Recursos de colección
Project Euclid (Hosted at Cornell University Library) (191.996 recursos)
Journal of Differential Geometry
Journal of Differential Geometry
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...
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...
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.
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.
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.
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.
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...
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...
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.
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 $\mathcal{M}$...
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.
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...
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...
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...
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.
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.
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...
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 $X$.
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.
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.