mathbb{R})$. We construct a parametrization of the Hitchin component that is well-adapted to a geodesic lamination $\lambda$ on the surface. This is a natural extension of Thurston’s parametrization of the Teichmüller space $\mathcal{T}(S)$ by shearing coordinates associated with $\lambda$, corresponding to the case $n=2$. However, significantly new ideas are needed in this higher-dimensional case. The article concludes with a few applications.
</p>projecteuclid.org/euclid.acta/1517426684_20180131142450Wed, 31 Jan 2018 14:24 ESTEnumeration of points, lines, planes, etc.https://projecteuclid.org/euclid.acta/1517426685<strong>June Huh</strong>, <strong>Botong Wang</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 218, Number 2, 297--317.</p><p><strong>Abstract:</strong><br/>
One of the earliest results in enumerative combinatorial geometry is the following theorem of de Bruijn and Erdős: Every set of points $E$ in a projective plane determines at least $\lvert E \rvert$ lines, unless all the points are contained in a line. The result was extended to higher dimensions by Motzkin and others, who showed that every set of points $E$ in a projective space determines at least $\lvert E \rvert$ hyperplanes, unless all the points are contained in a hyperplane. Let $E$ be a spanning subset of an $r$-dimensional vector space. We show that, in the partially ordered set of subspaces spanned by subsets of $E$, there are at least as many $(r-p)$-dimensional subspaces as there are $p$-dimensional subspaces, for every $p$ at most $\frac{1}{2} r$. This confirms the “top-heavy” conjecture by Dowling and Wilson for all matroids realizable over some field. The proof relies on the decomposition theorem package for $\ell$-adic intersection complexes.
</p>projecteuclid.org/euclid.acta/1517426685_20180131142450Wed, 31 Jan 2018 14:24 ESTThe tempered spectrum of a real spherical spacehttps://projecteuclid.org/euclid.acta/1517426686<strong>Friedrich Knop</strong>, <strong>Bernhard Krötz</strong>, <strong>Henrik Schlichtkrull</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 218, Number 2, 319--383.</p><p><strong>Abstract:</strong><br/>
Let $G/H$ be a unimodular real spherical space which is either absolutely spherical, i.e. the real form of a complex spherical space, or of wave-front type. It is shown that every tempered representation for $G/H$ embeds into a twisted discrete series for a boundary degeneration of $G/H$. If $G/H$ is of wave-front type it follows that the tempered representation is parabolically induced by a twisted discrete series representation for a real spherical space formed by a Levi subgroup.
</p>projecteuclid.org/euclid.acta/1517426686_20180131142450Wed, 31 Jan 2018 14:24 ESTAsymptotic behavior of flows by powers of the Gaussian curvaturehttps://projecteuclid.org/euclid.acta/1517430210<strong>Simon Brendle</strong>, <strong>Kyeongsu Choi</strong>, <strong>Panagiota Daskalopoulos</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 219, Number 1, 1--16.</p><p><strong>Abstract:</strong><br/>
We consider a $1$-parameter family of strictly convex hypersurfaces in $\mathbb{R}^{n+1}$ moving with speed $-K^{\alpha} ν$, where ν denotes the outward-pointing unit normal vector and $\alpha \geqslant 1 / (n+2)$. For $\alpha \gt 1 / (n+2)$, we show that the flow converges to a round sphere after rescaling. In the affine invariant case $\alpha = 1 / (n+2)$, our arguments give an alternative proof of the fact that the flow converges to an ellipsoid after rescaling.
</p>projecteuclid.org/euclid.acta/1517430210_20180131152337Wed, 31 Jan 2018 15:23 ESTCorrection to “On the density of geometrically finite Kleinian groups”https://projecteuclid.org/euclid.acta/1517430211<strong>Jeffrey F. Brock</strong>, <strong>Kenneth W. Bromberg</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 219, Number 1, 17--19.</p><p><strong>Abstract:</strong><br/>
This erratum corrects an error arising in the proof of Proposition 6.4 in the article “On the density of geometrically finite Kleinian groups” [Brock, J. F. & Bromberg, K. W., Acta Math. , 192 (2004), 33–93].
</p>projecteuclid.org/euclid.acta/1517430211_20180131152337Wed, 31 Jan 2018 15:23 ESTBernstein- and Markov-type inequalities for rational functionshttps://projecteuclid.org/euclid.acta/1517430212<strong>Sergei Kalmykov</strong>, <strong>Béla Nagy</strong>, <strong>Vilmos Totik</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 219, Number 1, 21--63.</p><p><strong>Abstract:</strong><br/>
Asymptotically sharp Bernstein- and Markov-type inequalities are established for rational functions on $C^2$ smooth Jordan curves and arcs. The results are formulated in terms of the normal derivatives of certain Green’s functions with poles at the poles of the rational functions in question. As a special case (when all the poles are at infinity) the corresponding results for polynomials are recaptured.
</p>projecteuclid.org/euclid.acta/1517430212_20180131152337Wed, 31 Jan 2018 15:23 ESTSingular Ricci flows Ihttps://projecteuclid.org/euclid.acta/1517430213<strong>Bruce Kleiner</strong>, <strong>John Lott</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 219, Number 1, 65--134.</p><p><strong>Abstract:</strong><br/> We introduce singular Ricci flows, which are Ricci flow spacetimes subject to certain asymptotic conditions. These provide a solution to the long-standing problem of finding a good notion of Ricci flow through singularities, in the $3$-dimensional case. We prove that Ricci flow with surgery, starting from a fixed initial condition, subconverges to a singular Ricci flow as the surgery parameter tends to zero. We establish a number of geometric and analytical properties of singular Ricci flows. </p>projecteuclid.org/euclid.acta/1517430213_20180131152337Wed, 31 Jan 2018 15:23 ESTQuantum indices and refined enumeration of real plane curveshttps://projecteuclid.org/euclid.acta/1517430214<strong>Grigory Mikhalkin</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 219, Number 1, 135--180.</p><p><strong>Abstract:</strong><br/>
We associate a half-integer number, called the quantum index, to algebraic curves in the real plane satisfying to certain conditions. The area encompassed by the logarithmic image of such curves is equal to $\pi^2$ times the quantum index of the curve, and thus has a discrete spectrum of values. We use the quantum index to refine enumeration of real rational curves in a way consistent with the Block–Göttsche invariants from tropical enumerative geometry.
</p>projecteuclid.org/euclid.acta/1517430214_20180131152337Wed, 31 Jan 2018 15:23 ESTGauduchon metrics with prescribed volume formhttps://projecteuclid.org/euclid.acta/1517430215<strong>Gábor Székelyhidi</strong>, <strong>Valentino Tosatti</strong>, <strong>Ben Weinkove</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 219, Number 1, 181--211.</p><p><strong>Abstract:</strong><br/>
We prove that on any compact complex manifold one can find Gauduchon metrics with prescribed volume form. This is equivalent to prescribing the Chern–Ricci curvature of the metrics, and thus solves a conjecture of Gauduchon from 1984.
</p>projecteuclid.org/euclid.acta/1517430215_20180131152337Wed, 31 Jan 2018 15:23 ESTGlobal solutions of the gravity-capillary water-wave system in three dimensionshttps://projecteuclid.org/euclid.acta/1525299701<strong>Yu Deng</strong>, <strong>Alexandru D. Ionescu</strong>, <strong>Benoît Pausader</strong>, <strong>Fabio Pusateri</strong>. <p><strong>Source: </strong>Acta Mathematica, Volume 219, Number 2, 213--402.</p><p><strong>Abstract:</strong><br/>
In this paper we prove global regularity for the full water-wave system in three dimensions for small data, under the influence of both gravity and surface tension. This problem presents essential difficulties which were absent in all of the earlier global regularity results for other water-wave models.
To construct global solutions, we use a combination of energy estimates and matching dispersive estimates. There is a significant new difficulty in proving energy estimates in our problem, namely the combination of slow pointwise decay of solutions (no better than ${\lvert t \rvert}^{- 5/6}$) and the presence of a large, codimension-$1$, set of quadratic time-resonances. To deal with such a situation, we propose here a new mechanism, which exploits a non-degeneracy property of the time-resonant hypersurfaces and some special structure of the quadratic part of the non-linearity, connected to the conserved energy of the system.
The dispersive estimates rely on analysis of the Duhamel formula in the Fourier space. The main contributions come from the set of space-time resonances, which is a large set of dimension $1$. To control the corresponding bilinear interactions, we use harmonic analysis techniques, such as orthogonality arguments in the Fourier space and atomic decompositions of functions. Most importantly, we construct and use a refined norm which is well adapted to the geometry of the problem.
</p>projecteuclid.org/euclid.acta/1525299701_20180502182144Wed, 02 May 2018 18:21 EDT