Mostrando recursos 1 - 20 de 30

  1. The $C_0$-inextendibility of the Schwarzschild spacetime and the spacelike diameter in Lorentzian geometry

    Sbierski, Jan
    The maximal analytic Schwarzschild spacetime is manifestly inextendible as a Lorentzian manifold with a twice continuously differentiable metric. In this paper, we prove the stronger statement that it is even inextendible as a Lorentzian manifold with a continuous metric. To capture the obstruction to continuous extensions through the curvature singularity, we introduce the notion of the spacelike diameter of a globally hyperbolic region of a Lorentzian manifold with a merely continuous metric and give a sufficient condition for the spacelike diameter to be finite. The investigation of low-regularity inextendibility criteria is motivated by the strong cosmic censorship conjecture.

  2. Resolution of the canonical fiber metrics for a Lefschetz fibration

    Melrose, Richard; Zhu, Xuwen
    We consider the family of constant curvature fiber metrics for a Lefschetz fibration with regular fibers of genus greater than one. A result of Obitsu and Wolpert is refined by showing that on an appropriate resolution of the total space, constructed by iterated blow-up, this family is log-smooth, i.e., polyhomogeneous with integral powers but possible multiplicities, at the preimage of the singular fibers in terms of parameters of size comparable to the logarithm of the length of the shrinking geodesic.

  3. Exotic hyperbolic gluings

    Chruściel, Piotr T.; Delay, Erwann
    We carry out “exotic gluings” à la Carlotto–Schoen for asymptotically hyperbolic general relativistic initial data sets. In particular, we obtain a direct construction of non-trivial initial data sets which are exactly hyperbolic in large regions extending to conformal infinity.

  4. Green function, Painlevé VI equation, and Eisenstein series of weight one

    Chen, Zhijie; Kuo, Ting-Jung; Lin, Chang-Shou; Wang, Chin-Lung
    The behavior and the location of singular points of a solution to Painlevé VI equation could encode important geometric properties. For example, Hitchin’s formula indicates that singular points of algebraic solutions are exactly the zeros of Eisenstein series of weight one. In this paper, we study the problem: How many singular points of a solution $\lambda (t)$ to the Painlevé VI equation with parameter $(\frac{1}{8}, \frac{-1}{8}, \frac{1}{8}, \frac{3}{8})$ might have in $\mathbb{C} \setminus \lbrace 0, 1\rbrace$? Here $t_0 \in \mathbb{C} \setminus \lbrace 0, 1\rbrace$ is called a singular point of $\lambda (t)$ if $\lambda (t_0) \in \lbrace 0, 1, t_0,...

  5. Teichmüller harmonic map flow into nonpositively curved targets

    Rupflin, Melanie; Topping, Peter M.
    The Teichmüller harmonic map flow deforms both a map from an oriented closed surface $M$ into an arbitrary closed Riemannian manifold, and a constant curvature metric on $M$, so as to reduce the energy of the map as quickly as possible [16]. The flow then tries to converge to a branched minimal immersion when it can [16, 18]. The only thing that can stop the flow is a finite-time degeneration of the metric on $M$ where one or more collars are pinched. In this paper we show that finite-time degeneration cannot happen in the case that the target has nonpositive...

  6. Teichmüller harmonic map flow into nonpositively curved targets

    Rupflin, Melanie; Topping, Peter M.
    The Teichmüller harmonic map flow deforms both a map from an oriented closed surface $M$ into an arbitrary closed Riemannian manifold, and a constant curvature metric on $M$, so as to reduce the energy of the map as quickly as possible [16]. The flow then tries to converge to a branched minimal immersion when it can [16, 18]. The only thing that can stop the flow is a finite-time degeneration of the metric on $M$ where one or more collars are pinched. In this paper we show that finite-time degeneration cannot happen in the case that the target has nonpositive...

  7. Teichmüller harmonic map flow into nonpositively curved targets

    Rupflin, Melanie; Topping, Peter M.
    The Teichmüller harmonic map flow deforms both a map from an oriented closed surface $M$ into an arbitrary closed Riemannian manifold, and a constant curvature metric on $M$, so as to reduce the energy of the map as quickly as possible [16]. The flow then tries to converge to a branched minimal immersion when it can [16, 18]. The only thing that can stop the flow is a finite-time degeneration of the metric on $M$ where one or more collars are pinched. In this paper we show that finite-time degeneration cannot happen in the case that the target has nonpositive...

  8. Min–max for phase transitions and the existence of embedded minimal hypersurfaces

    Guaraco, Marco A. M.
    Strong parallels can be drawn between the theory of minimal hypersurfaces and the theory of phase transitions. Borrowing ideas from the former we extend recent results on the regularity of stable phase transition interfaces to the finite Morse index case. As an application we present a PDE-based proof of the celebrated theorem of Almgren–Pitts, on the existence of embedded minimal hypersurfaces in compact manifolds. We compare our results with other min–max theories.

  9. Min–max for phase transitions and the existence of embedded minimal hypersurfaces

    Guaraco, Marco A. M.
    Strong parallels can be drawn between the theory of minimal hypersurfaces and the theory of phase transitions. Borrowing ideas from the former we extend recent results on the regularity of stable phase transition interfaces to the finite Morse index case. As an application we present a PDE-based proof of the celebrated theorem of Almgren–Pitts, on the existence of embedded minimal hypersurfaces in compact manifolds. We compare our results with other min–max theories.

  10. Min–max for phase transitions and the existence of embedded minimal hypersurfaces

    Guaraco, Marco A. M.
    Strong parallels can be drawn between the theory of minimal hypersurfaces and the theory of phase transitions. Borrowing ideas from the former we extend recent results on the regularity of stable phase transition interfaces to the finite Morse index case. As an application we present a PDE-based proof of the celebrated theorem of Almgren–Pitts, on the existence of embedded minimal hypersurfaces in compact manifolds. We compare our results with other min–max theories.

  11. Deformations of semisimple Poisson pencils of hydrodynamic type are unobstructed

    Carlet, Guido; Posthuma, Hessel; Shadrin, Sergey
    We prove that the bihamiltonian cohomology of a semisimple pencil of Poisson brackets of hydrodynamic type vanishes for almost all degrees. This implies the existence of a full dispersive deformation of a semisimple bihamiltonian structure of hydrodynamic type starting from any infinitesimal deformation.

  12. Deformations of semisimple Poisson pencils of hydrodynamic type are unobstructed

    Carlet, Guido; Posthuma, Hessel; Shadrin, Sergey
    We prove that the bihamiltonian cohomology of a semisimple pencil of Poisson brackets of hydrodynamic type vanishes for almost all degrees. This implies the existence of a full dispersive deformation of a semisimple bihamiltonian structure of hydrodynamic type starting from any infinitesimal deformation.

  13. Deformations of semisimple Poisson pencils of hydrodynamic type are unobstructed

    Carlet, Guido; Posthuma, Hessel; Shadrin, Sergey
    We prove that the bihamiltonian cohomology of a semisimple pencil of Poisson brackets of hydrodynamic type vanishes for almost all degrees. This implies the existence of a full dispersive deformation of a semisimple bihamiltonian structure of hydrodynamic type starting from any infinitesimal deformation.

  14. Non-existence of time-periodic vacuum space-times

    Alexakis, Spyros; Schlue, Volker
    We prove that smooth asymptotically flat solutions to the Einstein vacuum equations which are assumed to be periodic in time, are in fact stationary in a neighborhood of infinity. Our result applies under physically relevant regularity assumptions purely at the level of the initial data. In particular, our work removes the assumption of analyticity up to null infinity in Bičák, Scholtz, and Tod (2010). The proof relies on extending a suitably constructed “candidate” Killing vector field from null infinity, via Carleman-type estimates obtained in Alexakis, Schlue, and Shao (2013).

  15. Non-existence of time-periodic vacuum space-times

    Alexakis, Spyros; Schlue, Volker
    We prove that smooth asymptotically flat solutions to the Einstein vacuum equations which are assumed to be periodic in time, are in fact stationary in a neighborhood of infinity. Our result applies under physically relevant regularity assumptions purely at the level of the initial data. In particular, our work removes the assumption of analyticity up to null infinity in Bičák, Scholtz, and Tod (2010). The proof relies on extending a suitably constructed “candidate” Killing vector field from null infinity, via Carleman-type estimates obtained in Alexakis, Schlue, and Shao (2013).

  16. Non-existence of time-periodic vacuum space-times

    Alexakis, Spyros; Schlue, Volker
    We prove that smooth asymptotically flat solutions to the Einstein vacuum equations which are assumed to be periodic in time, are in fact stationary in a neighborhood of infinity. Our result applies under physically relevant regularity assumptions purely at the level of the initial data. In particular, our work removes the assumption of analyticity up to null infinity in Bičák, Scholtz, and Tod (2010). The proof relies on extending a suitably constructed “candidate” Killing vector field from null infinity, via Carleman-type estimates obtained in Alexakis, Schlue, and Shao (2013).

  17. Index to Volume 107


  18. An extension of a theorem of Wu–Yau

    Tosatti, Valentino; Yang, Xiaokui
    We show that a compact Kähler manifold with nonpositive holomorphic sectional curvature has nef canonical bundle. If the holomorphic sectional curvature is negative then it follows that the canonical bundle is ample, confirming a conjecture of Yau. The key ingredient is the recent solution of this conjecture in the projective case by Wu–Yau.

  19. Isoperimetric inequality for the third eigenvalue of the Laplace–Beltrami operator on $\mathbb{S}^2$

    Nadirashvili, Nikolai; Sire, Yannick
    We prove Hersch’s type isoperimetric inequality for the third positive eigenvalue on $\mathbb{S}^2$. Our method builds on the theory we developed to construct extremal metrics on Riemannian surfaces in conformal classes for any eigenvalue.

  20. Barycenter technique and the Riemann mapping problem of Cherrier–Escobar

    Mayer, Martin; Ndiaye, Cheikh Birahim
    We solve in the affirmative the remaining cases of the Riemann mapping problem of Cherrier–Escobar [35][38] first raised by Cherrier [35] in 1984. Indeed, performing a suitable scheme of the barycenter technique of Bahri–Coron [14] via the Almaraz–Chen’s bubbles [3][34], we completely solve all the cases left open after the work of Chen [34]. Hence, combining our work with the ones of Almaraz [2], Chen [34], Cherrier [35], Escobar [38][40] and Marques [55][56], we have that every compact Riemannian manifold with boundary, of dimension greater or equal than three, and with finite Sobolev quotient, carries a conformal scalar flat metric...

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