Showing papers in "Annals of Global Analysis and Geometry in 2001"
TL;DR: In this article, a complete six-dimensional nearly Kahlermanifold together with the first canonical Hermitian connection is considered and it is shown that if the holonomy of this connection is reducible, then the manifold endowed with a modified metric and almost complex structure is aKahlerian twistor space.
Abstract: We consider a complete six-dimensional nearly Kahlermanifold together with the first canonical Hermitian connection. We showthat if the holonomy of this connection is reducible, then the manifoldendowed with a modified metric and almost complex structure is aKahlerian twistor space. This result was conjectured byReyes-Carrion.
59 citations
TL;DR: In this article, a class of subsets of Ω(n − 1)-current supported by the unit normal bundle and its properties are established, which can be represented as locally finite unions of sets with positive reach and a translative integral geometrical formula for curvature measures is proved.
Abstract: A class of subsets of ℝ
d
which can berepresented as locally finite unions of sets with positive reach isconsidered. It plays a role in PDE's on manifolds with singularities.For such a set, the unit normal cycle (determining the d − 1curvature measures) is introduced as a (d − 1)-currentsupported by the unit normal bundle and its properties are established.It is shown that, under mild additional assumptions, the unit normalcycle (and, hence, also the curvature measures) of such a set can beapproximated by that of a close parallel body or, alternatively, by themirror image of that of the closure of the complement of the parallelbody (which has positive reach). Finally, the mixed curvature measuresof two sets of this class are introduced and a translative integralgeometric formula for curvature measures is proved.
52 citations
TL;DR: In this article, different blow-up constructions on a symplectic orbifold have been studied for the case of a Hamiltonian torus action not necessarily quasi-free, and the wall-crossing theorem of Guillemin and Sternberg has been generalized to the manifold case.
Abstract: In the first part of this paper we study different blow-upconstructions on symplectic orbifolds. Unlike the manifold case,we can define different blow-ups by using different circleactions. In the second part, we use some of these constructions todescribe the behavior of reduced spaces of a Hamiltonian circleaction on a symplectic orbifold, when passing a critical level ofits Hamiltonian function. Using these descriptions, we generalize,in the manifold case, the wall-crossing theorem of Guillemin and Sternberg to the case of a Hamiltonian torus action not necessarily quasi-free and also the Duistermaat–Heckman theorem to intervalsof values of the Hamiltonian function containing critical values.
51 citations
TL;DR: Lower bounds for the length of the zero set of an eigenfunction of the Laplace operator on a Riemann surface were shown in this article for non-negative curvature.
Abstract: We prove lower bounds for the length of the zero set of aneigenfunction of the Laplace operator on a Riemann surface; inparticular, in non-negative curvature, or when the associated eigenvalueis large, we give a lower bound which involves only the square root ofthe eigenvalue and the area of the manifold (modulo a numericalconstant, this lower bound is sharp).
44 citations
TL;DR: In this paper, the problem of classifying Legendrian knots in overtwisted contact structures on S3 was studied, and it was shown that topologically isotopic Legendrian knot have to be Legendrian isotopic if they have equal values of the well-known invariants rot and tb.
Abstract: We study the problem of classifying Legendrian knots in overtwisted contact structures on S3. The question is whether topologically isotopic Legendrian knots have to be Legendrian isotopic if they have equal values of the well-known invariants rot and tb. We give positive answer in the case that there is an overtwisted disc intersecting none of the knots and we construct an example of a knot intersecting each overtwisted disc (this provides a counterexample to the conjecture of Eliashberg). Our proof needs some results on the structure of the group of contactomorphisms of S3. We divide the subgroup Cont+(S3, ξ) of coorientation-preserving contactomorphisms for an overtwisted contact distribution ξ into two classes.
44 citations
TL;DR: In this paper, a class of rank one Einstein solvmanifolds whose derived algebras are two-step nilpotent was constructed, including a continuous subfamily of manifolds with negative Ricci curvature.
Abstract: We construct new homogeneous Einstein spaces with negative Ricci curvature in two ways: First, we give a method for classifying and constructing a class of rank one Einstein solvmanifolds whose derived algebras are two-step nilpotent. As an application, we describe an explicit continuous family of ten- dimensional Einstein manifolds with a two-dimensional parameter space, including a continuous subfamily of manifolds with negative sectional curvature. Secondly, we obtain new examples of non-symmetric Einstein solvmanifolds by modifying the algebraic structure of non-compact irreducible symmetric spaces of rank greater than one, preserving the (constant) Ricci curvature.
43 citations
TL;DR: In this paper, it was proved that the second realcontinuous bounded cohomology group of a connected locally compact group is finite-dimensional, and their applications to the description of their second continuous bounded cohology groups are indicated.
Abstract: Results on continuous pseudocharacters on locally compactgroups are presented and their applications to the description oftheir second continuous bounded cohomology groups areindicated. In particular, it is proved that the second realcontinuous bounded cohomology group of a connected locallycompact group is finite-dimensional.
43 citations
TL;DR: An overview of the various transformations of isothermic surfaces and their interrelations is given using aquaternionic formalism in this paper, where applications to the theory of cmc-1 surfaces in hyperbolic space are given and relations between the two theories are discussed.
Abstract: An overview of the various transformations of isothermic surfaces and their interrelations is given using aquaternionic formalism. Applications to the theory of cmc-1 surfaces inhyperbolic space are given and relations between the two theories are discussed. Within this context, we give Mobius geometric characterizations for cmc-1 surfaces in hyperbolic space and theirminimal cousins.
43 citations
TL;DR: In this article, the Ricci-flat ALE Kahler metrics on X were studied and it was shown that if G ⊂ SU(m) and X is a crepant resolution of ℂm/G, then there is a unique Ricciflat ALE-Kahler metric in each Kahler class.
Abstract: Let G be a finite subgroup of U(m) such thatℂm/G has an isolated singularity at 0. Let X be a resolution of ℂm/G, andg a Kahler metric on X. We callg Asymptotically Locally Euclidean (ALE) if it isasymptotic in a certain way to the Euclidean metric onℂm/G. In this paper we study Ricci-flat ALE Kahler metrics on X. We show that if G ⊂ SU(m) and X is a crepant resolution of ℂm/G, then there is a unique Ricci-flat ALE Kahler metric in each Kahlerclass. This is proved using a version of the Calabi conjecture for ALEmanifolds. We also show the metrics have holonomy SU(m).
37 citations
TL;DR: In this article, a set of necessary and sufficient conditions for an ann-dimensional semi-Riemannian manifold to be conformal to an Einsteinspace is presented.
Abstract: This paper presents a set of necessary and sufficient conditions for ann-dimensional semi-Riemannian manifold to be conformal to an Einstein space We extend results due to C N Kozameh, E T Newman and K P Tod who solved the problem in the four-dimensional Lorentz case formanifolds with nondegenerate Weyl tensor, ie for spacetimes withJ ≠ 0 In particular, in n-dimension we will find tensorialconditions if the Weyl tensor operates injectively on the alternatingtwo-forms Moreover, in the four-dimensional Riemannian case we canalways decide whether a manifold is locally conformal to an Einsteinspace
37 citations
TL;DR: In this paper, the authors give optimal lower bounds for the hypersurface Dirac operator in terms of the Yamabe number, the energy-momentum tensor and the mean curvature.
Abstract: We give optimal lower bounds for the hypersurface Diracoperator in terms of the Yamabe number, the energy-momentum tensor andthe mean curvature. In the limiting case, we prove that the hypersurfaceis an Einstein manifold with constant mean curvature.
TL;DR: For a parameter λ > 0, the existence of smooth solutions to heat flow for a self-dual Yang-Mills-Higgs field on a holomorphic vector bundle over a Kahler manifold was established in this article.
Abstract: For a parameter λ > 0, we study a type of vortex equations, which generalize the well-known Hermitian–Einstein equation, for a connection A and a section φ of a holomorphic vector bundle E over a Kahler manifold X. We establish a global existence of smooth solutions to heat flow for a self-dual Yang–Mills–Higgs field on E. Assuming the λ-stability of (E, φ), we prove the existence of the Hermitian Yang–Mills–Higgs metric on the holomorphic bundle E by studying the limiting behaviour of the gauge flow.
TL;DR: In this paper, the submanifold Dirac operator is shown to be an EPT if the thenormal bundle is flat and the Yamabe number or the energy-momentum tensor is known.
Abstract: In this paper we generalize the results of Part I to the submanifoldDirac operator. In particular, we give optimal lower bounds for thesubmanifold Dirac operator in terms of the mean curvature and othergeometric invariants as the Yamabe number or the energy-momentum tensor.In the limiting case, we prove that the submanifold is Einstein if thenormal bundle is flat.
TL;DR: In this paper, the authors show that the class of Riemannian manifolds with unbounded ν is larger than originally thought and contains many metrically and diffeomorphically ''exotic'' examples.
Abstract: A Riemannian manifold (\(\mathcal{M}^n \)n, g) is said to be the center of thecomplex manifold \(\mathcal{X}^n \)n if \(\mathcal{M}\) is the zero set of a smooth strictly plurisubharmonic exhaustion function ν2 on \(\mathcal{X}\) such that ν is plurisubharmonic and solves theMonge–Ampere equation (∂\(\bar \partial \)ν)n = 0 off \(\mathcal{M}\), and g is induced by the canonical Kahler metric withfundamental two-form \( - \sqrt { - 1} \) ∂\(\bar \partial \)ν2. Insisting that ν be unbounded puts severe restrictions on \(\mathcal{X}\) as acomplex manifold as well as on (\(\mathcal{M}\), g). It is an open problemto determine the class Riemannian manifolds that are centers of complexmanifolds with unbounded ν. Before the present work, the list of knownexamples of manifolds in that class was small. In the main result of thispaper we show, by means of the moment map corresponding to isometric actionsand the associated bundle construction, that such class is larger than originally thought and contains many metrically and diffeomorphically`exotic' examples.
TL;DR: In this paper, the Ricci-flat QALE Kahler metrics are shown to be asymptotic to the Euclidean metric on ℂm/G away from its singular set.
Abstract: Let G be a finite subgroup of U(m),and X a resolution of ℂm/G. We define aspecial class of Kahler metrics g on Xcalled Quasi Asymptotically Locally Euclidean (QALE) metrics. Thesesatisfy a complicated asymptotic condition, implying that gis asymptotic to the Euclidean metric on ℂm/G away fromits singular set. When ℂm/Ghas an isolated singularity,QALE metrics are just ALE metrics. Our main result is an existencetheorem for Ricci-flat QALE Kahler metrics: if G is afinite subgroup of SU(m) and X a crepant resolution of ℂm/G, then there is a unique Ricci-flat QALE Kahler metric on X in each Kahler class.This is proved using a version of the Calabi conjecture for QALEmanifolds. We also determine the holonomy group of the metrics in termsof G.
TL;DR: In this paper, it was shown that theWhitney sphere is the only Willmore Lagrangian surface of genus zero in ℝ4 and established some existence and uniqueness results about WLG tori in √ 4 ≈ √ 2.
Abstract: We make a contribution to the study of Willmore surfaces infour-dimensional Euclidean space ℝ4 by making useof the identification of ℝ4 with two-dimensionalcomplex Euclidean space ℂ2. We prove that theWhitney sphere is the only Willmore Lagrangian surface of genus zero inℝ4 and establish some existence and uniquenessresults about Willmore Lagrangian tori in ℝ4≡ ℂ2.
TL;DR: In this article, the authors constructed explicit families of Lagrangian 3-folds in C3, which are diffeomorphic to R3 or C2, and showed a one-to-one correspondence between sets of evolution data with m = 3 and homogeneous symplectic 2-manifolds P.
Abstract: This is the third in a series of papers constructing explicit examples of special Lagrangian submanifolds in C
m
. The previous paper (Math. Ann.
320 (2001), 757–797), defined the idea of evolution data, which includes an (m − 1)-submanifold P in R
n
, and constructed a family of special Lagrangian m-folds N in C
m
, which are swept out by the image of P under a 1-parameter family of affine maps φ
t
: R
n
→ C
m
, satisfying a first-order o.d.e. in t. In this paper we use the same idea to construct special Lagrangian 3-folds in C3. We find a one-to-one correspondence between sets of evolution data with m = 3 and homogeneous symplectic 2-manifolds P. This enables us to write down several interesting sets of evolution data, and so to construct corresponding families of special Lagrangian 3-folds in C3.Our main results are a number of new families of special Lagrangian 3-foldsin C3, which we write very explicitly in parametric form. Generically these are nonsingular as immersed 3-submanifolds, and diffeomorphic to R3 or
% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuGqLXgBG0evaeXatLxBI9gBam% XvP5wqSXMqHnxAJn0BKvguHDwzZbqegm0B1jxALjhiov2DaeHbuLwB% Lnhiov2DGi1BTfMBaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFf% euY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9% q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqaba% WaaqaafaaakeaaruWrPDwAaGGbciab-nfatbaa!3D86!
1× R2. Some of the 3-folds are singular, and we describe their singularities, which we believe are of a new kind.We hope these 3-folds will be helpful in understanding singularities ofcompact special Lagrangian 3-folds in Calabi–Yau 3-folds. This will beimportant in resolving the SYZ conjecture in Mirror Symmetry.
TL;DR: In this paper, it was shown that there is a γ ∈ D which minimizes the functional complexity of the functional area minimizing surface with boundary γ(0, 1) if a ∈ (0, ∞) is suitably chosen.
Abstract: Let D:= {γ ∈ C
3 (
$$\mathbb{R},{\text{ }}\mathbb{R}$$
3) ∣ γ (s) = γ(s+1), ∣
$$\dot \gamma $$
∣ ≡ 1 γ ([0,1]) is simple closed curve}. In this paper we show that there is γ ∈ D which minimizes the functional
$$E_{\gamma 0} \left( \gamma \right): = \int_0^1 {\left| {\ddot \gamma \left( s \right) - \ddot \gamma _0 \left( s \right)} \right|^2 } ds + $$
+ a(area minimizing surface with boundary γ([0,1])), γ0 ∈ D if a ∈ (0,∞) is suitably chosen. where γ0 ∈ D if a ∈ (0, ∞) is suitably chosen.
TL;DR: In this paper, the authors studied the problem of finding constant mean curvature graphs over a domain of a totally geodesic hyperplane and an equidistant hypersurface Q of hyperbolic space.
Abstract: We study the problem of finding constant mean curvature graphs over a domainof a totally geodesic hyperplane and an equidistant hypersurface Q of hyperbolic space. We find the existence of graphs of constant mean curvature H over mean convex domains � ⊂ Q and with boundary ∂� for −H∂� 0 is the mean curvature of the boundary ∂� . Here h is the mean curvature respectively of the geodesic hyperplane (h = 0) and of the equidistant hypersurface (0 < |h| < 1). The lower bound on H is optimal.
TL;DR: In this article, a global conformal representation for flat surfaces with a flat normal bundle in the standard flat Lorentzian space form is given, where flat surfaces in hyperbolic 3-space, the de Sitter 3 space, the null cone, and other numerous examples are described.
Abstract: We give a global conformal representation for flat surfaces with a flat normal bundle in the standard flat Lorentzian space form
$$\mathbb{L}$$
4. Particularly, flat surfaces in hyperbolic 3-space, the de Sitter 3-space, the null cone, and other numerous examples aredescribed.
TL;DR: In this paper, the convex hull property for properly immersed minimal hypersurfaces in a cone of Ω n ≥ 3 has been proved, and the existence of new barriers for the maximum principle application in non-compact truncated tetrahedral domains has been shown.
Abstract: We prove the convex hull property for properly immersed minimal hypersurfaces in a cone of ℝ
n
. We deal with the existence of new barriers for the maximum principle application in noncompact truncated tetrahedral domains of ℝ3, describing the space of such domainsadmitting barriers of this kind. Nonexistence results for nonflatminimal surfaces whose boundary lies in opposite faces of a tetrahedraldomain are obtained. Finally, new simple closed subsets of ℝ3 whichhave the property of intersecting any properly immersed minimal surfaceare shown.
TL;DR: In this paper, a differential complex of coeffective type for any Spin(7)-manifold M locally conformal to a Riemannian 8-manifolds with holonomy contained in spin(7) is introduced, and local properties of this complex, such as ellipticity and acyclicity are studied.
Abstract: We introduce a differential complex of coeffective type for anySpin(7)-manifold M locally conformal to aRiemannian 8-manifold with holonomy contained in Spin(7).Local properties of this complex, such as ellipticity and acyclicity,are studied. The relationship between the coeffective cohomology ofM and the topology of the manifold is discussed in the caseof M having a subgroup of Spin(7) as aholonomy group.
TL;DR: In this article, it was shown that one may enlarge a Kahler metric with positive Ricci curvature without makingsmaller located on the manifold M. In certain cases, these estimates are sharp: they give examples where equality is obtained.
Abstract: In this note, we look at estimates for the scalar curvatureof a Riemannian manifold M which are related to spin c Dirac operators: We show that one may not enlarge a Kahler metric with positive Ricci curvature without makingsmaller somewhere on M. We also give explicit upper bounds for minfor arbitrary Riemannian metrics on certain submanifolds of complex projective space. In certain cases, these estimates are sharp: we give examples where equality is obtained.
TL;DR: In this article, the Yamabe invariant of Riemannian manifolds admits metrics of positive scalar curvature was studied and a technique to find positive lower bounds for the invariant was proposed.
Abstract: We study the Yamabe invariant of manifolds which admit metrics of positive scalar curvature. Analysing `best Sobolev constants'we give a technique to find positive lower bounds for the invariant.We apply these ideas to show that for any compact Riemannian manifold (N n ,g) of positive scalarcurvature there is a positive constant K =K(N, g), which depends only on (N, g), such that for any compact manifold M m , the Yamabe invariantof M m × N n is no less than K times the invariant ofSn + m. We will find some estimates for the constant K in the case N =S n .
TL;DR: In this article, the twistor spaces of oriented Riemannian four-manifolds were studied as a source of almost Hermitian *-Einstein manifolds.
Abstract: In this paper we study the twistor spaces of oriented Riemannianfour-manifolds as a source of almost-Hermitian *-Einstein manifoldsand show that some results in dimension four related to the RiemannianGoldberg–Sachs theorem cannot be extended to higher dimensions.
TL;DR: In this article, the authors obtained curvature estimates for certain stable minimal hypersurfaces in R4 and R5 without using volume bounds, and they showed that if M is a complete stable hypersurface in R 4 or R 5, then M is hyperplane whenM intersects each extrinsic ball in, at most, N-components.
Abstract: We obtain curvature estimates for certain stable minimalhypersurfaces in R4 and R5without using volume bounds. It follows that if M is acomplete stable minimal hypersurface in R4 orR5, then M is a hyperplane whenM intersects each extrinsic ball in, at most,N-components.
TL;DR: In this paper, a desingularization result for minimal surfaces in Euclidean space using Weierstrass representation was proved. And they solved the period problem using the implicit function theorem at a degenerate point.
Abstract: We prove a desingularization result for minimal surfaces inEuclidean space using Weierstrass representation. We solve the periodproblem using the implicit function theorem at a degenerate point.
TL;DR: In this article, the maximum principle at infinity generalizes Hopf's maximum principle for hypersurfaces with constant mean curvature in R 3 with bounded Gaussian curvature for parabolic surfaces.
Abstract: Maximum principles at infinity generalize Hopf's maximum principle for hypersurfaces with constant mean curvature in R
n
. We establish such a maximum principle for parabolic surfaces in R3 with nonzero constant mean curvature and bounded Gaussian curvature.
TL;DR: For a complete solution to the Ricci-Kahler flow where the curvature, the potential and scalar curvature functions and their gradients are bounded depending on time, the absolute value of both the scalar and the gradients of a modified potential function are bounded by C/t as discussed by the authors.
Abstract: We show that for a complete solution to theRicci–Kahler flow where the curvature, the potential andscalar curvature functions and their gradients are bounded depending ontime, the absolute value of both the scalar curvature and the gradientsquared of a modified potential function are bounded byC/t.
TL;DR: In this article, the Dirac operator on fibrations over S 1 which have up to holonomy a warped product metric was considered and lower bounds for the eigenvalues on M were given.
Abstract: We consider the Dirac operator on fibrations overS1 which have up to holonomy a warped product metric. Wegive lower bounds for the eigenvalues on M and if the Diracoperator on the typical fibre F has a kernel, we calculatethe corresponding part of the spectrum on M explicitly.