Mostrando recursos 1 - 20 de 41

  1. On realization of tangent cones of homologically area-minimizing compact singular submanifolds

    Zhang, Yongsheng
    We show that every oriented area-minimizing cone in “A Sufficient Criterion for a Cone to be Area-Minimizing” [G.R. Lawlor, Mem. of the Amer. Math. Soc., Vol. 91, 1991] can be realized as a tangent cone at a singular point of some homologically area-minimizing singular compact submanifold.

  2. Nonlocal $s$-minimal surfaces and Lawson cones

    Dávila, Juan; del Pino, Manuel; Wei, Juncheng
    The nonlocal $s$-fractional minimal surface equation for $\Sigma = \partial E$ where $E$ is an open set in $\mathbb{R}^N$ is given by ¶ \[ H_\Sigma^ s (p) := \int_{\mathbb{R}^N} \frac{\chi_E(x) - \chi_{E^c}(x)} {{\lvert x-p \rvert}^{N+s}}\, dx \ =\ 0 \; \textrm{for all} \; p\in\Sigma \textrm{ .} \] ¶ Here $0 \lt s \lt 1 , \chi$ designates characteristic function, and the integral is understood in the principal value sense. The classical notion of minimal surface is recovered by letting $s \to 1$. In this paper we exhibit the first concrete examples (beyond the plane) of nonlocal $s$-minimal surfaces. When $s$ is close...

  3. K-semistability for irregular Sasakian manifolds

    Collins, Tristan C.; Székelyhidi, Gábor
    We introduce a notion of K-semistability for Sasakian manifolds. This extends to the irregular case of the orbifold K-semistability of Ross–Thomas. Our main result is that a Sasakian manifold with constant scalar curvature is necessarily K-semistable. As an application, we show how one can recover the volume minimization results of Martelli–Sparks–Yau, and the Lichnerowicz obstruction of Gauntlett–Martelli–Sparks–Yau from this point of view.

  4. Rigidity of equality of Lyapunov exponents for geodesic flows

    Butler, Clark
    We study the relationship between the Lyapunov exponents of the geodesic flow of a closed negatively curved manifold and the geometry of the manifold. We show that if each periodic orbit of the geodesic flow has exactly one Lyapunov exponent on the unstable bundle then the manifold has constant negative curvature. We also show under a curvature pinching condition that equality of all Lyapunov exponents with respect to volume on the unstable bundle also implies that the manifold has constant negative curvature. We then study the degree to which one can emulate these rigidity theorems for the hyperbolic spaces of...

  5. Picard groups of Poisson manifolds

    Bursztyn, Henrique; Fernandes, Rui Loja
    For a Poisson manifold $M$ we develop systematic methods to compute its Picard group $\mathrm{Pic}(M)$, i.e., its group of self Morita equivalences. We establish a precise relationship between $\mathrm{Pic}(M)$, and the group of gauge transformations up to Poisson diffeomorphisms showing, in particular, that their connected components of the identity coincide; this allows us to introduce the Picard Lie algebra of $M$ and to study its basic properties. Our methods lead to the proof of a conjecture from “Poisson sigma models and symplectic groupoids, in Quantization of Singular Symplectic Quotients” [A.S. Cattaneo, G. Felder, Progress in Mathematics 198 (2001), 41] stating...

  6. Index to Volume 108


  7. Mean curvature flows in manifolds of special holonomy

    Tsai, Chung-Jun; Wang, Mu-Tao
    We study the uniqueness of minimal submanifolds and the stability of the mean curvature flow in several well-known model spaces of manifolds of special holonomy. These include the Stenzel metric on the cotangent bundle of spheres, the Calabi metric on the cotangent bundle of complex projective spaces, and the Bryant–Salamon metrics on vector bundles over certain Einstein manifolds. In particular, we show that the zero sections, as calibrated submanifolds with respect to their respective ambient metrics, are unique among compact minimal submanifolds and are dynamically stable under the mean curvature flow. The proof relies on intricate interconnections of the Ricci...

  8. Heat flows on hyperbolic spaces

    Lemm, Marius; Markovic, Vladimir
    In this paper we develop new methods for studying the convergence problem for the heat flow on negatively curved spaces and prove that any quasiconformal map of the sphere $\mathbb{S}^{n-1} , n \geq 3$, can be extended to the $n$-dimensional hyperbolic space such that the heat flow starting with this extension converges to a quasi-isometric harmonic map. This implies the Schoen–Li–Wang conjecture that every quasiconformal map of $\mathbb{S}^{n-1} , n \geq 3$, can be extended to a harmonic quasi-isometry of the $n$-dimensional hyperbolic space.

  9. On the microlocal analysis of the geodesic X-ray transform with conjugate points

    Holman, Sean; Uhlmann, Gunther
    We study the microlocal properties of the geodesic X-ray transform $\mathcal{X}$ on a manifold with boundary allowing the presence of conjugate points. Assuming that there are no self-intersecting geodesics and all conjugate pairs are nonsingular we show that the normal operator $\mathcal{N} = \mathcal{X}^t \circ \mathcal{X}$ can be decomposed as the sum of a pseudodifferential operator of order $-1$ and a sum of Fourier integral operators. We also apply this decomposition to prove inversion of $\mathcal{X}$ is only mildly ill-posed when all conjugate points are of order $1$, and a certain graph condition is satisfied, in dimension three or higher.

  10. On the Björling problem for Willmore surfaces

    Brander, David; Wang, Peng
    We solve the analogue of Björling’s problem for Willmore surfaces via a harmonic map representation. For the umbilic-free case the problem and solution are as follows: given a real analytic curve $y_0$ in $\mathbb{S}^3$, together with the prescription of the values of the surface normal and the dual Willmore surface along the curve, lifted to the light cone in Minkowski $5$-space $\mathbb{R}^5_1$, we prove, using isotropic harmonic maps, that there exists a unique pair of dual Willmore surfaces $y$ and $\hat{y}$ satisfying the given values along the curve. We give explicit formulae for the generalized Weierstrass data for the surface...

  11. Comparing the Morse index and the first Betti number of minimal hypersurfaces

    Ambrozio, Lucas; Carlotto, Alessandro; Sharp, Ben
    By extending and generalizing previous work by Ros and Savo, we describe a method to show that in certain positively curved ambient manifolds the Morse index of every closed minimal hypersurface is bounded from below by a linear function of its first Betti number. The technique is flexible enough to prove that such a relation between the index and the topology of minimal hypersurfaces holds, for example, on all compact rank one symmetric spaces, on products of the circle with spheres of arbitrary dimension and on suitably pinched submanifolds of the Euclidean spaces. These results confirm a general conjecture due...

  12. 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.

  13. 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.

  14. 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.

  15. 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,...

  16. 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...

  17. 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...

  18. 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...

  19. 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.

  20. 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.

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