Mostrando recursos 1 - 20 de 27

  1. Global solutions of the gravity-capillary water-wave system in three dimensions

    Deng, Yu; Ionescu, Alexandru D.; Pausader, Benoît; Pusateri, Fabio
    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...

  2. Gauduchon metrics with prescribed volume form

    Székelyhidi, Gábor; Tosatti, Valentino; Weinkove, Ben
    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.

  3. Quantum indices and refined enumeration of real plane curves

    Mikhalkin, Grigory
    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.

  4. Singular Ricci flows I

    Kleiner, Bruce; Lott, John
    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.

  5. Bernstein- and Markov-type inequalities for rational functions

    Kalmykov, Sergei; Nagy, Béla; Totik, Vilmos
    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.

  6. Correction to “On the density of geometrically finite Kleinian groups”

    Brock, Jeffrey F.; Bromberg, Kenneth W.
    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].

  7. Asymptotic behavior of flows by powers of the Gaussian curvature

    Brendle, Simon; Choi, Kyeongsu; Daskalopoulos, Panagiota
    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.

  8. The tempered spectrum of a real spherical space

    Knop, Friedrich; Krötz, Bernhard; Schlichtkrull, Henrik
    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.

  9. Enumeration of points, lines, planes, etc.

    Huh, June; Wang, Botong
    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...

  10. Hitchin characters and geodesic laminations

    Bonahon, Francis; Dreyer, Guillaume
    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.

  11. Equivariant Dirac operators and differentiable geometric invariant theory

    Paradan, Paul-Emile; Vergne, Michèle

  12. Equivariant Dirac operators and differentiable geometric invariant theory

    Paradan, Paul-Emile; Vergne, Michèle
    In this paper, we give a geometric expression for the multiplicities of the equivariant index of a spin$^c$ Dirac operator.

  13. A hyper-Kähler compactification of the intermediate Jacobian fibration associated with a cubic 4-fold

    Laza, Radu; Saccà, Giulia; Voisin, Claire

  14. A hyper-Kähler compactification of the intermediate Jacobian fibration associated with a cubic 4-fold

    Laza, Radu; Saccà, Giulia; Voisin, Claire
    Let $X$ be a general cubic $4$-fold. It was observed by Donagi and Markman that the relative intermediate Jacobian fibration $\mathcal{J}_U/U$ (with $U=(\mathbb{P}^5)^\vee\setminus X^\vee$) associated with the family of smooth hyperplane sections of $X$ carries a natural holomorphic symplectic form making the fibration Lagrangian. In this paper, we obtain a smooth projective compactification $\overline{\mathcal{J}}$ of $\mathcal{J}_U$ with the property that the holomorphic symplectic form on $\mathcal{J}_U$ extends to a holomorphic symplectic form on $\overline{\mathcal{J}}$. In particular, $\overline{\mathcal{J}}$ is a $10$-dimensional compact hyper-Kähler manifold, which we show to be deformation equivalent to the exceptional example of O'Grady. This proves a conjecture...

  15. Tits geometry and positive curvature

    Fang, Fuquan; Grove, Karsten; Thorbergsson, Gudlaugur

  16. Tits geometry and positive curvature

    Fang, Fuquan; Grove, Karsten; Thorbergsson, Gudlaugur
    There is a well-known link between (maximal) polar representations and isotropy representations of symmetric spaces provided by Dadok. Moreover, the theory by Tits and Burns–Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least $3$ and irreducible topological spherical buildings of rank at least $3$. ¶ We discover and exploit a rich structure of a (connected) chamber system of finite (Coxeter) type $\mathsf{M}$ associated with any polar action of cohomogeneity at least $2$ on any simply connected closed positively curved manifold. Although this chamber system is typically not a Tits geometry of type $\mathsf{M}$, its...

  17. Local Hodge theory of Soergel bimodules

    Williamson, Geordie
    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.

  18. Maximum independent sets on random regular graphs

    Ding, Jian; Sly, Allan; Sun, Nike
    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.

  19. Global bifurcation of steady gravity water waves with critical layers

    Constantin, Adrian; Strauss, Walter; Vărvărucă, Eugen
    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.

  20. Approximation of the Dirichlet problem on a half space

    Schaeffer, David G.

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