Mostrando recursos 1 - 20 de 2.311

  1. Toward a classification of killing vector fields of constant length on pseudo-Riemannian normal homogeneous spaces

    Wolf, Joseph A.; Podestà, Fabio; Xu, Ming
    In this paper we develop the basic tools for a classification of Killing vector fields of constant length on pseudo-Riemannian homogeneous spaces. This extends a recent paper of M. Xu and J. A. Wolf, which classified the pairs $(M,\xi)$ where $M = G/H$ is a Riemannian normal homogeneous space, G is a compact simple Lie group, and $\xi \in \mathfrak{g}$ defines a nonzero Killing vector field of constant length on $M$. The method there was direct computation. Here we make use of the moment map $M \to \mathfrak{g}^{*}$ and the flag manifold structure of $\mathrm{Ad} (G) \xi$ to give a...

  2. Symmetry gaps in Riemannian geometry and minimal orbifolds

    van Limbeek, Wouter
    We study the size of the isometry group $\mathrm{Isom}(M,g)$ of Riemannian manifolds $(M,g)$ as $g$ varies. For $M$ not admitting a circle action, we show that the order of $\mathrm{Isom}(M,g)$ can be universally bounded in terms of the bounds on Ricci curvature, diameter, and injectivity radius of $M$. This generalizes results known for negative Ricci curvature to all manifolds. ¶ More generally we establish a similar universal bound on the index of the deck group $\pi_1 (M)$ in the isometry group $\mathrm{Isom}(\widetilde{M},\widetilde{g})$ of the universal cover $\widetilde{M}$ in the absence of suitable actions by connected groups. We apply this to characterize...

  3. Boundary torsion and convex caps of locally convex surfaces

    Ghomi, Mohammad
    We prove that the torsion of any closed space curve which bounds a simply connected locally convex surface vanishes at least $4$ times. This answers a question of Rosenberg related to a problem of Yau on characterizing the boundary of positively curved disks in Euclidean space. Furthermore, our result generalizes the $4$ vertex theorem of Sedykh for convex space curves, and thus constitutes a far reaching extension of the classical $4$ vertex theorem. The proof involves studying the arrangement of convex caps in a locally convex surface, and yields a Bose type formula for these objects.

  4. Non-properly embedded $H$-planes in $\mathbb{H}^3$

    Coskunuzer, Baris; Meeks, William H.; Tinaglia, Giuseppe
    For any $H \in [0, 1)$, we construct complete, non-proper, stable, simply-connected surfaces with constant mean curvature $H$ embedded in hyperbolic three-space.

  5. On weakly maximal representations of surface groups

    Ben Simon, Gabi; Burger, Marc; Hartnick, Tobias; Iozzi, Alessandra; Wienhard, Anna
    We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called weakly maximal representations. We prove that weakly maximal representations are discrete and injective and we describe the structure of the Zariski closure of their image. Furthermore, we prove that the set of weakly maximal representations is a closed subset of the representation variety and describe its relation to other geometrically significant subsets of the representations variety.

  6. Space of nonnegatively curved metrics and pseudoisotopies

    Belegradek, Igor; Farrell, F. Thomas; Kapovitch, Vitali
    Let $V$ be an open manifold with complete nonnegatively curved metric such that the normal sphere bundle to a soul has no section. We prove that the souls of nearby nonnegatively curved metrics on $V$ are smoothly close. Combining this result with some topological properties of pseudoisotopies we show that for many $V$ the space of complete nonnegatively curved metrics has infinite higher homotopy groups.

  7. Min–max hypersurface in manifold of positive Ricci curvature

    Zhou, Xin
    In this paper, we study the shape of the min–max minimal hypersurface produced by Almgren–Pitts–Schoen–Simon in a Riemannian manifold $(M^{n+1}, g)$ of positive Ricci curvature for all dimensions. The min–max hypersurface has a singular set of Hausdorff codimension $7$. We characterize the Morse index, area and multiplicity of this singular min–max hypersurface. In particular, we show that the min–max hypersurface is either orientable and has Morse index one, or is a double cover of a non-orientable stable minimal hypersurface. ¶ As an essential technical tool, we prove a stronger version of the discretization theorem. The discretization theorem, first developed by Marques–Neves in their proof of the Willmore conjecture, is...

  8. Minkowski formulae and Alexandrov theorems in spacetime

    Wang, Mu-Tao; Wang, Ye-Kai; Zhang, Xiangwen
    The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit “hidden symmetry” from conformal Killing–Yano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codimension-two submanifold with constant normalized null expansion (null mean curvature) must lie in a shear-free (umbilical) null hypersurface. These results are generalized for higher order curvature invariants. In particular, the notion of mixed higher order mean curvature is introduced to highlight the special null geometry of the submanifold. Finally, Alexandrov type theorems are established for spacelike submanifolds with constant mixed higher order mean curvature, which are generalizations of hypersurfaces of constant...

  9. On normalized differentials on hyperelliptic curves of infinite genus

    Kappeler, Thomas; Topalov, Peter
    We develop a new approach for constructing normalized differentials on hyperelliptic curves of infinite genus and obtain uniform asymptotic estimates for the distribution of their zeros.

  10. Invariant distributions and X-ray transform for Anosov flows

    Guillarmou, Colin
    For Anosov flows preserving a smooth measure on a closed manifold $\mathcal{M}$, we define a natural self-adjoint operator $\Pi$ which maps into the space of flow invariant distributions in $\cap_{r \lt 0} H^r (\mathcal{M})$ and whose kernel is made of coboundaries in $\cup_{s \gt 0} H^s (\mathcal{M})$. We describe relations to the Livsic theorem and recover regularity properties of cohomological equations using this operator. For Anosov geodesic flows on the unit tangent bundle $\mathcal{M}= SM$ of a compact manifold $\mathcal{M}$, we apply this theory to study X-ray transform on symmetric tensors on $\mathcal{M}$. In particular, we prove existence of flow invariant distributions on $SM$ with prescribed push-forward on $\mathcal{M}$...

  11. A new tensorial conservation law for Maxwell fields on the Kerr background

    Andersson, Lars; Bäckdahl, Thomas; Blue, Pieter
    A new, conserved, symmetric tensor field for a source-free Maxwell test field on a four-dimensional spacetime with a conformal Killing–Yano tensor, satisfying a certain compatibility condition, is introduced. In particular, this construction works for the Kerr spacetime.

  12. On the Narasimhan–Seshadri correspondence for real and quaternionic vector bundles

    Schaffhauser, Florent
    Let $(M,\sigma)$ be a compact Klein surface of genus $g \geq 2$ and let $E$ be a smooth Hermitian vector bundle on $M$. Let $\tau$ be a Real or Quaternionic structure on $E$ and denote respectively by $\mathcal{G}^{\tau}_{\mathbb{C}}$ and $\mathcal{G}^{\tau}_{E}$ the groups of complex linear and unitary automorphisms of $E$ that commute to $\tau$. In this paper, we study the action of $\mathcal{G}^{\tau}_{\mathbb{C}}$ on the space $\mathcal{A}^{\tau}_{E}$ of $\tau$-compatible unitary connections on $E$ and show that the closure of a semi-stable $\mathcal{G}^{\tau}_{\mathbb{C}}$-orbit contains a unique $\mathcal{G}^{\tau}_{E}$-orbit of projectively flat connections. We then use this invariant-theoretic perspective to prove a version of the Narasimhan–Seshadri correspondence in this context: $S$-equivalence classes of semi-stable...

  13. Bogomolov–Tian–Todorov theorems for Landau–Ginzburg models

    Katzarkov, Ludmil; Kontsevich, Maxim; Pantev, Tony
    In this paper we prove the smoothness of the moduli space of Landau–Ginzburg models. We formulate and prove a Bogomolov–Tian–Todorov theorem for the deformations of Landau–Ginzburg models, develop the necessary Hodge theory for varieties with potentials, and prove a double degeneration statement needed for the unobstructedness result. We discuss the various definitions of Hodge numbers for non-commutative Hodge structures of Landau–Ginzburg type and the role they play in mirror symmetry. We also interpret the resulting families of de Rham complexes attracted to a potential in terms of mirror symmetry for one parameter families of symplectic Fano manifolds and argue that modulo a natural triviality property the moduli spaces of...

  14. Width, Ricci curvature, and minimal hypersurfaces

    Glynn-Adey, Parker; Liokumovich, Yevgeny
    Let $(M,g_0)$ be a closed Riemannian manifold of dimension $n$, for $3 \leq n \leq 7$, and non-negative Ricci curvature. Let $g = \phi^2 g_0$ be a metric in the conformal class of $g_0$. We show that there exists a smooth closed embedded minimal hypersurface in $(M,g)$ of volume bounded by $C(n)V^{\frac{n-1}{n}}$, where $V$ is the total volume of $(M,g)$. When $Ric(M,g_0) \geq -(n-1)$ we obtain a similar bound with constant $C$ depending only on n and the volume of $(M,g_0)$. Our second result concerns manifolds $(M,g)$ of positive Ricci curvature and dimension at most seven. We obtain an effective version of a theorem of F. C. Marques...

  15. Erratum for “The degree theorem in higher rank”

    Connell, Chris; Farb, Benson
    The purpose of this erratum is to correct a mistake in the proof of Theorem 4.1 of the article “The degree theorem in higher rank”, J. Diff. Geom., Vol. 65 (2003), pp. 19–59.

  16. Isoperimetric structure of asymptotically conical manifolds

    Chodosh, Otis; Eichmair, Michael; Volkmann, Alexander
    We study the isoperimetric structure of Riemannian manifolds that are asymptotic to cones with non-negative Ricci curvature. Specifically, we generalize to this setting the seminal results of G. Huisken and S.–T. Yau on the existence of a canonical foliation by volume-preserving stable constant mean curvature surfaces at infinity of asymptotically flat manifolds as well as the results of the second-named author with S. Brendle and J. Metzger on the isoperimetric structure of asymptotically flat manifolds. We also include an observation on the isoperimetric cone angle of such manifolds. This result is a natural analogue of the positive mass theorem in this setting.

  17. The cohomological crepant resolution conjecture for the Hilbert–Chow morphisms

    Li, Wei-Ping; Qin, Zhenbo
    We prove that Ruan’s Cohomological Crepant Resolution Conjecture holds for the Hilbert–Chow morphisms. There are two main ideas in the proof. The first one is to use the representation theoretic approach proposed in [QW] which involves vertex operator techniques. The second is to prove certain universality structures about the $3$-pointed genus-$0$ extremal Gromov–Witten invariants of the Hilbert schemes by using the indexing techniques from [LiJ], the product formula from [Beh2] and the co-section localization from [KL1, KL2, LL]. We then reduce Ruan’s Conjecture from the case of an arbitrary surface to the case of smooth projective toric surfaces which has...

  18. Double quintic symmetroids, Reye congruences, and their derived equivalence

    Hosono, Shinobu; Takagi, Hiromichi
    Let $\mathscr{Y}$ be the double cover of the quintic symmetric determinantal hypersurface in $\mathbb{P}^{14}$. We consider Calabi–Yau threefolds $Y$ defined as smooth linear sections of $\mathscr{Y}$. In our previous works, we have shown that these Calabi–Yau threefolds $Y$ are naturally paired with Reye congruence Calabi–Yau threefolds $X$ by the projective duality of $\mathscr{Y}$, and observed that these Calabi–Yau threefolds have several interesting properties from the viewpoint of mirror symmetry and also projective geometry. In this paper, we prove the derived equivalence between the linear sections $Y$ of $\mathscr{Y}$ and the corresponding Reye congruences $X$.

  19. Kähler manifolds of semi-negative holomorphic sectional curvature

    Heier, Gordon; Lu, Steven S. Y.; Wong, Bun
    In an earlier work, we investigated some consequences of the existence of a Kähler metric of negative holomorphic sectional curvature on a projective manifold. In the present work, we extend our results to the case of semi-negative (i.e., non-positive) holomorphic sectional curvature. In doing so, we define a new invariant that records the largest codimension of maximal subspaces in the tangent spaces on which the holomorphic sectional curvature vanishes. Using this invariant, we establish lower bounds for the nef dimension and, under certain additional assumptions, for the Kodaira dimension of the manifold. In dimension two, a precise structure theorem is obtained.

  20. On the topology and index of minimal surfaces

    Chodosh, Otis; Maximo, Davi
    We show that for an immersed two-sided minimal surface in $\mathbb{R}^3$, there is a lower bound on the index depending on the genus and number of ends. Using this, we show the nonexistence of an embedded minimal surface in $\mathbb{R}^3$ of index $2$, as conjectured by Choe. Moreover, we show that the index of an immersed two-sided minimal surface with embedded ends is bounded from above and below by a linear function of the total curvature of the surface.

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