Mostrando recursos 1 - 20 de 65

  1. Index to Volume 109


  2. The local picture theorem on the scale of topology

    Meeks, William H.; Pérez, Joaquín; Ros, Antonio
    We prove a descriptive theorem on the extrinsic geometry of an embedded minimal surface of injectivity radius zero in a homogeneously regular Riemannian three-manifold, in a certain small intrinsic neighborhood of a point of almost-minimal injectivity radius. This structure theorem includes a limit object which we call a minimal parking garage structure on $\mathbb{R}^3$, whose theory we also develop.

  3. The intersection of a hyperplane with a lightcone in the Minkowski spacetime

    Le, Pengyu
    Klainerman, Luk and Rodnianski derived an anisotropic criterion for formation of trapped surfaces in vacuum, extending the original trapped surface formation theorem of Christodoulou. The effort to understand their result led us to study the intersection of a hyperplane with a lightcone in the Minkowski spacetime. For the intrinsic geometry of the intersection, depending on the hyperplane being spacelike, null or timelike, it has the constant positive, zero or negative Gaussian curvature. For the extrinsic geometry of the intersection, we find that it is a noncompact marginal trapped surface when the hyperplane is null. In this case, we find a...

  4. Looijenga’s conjecture via integral-affine geometry

    Engel, Philip
    A cusp singularity is a surface singularity whose minimal resolution is a cycle of smooth rational curves meeting transversely. Cusp singularities come in naturally dual pairs. In 1981, Looijenga proved that whenever a cusp singularity is smoothable, the minimal resolution of the dual cusp is an anticanonical divisor of some smooth rational surface. He conjectured the converse. Recent work of Gross, Hacking, and Keel has proven Looijenga’s conjecture using methods from mirror symmetry. This paper provides an alternative proof of Looijenga’s conjecture based on a combinatorial criterion for smoothability given by Friedman and Miranda in 1983.

  5. A discrete uniformization theorem for polyhedral surfaces II

    Gu, Xianfeng; Guo, Ren; Luo, Feng; Sun, Jian; Wu, Tianqi
    A notion of discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying Gauss–Bonnet formula. Furthermore, the hyperbolic polyhedral metric with given curvature can be obtained using a discrete Yamabe flow with surgery. In particular, each hyperbolic polyhedral metric on a closed surface with negative Euler characteristic is discrete conformal to a unique hyperbolic metric.

  6. Subspace concentration of dual curvature measures of symmetric convex bodies

    Böröczky, Károly J.; Henk, Martin; Pollehn, Hannes
    We prove a tight subspace concentration inequality for the dual curvature measures of a symmetric convex body.

  7. The geometric torsion conjecture for abelian varieties with real multiplication

    Bakker, Benjamin; Tsimerman, Jacob
    The geometric torsion conjecture asserts that the torsion part of the Mordell–Weil group of a family of abelian varieties over a complex quasi-projective curve is uniformly bounded in terms of the genus of the curve. We prove the conjecture for abelian varieties with real multiplication, uniformly in the field of multiplication. Fixing the field, we, furthermore, show that the torsion is bounded in terms of the gonality of the base curve, which is the closer analog of the arithmetic conjecture. The proof is a hybrid technique employing both the hyperbolic and algebraic geometry of the toroidal compactifications of the Hilbert...

  8. Fully non-linear elliptic equations on compact Hermitian manifolds

    Székelyhidi, Gábor
    We derive a priori estimates for solutions of a general class of fully non-linear equations on compact Hermitian manifolds. Our method is based on ideas that have been used for different specific equations, such as the complex Monge–Ampère, Hessian and inverse Hessian equations. As an application we solve a class of Hessian quotient equations on Kähler manifolds assuming the existence of a suitable subsolution. The method also applies to analogous equations on compact Riemannian manifolds.

  9. LG/CY correspondence for elliptic orbifold curves via modularity

    Shen, Yefeng; Zhou, Jie
    We prove the Landau–Ginzburg/Calabi–Yau correspondence between the Gromov–Witten theory of each elliptic orbifold curve and its Fan–Jarvis–Ruan–Witten theory counterpart via modularity. We show that the correlation functions in these two enumerative theories are different representations of the same set of quasi-modular forms, expanded around different points on the upper-half plane. We relate these two representations by the Cayley transform.

  10. Foliations by spheres with constant expansion for isolated systems without asymptotic symmetry

    Nerz, Christopher
    Motivated by the foliation by stable spheres with constant mean curvature constructed by Huisken–Yau, Metzger proved that every initial data set can be foliated by spheres with constant expansion (CE) if the manifold is asymptotically equal to the standard $[t=0]$-timeslice of the Schwarzschild solution. In this paper, we generalize his result to asymptotically flat initial data sets and weaken additional smallness assumptions made by Metzger. Furthermore, we prove that the CE-surfaces are in a well-defined sense (asymptotically) independent of time if the linear momentum vanishes.

  11. A discrete uniformization theorem for polyhedral surfaces

    Gu, Xianfeng David; Luo, Feng; Sun, Jian; Wu, Tianqi
    A discrete conformality for polyhedral metrics on surfaces is introduced in this paper. It is shown that each polyhedral metric on a compact surface is discrete conformal to a constant curvature polyhedral metric which is unique up to scaling. Furthermore, the constant curvature metric can be found using a finite dimensional variational principle.

  12. Concerning Toponogov’s theorem and logarithmic improvement of estimates of eigenfunctions

    Blair, Matthew D.; Sogge, Christopher D.
    We use Toponogov’s triangle comparison theorem from Riemannian geometry along with quantitative scale oriented variants of classical propagation of singularities arguments to obtain logarithmic improvements of the Kakeya–Nikodym norms introduced in [22] for manifolds of nonpositive sectional curvature. Using these and results from our paper [4] we are able to obtain log-improvements of $L^p (M)$ estimates for such manifolds when $2 \lt p \lt \frac{2(n+1)}{n-1}$. These in turn imply $(\log \lambda)^{\sigma_n} , \sigma_n \approx n$, improved lower bounds for $L^1$-norms of eigenfunctions of the estimates of the second author and Zelditch [28], and using a result from Hezari and the...

  13. 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.

  14. 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...

  15. 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.

  16. 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...

  17. 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...

  18. Index to Volume 108


  19. 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...

  20. 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.

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