Showing papers in "Annals of Global Analysis and Geometry in 2009"
TL;DR: In this article it was shown that an almost paracontact structure admits a connection with totally skew-symmetric torsion if and only if the Nijenhuis tensor of the structure is skew symmetric and the defining vector field is Killing.
Abstract: The canonical paracontact connection is defined and it is shown that its torsion is the obstruction the paracontact manifold to be paraSasakian. A $${\mathcal{D}}$$
-homothetic transformation is determined as a special gauge transformation. The η-Einstein manifold are defined, it is proved that their scalar curvature is a constant, and it is shown that in the paraSasakian case these spaces can be obtained from Einstein paraSasakian manifolds with $${\mathcal{D}}$$
-homothetic transformations. It is shown that an almost paracontact structure admits a connection with totally skew-symmetric torsion if and only if the Nijenhuis tensor of the paracontact structure is skew-symmetric and the defining vector field is Killing.
229 citations
TL;DR: In this paper, the authors define a new class of submanifolds called pseudo-bubbles, defined by an equation weaker than constancy of mean curvature, and show that in a neighborhood of each point of a Riemannian manifold, there is a unique family of concentric pseudo bubble which contains all the pseudo-bubbles C 2,α 1,α 2 -close to small spheres, which permit them to reduce the isoperimetric problem for small volumes to a variational problem in finite dimension.
Abstract: We define a new class of submanifolds called pseudo-bubbles, defined by an equation weaker than constancy of mean curvature. We show that in a neighborhood of each point of a Riemannian manifold, there is a unique family of concentric pseudo-bubbles which contains all the pseudo-bubbles C
2,α
-close to small spheres. This permit us to reduce the isoperimetric problem for small volumes to a variational problem in finite dimension.
62 citations
TL;DR: In this paper, a local gradient estimate for the positive solution to the following parabolic equation was derived, where a, b are real constants, and M is a complete non-compact Riemannian manifold.
Abstract: In this paper, we derive a local gradient estimate for the positive solution to the following parabolic equation
$$u_t=\Delta u+au\, {\rm log}\, u+bu\quad {\rm in}\,M$$
, where a, b are real constants, M is a complete noncompact Riemannian manifold. As a corollary, we give a local gradient estimate for the corresponding elliptic equation:
$$\Delta u+au\,{\rm log}\, u+bu=0\quad {\rm in}\,M$$
, which improves and extends the result of Ma (J Funct Anal 241:374–382, 2006) and get a bound for the positive solution to this elliptic equation.
61 citations
TL;DR: In this article, conformal biharmonicity and holomorphicity of conformal immersions of surfaces are studied. But the authors focus on the transformation of Jacobi operator and bitension field under conformal change of metrics.
Abstract: This paper studies conformal biharmonic immersions. We first study the transformations of Jacobi operator and the bitension field under conformal change of metrics. We then obtain an invariant equation for a conformal biharmonic immersion of a surface into Euclidean 3-space. As applications, we construct a two-parameter family of non-minimal conformal biharmonic immersions of cylinder into \({\mathbb{R}^3}\) and some examples of conformal biharmonic immersions of four-dimensional Euclidean space into sphere and hyperbolic space, thus providing many simple examples of proper biharmonic maps with rich geometric meanings. These suggest that there are abundant proper biharmonic maps in the family of conformal immersions. We also explore the relationship between biharmonicity and holomorphicity of conformal immersions of surfaces.
41 citations
TL;DR: In this paper, a weakly Dirac-harmonic map from a Riemann spin surface to a compact hypersurface is shown to be smooth, and it is shown that the map is a weak Dirac map.
Abstract: We prove that a weakly Dirac-harmonic map from a Riemann spin surface to a compact hypersurface \({N \subset \mathbb{R}^{d+1}}\) is smooth.
41 citations
TL;DR: In this paper, a pseudo-differential Weyl calculus on nilpotent Lie groups was developed to deal with magnetic perturbations of right invariant vector fields. But the pseudo-Differential Wey calculus is not suitable for the case of right-invariant vector field perturbation.
Abstract: We develop a pseudo-differential Weyl calculus on nilpotent Lie groups, which allows one to deal with magnetic perturbations of right invariant vector fields. For this purpose, we investigate an infinite-dimensional Lie group constructed as the semidirect product of a nilpotent Lie group and an appropriate function space thereon. We single out an appropriate coadjoint orbit in the semidirect product and construct our pseudo-differential calculus as a Weyl quantization of that orbit.
32 citations
TL;DR: In this paper, it was shown that only open parts of cylinders over a geodesic are both minimal and flat, and that the condition that the projection of the direction tangent to the tangent space of the surface is a principal direction, is the same as the condition for the surface to be normally flat.
Abstract: We show a way to choose nice coordinates on a surface in \({\mathbb{S}^2 \times \mathbb{R}}\) and use this to study minimal surfaces. We show that only open parts of cylinders over a geodesic in \({\mathbb{S}^2}\) are both minimal and flat. We also show that the condition that the projection of the direction tangent to \({\mathbb{R}}\) onto the tangent space of the surface is a principal direction, is equivalent to the condition that the surface is normally flat in \({\mathbb{E}^4}\) . We present classification theorems under the extra assumption of minimality or flatness.
32 citations
TL;DR: In this paper, the limiting behavior of the Brown-York mass and the Hawking mass along nearly round surfaces at infinity of an asymptotically flat manifold was studied, and it was shown that these surfaces can be defined in an intrinsic way.
Abstract: In this article, we study the limiting behavior of the Brown-York mass and Hawking mass along nearly round surfaces at infinity of an asymptotically flat manifold. Nearly round surfaces can be defined in an intrinsic way. Our results show that the ADM mass of an asymptotically flat three manifold can be approximated by some geometric invariants of a family of nearly round surfaces, which approach to infinity of the manifold.
30 citations
TL;DR: In this paper, the first eigenvalue of the Laplacian acting on differential forms on a compact Riemannian domain, for the absolute or relative boundary conditions, was studied.
Abstract: We study the first eigenvalue of the Laplacian acting on differential forms on a compact Riemannian domain, for the absolute or relative boundary conditions. We prove a series of lower bounds when the domain is starlike or p-convex and the ambient manifold has pinched negative curvature. The bounds are sharp for starlike domains. We then compute the asymptotics of the first eigenvalue of hyperbolic balls of large radius. Finally, we give lower bounds also for Euclidean domains.
28 citations
TL;DR: In this article, the authors give a criterion for a finitely generated group Γ to have the fixed point property for a certain class of Hadamard spaces, and prove a fixed-point theorem for random-group actions on the same class.
Abstract: In this paper, using the generalized version of the theory of combinatorial harmonic maps, we give a criterion for a finitely generated group Γ to have the fixed-point property for a certain class of Hadamard spaces, and prove a fixed-point theorem for random-group actions on the same class of Hadamard spaces. We also study the existence of an equivariant energy-minimizing map from a Γ-space to the limit space of a sequence of Hadamard spaces with the isometric actions of a finitely generated group Γ. As an application, we present the existence of a constant which may be regarded as a Kazhdan constant for isometric discrete-group actions on a family of Hadamard spaces.
26 citations
TL;DR: In this article, the authors study the interrelationship between topological and analytical properties of Sobolev bundles and describe some of their applications to variational problems on principal bundles and prove a topological compactness result for minimizing sequences of a conformally invariant Yang-Mills functional.
Abstract: We study the interrelationship between topological and analytical properties of Sobolev bundles and describe some of their applications to variational problems on principal bundles. We in particular show that the category of Sobolev principal G-bundles of class W
2,m/2 defined over M
m
is equivalent to the category of smooth principal G-bundles on M and give a characterization of the weak sequential closure of smooth principal G-bundles with prescribed isomorphism class. We also prove a topological compactness result for minimizing sequences of a conformally invariant Yang-Mills functional.
TL;DR: In this article, the incompleteness in the proof of the theorem appearing in the title [published in Szabo (Ann Glob Anal Geom, to appear, 2008)] has been discovered.
Abstract: Just recently, incompleteness in the proof of the theorem appearing in the title [published in Szabo (Ann Glob Anal Geom, to appear, 2008)] has been discovered. Without this problematic part, the theorem is established only in the following restricted form: “A regular Finsler metric is Berwald if and only if it satisfies the dual Landsberg condition.” The incompleteness appears in proving that the original Landsberg condition implies the dual one.
TL;DR: In this article, a regularity result for critical points of the polyharmonic energy was proved for stationary harmonic and bi-harmonic maps, based on a Gagliardo-Nirenberg-type estimate and avoiding the moving frame technique.
Abstract: We prove a regularity result for critical points of the polyharmonic energy $${E(u)=\int_\Omega\vert
abla^k u\vert^2dx}$$
in $${W^{k,2p}(\Omega,{\mathcal N})}$$
with $${k\in{\mathbb N}}$$
and p > 1. Our proof is based on a Gagliardo–Nirenberg-type estimate and avoids the moving frame technique. In view of the monotonicity formulae for stationary harmonic and biharmonic maps, we infer partial regularity in theses cases.
TL;DR: In this paper, a variety of classes of functions can be realized as the mean curvature on the boundary of the standard n dimensional ball, n ≥ 3, with respect to some scalar flat metric.
Abstract: We provide a variety of classes of functions that can be realized as the mean curvature on the boundary of the standard n dimensional ball, n ≥ 3, with respect to some scalar flat metric. Because of the presence of some critical nonlinearity, blow up phenomena occur and existence results are highly nontrivial since one has to overcome topological obstructions. Our approach consists of, on one hand, developing a Morse theoretical approach to this problem through a Morse-type reduction of the associated Euler–Lagrange functional in a neighborhood of its critical points at Infinity and, on the other hand, extending to this problem some topological invariants introduced by A. Bahri in his study of Yamabe type problems on closed manifolds.
TL;DR: In this paper, the inverse spectral problem for weighted projective spaces using wave-trace methods was studied and it was shown that in many cases one can "hear" the weights of the projective space.
Abstract: We study the inverse spectral problem for weighted projective spaces using wave-trace methods. We show that in many cases one can “hear” the weights of a weighted projective space.
TL;DR: In this paper, the integrability of this extension to a Lie group extension for non-connected, possibly infinite-dimensional Lie groups K was analyzed, and sufficient conditions for integrinability in terms of data related only to the group K were provided.
Abstract: If K is a Lie group and q : P → M is a principal K-bundle over the compact manifold M, then any invariant symmetric V-valued bilinear form on the Lie algebra $${\mathfrak{k}}$$
of K defines a Lie algebra extension of the gauge algebra by a space of bundle-valued 1-forms modulo exact 1-forms. In this article, we analyze the integrability of this extension to a Lie group extension for non-connected, possibly infinite-dimensional Lie groups K. If K has finitely many connected components, we give a complete characterization of the integrable extensions. Our results on gauge groups are obtained by the specialization of more general results on extensions of Lie groups of smooth sections of Lie group bundles. In this more general context, we provide sufficient conditions for integrability in terms of data related only to the group K.
TL;DR: In this paper, the authors extend classical eigenvalue estimates for Dirac operators on compact manifolds to non-compact, even incomplete manifolds, with positive scalar curvature.
Abstract: We extend several classical eigenvalue estimates for Dirac operators on compact manifolds to noncompact, even incomplete manifolds. This includes Friedrich’s estimate for manifolds with positive scalar curvature as well as the author’s estimate on surfaces.
TL;DR: In this paper, it was shown that a complete K-contact manifold admitting both the Einstein-Weyl structures W± = (g, ±ω) is Sasakian.
Abstract: In this paper we study Einstein-Weyl structures in the framework of contact metric manifolds. First, we prove that a complete K-contact manifold admitting both the Einstein-Weyl structures W± = (g, ±ω) is Sasakian. Next, we show that a compact contact metric manifold admitting an Einstein-Weyl structure is either K-contact or the dual field of ω is orthogonal to the Reeb vector field, provided the Reeb vector field is an eigenvector of the Ricci operator. We also prove that a contact metric manifold admitting both the Einstein-Weyl structures and satisfying \({Q\varphi = \varphi Q}\) is either K-contact or Einstein. Finally, a couple of results on contact metric manifold admitting an Einstein-Weyl structure W = (g, fη) are presented.
TL;DR: In this paper, the authors prove that an immersed topological sphere with a positive function F on S2 which satisfies a convexity condition is the Wulff shape, up to translations and homotheties.
Abstract: Given a positive function F on S2 which satisfies a convexity condition, we define a function \({H^{F}_{\mathbb{C}}}\) for surfaces in \({\mathbb{R}^3}\) which is a generalization of the usual mean curvature function. We prove that an immersed topological sphere in \({\mathbb{R}^3}\) with \({H^{F}_{\mathbb{C}}}\) = constant is the Wulff shape, up to translations and homotheties.
TL;DR: In this article, the authors studied the geometry of a tangent sphere bundle of a Riemannian manifold and proved that TrM equipped with the standard contact metric structure is η-Einstein if and only if TrM is a space of constant sectional curvature.
Abstract: We study the geometry of a tangent sphere bundle of a Riemannian manifold (M, g). Let M be an n-dimensional Riemannian manifold and TrM be the tangent bundle of M of constant radius r. The main theorem is that TrM equipped with the standard contact metric structure is η-Einstein if and only if M is a space of constant sectional curvature \({\frac{1}{r^2}}\) or \({\frac{n-2}{r^2}}\).
TL;DR: In this paper, a sequel to Chan (Ann Glob Anal Geom, to appear) on simultaneous desingularizations of Calabi-Yau and special Lagrangian (SL) 3-folds with conical singularities is presented.
Abstract: This article is a sequel to Chan (Ann Glob Anal Geom, to appear) on simultaneous desingularizations of Calabi–Yau and special Lagrangian (SL) 3-folds with conical singularities. In Chan (Ann Glob Anal Geom, to appear) we treated the question of starting with a conically singular Calabi–Yau 3-fold and an SL 3-fold with conical singularities at the same points and deforming both together to get a smooth situation. In this article, we survey the major result from Chan (Ann Glob Anal Geom, to appear) and describe some examples from our earlier articles (Chan Q J Math 57:151–181, 2006, Q J Math, to appear) on Calabi–Yau desingularizations. We then provide many explicit examples of Asymptotically Conical (AC) SL submanifolds in two specific AC Calabi–Yau manifolds. Using the result in Chan (Ann Glob Anal Geom, to appear), we construct smooth examples of compact SL 3-folds in compact Calabi–Yau 3-folds by gluing those AC SL 3-folds into some conically singular SL 3-folds at the singular points.
TL;DR: In this article, the Yamabe invariant of manifolds with boundary satisfies a monotonicity property with respect to connected sums along the boundary, similar to the one in the closed case.
Abstract: We show that the Yamabe invariant of manifolds with boundary satisfies a monotonicity property with respect to connected sums along the boundary, similar to the one in the closed case. A consequence of our result is that handlebodies have maximal invariant.
TL;DR: In this paper, the authors classify compact Kahler manifolds with quasi-constant holomorphic sectional curvatures into three classes: compact, compact, and compact. Butt et al.
Abstract: The aim of this article is to classify compact Kahler manifolds with quasi-constant holomorphic sectional curvature.
TL;DR: In this article, it was shown that the number s(n) of disjoint minimal graphs supported on domains in R-n is bounded by e(n + 1)(2).
Abstract: We prove that the number s(n) of disjoint minimal graphs supported on domains in R-n is bounded by e(n + 1)(2). In the two-dimensional case, we show that s(2) <= 3.
TL;DR: In this paper, the assumptions Ric ≥ 0 and Ric ≤ 0 are modified by using Hessian and Laplacian of a smooth positive function such that, when this function is constant, these assumptions return to Ric ≥ H > 0.
Abstract: In the well-known vanishing theorems of Bochner, the assumptions Ric ≥ 0 and Ric ≤ 0 are modified by using Hessian and Laplacian of a smooth positive function such that, when this function is constant, these assumptions return to Ric ≥ 0 and Ric ≤ 0. We prove that the assertions and results of Bochner’s vanishing theorems still hold under these modified assumptions. Additionally, the assumption Ric ≥ H > 0 given in the eigenvalue estimate theorem of Lichnerowicz is also modified in the same way, and we obtain estimates for the first positive eigenvalue of the Laplace operator.
TL;DR: In this paper, it was shown that every irreducible compact Kahler manifold with harmonic Bochner curvature tensor and negative semi-definite Ricci tensor is a Kahler-Einstein manifold.
Abstract: We prove that every irreducible Kahler manifold with harmonic Bochner curvature tensor and constant scalar curvature is Kahler–Einstein and that every irreducible compact Kahler manifold with harmonic Bochner curvature tensor and negative semi-definite Ricci tensor is Kahler–Einstein.
TL;DR: In this article, Kapanadze et al. investigated the ellipticity of boundary value problems on a smooth manifold with boundary and edge on closed manifolds with edge, where the edge symbol is a special case of the interior and boundary symbol.
Abstract: The ellipticity of boundary value problems on a smooth manifold with boundary relies on a two-component principal symbolic structure \({(\sigma_{\psi},\sigma_\partial)}\), consisting of interior and boundary symbols. In the case of a smooth edge on manifolds with boundary, we have a third symbolic component, namely, the edge symbol \({\sigma_\wedge}\), referring to extra conditions on the edge, analogously as boundary conditions. Apart from such conditions ‘in integral form’ there may exist singular trace conditions, investigated in Kapanadze et al., Internal Equations and Operator Theory, 61, 241–279, 2008 on ‘closed’ manifolds with edge. Here, we concentrate on the phenomena in combination with boundary conditions and edge problem.
TL;DR: In this article, Mok-Siu-Yeung type formulas for horizontal pseudoharmonic maps from compact contact locally sub-symmetric spaces into strictly pseudoconvex CR manifolds were derived.
Abstract: We derive Mok-Siu-Yeung type formulas for horizontal maps from compact contact locally sub-symmetric spaces into strictly pseudoconvex CR manifolds and we obtain some rigidity theorems for the horizontal pseudoharmonic maps.
TL;DR: Chan et al. as mentioned in this paper extended this result to include desingularizations of special Lagrangian (SL) 3-folds with conical singularities that occur at the same points of the ambient Calabi-Yau manifold.
Abstract: In this paper we extend our previous results on resolving conically singular Calabi–Yau 3-folds (Chan, Quart. J. Math. 57:151–181, 2006; Quart. J. Math., to appear) to include the desingularizations of special Lagrangian (SL) 3-folds with conical singularities that occur at the same points of the ambient Calabi–Yau. The gluing construction of the SL 3-folds is achieved by applying Joyce’s analytic result (Joyce, Ann. Global. Anal. Geom. 26: 1–58, 2004, Thm. 5.3) on deforming Lagrangian submanifolds to nearby special Lagrangian submanifolds. Our result will in principle be able to construct more examples of compact SL submanifolds in compact Calabi–Yau manifolds. Various explicit examples and applications illustrating the result in this paper can be found in the sequel (Chan, Ann. Global. Anal. Geom., to appear).
TL;DR: In this paper, the eigenvalues of the Dirichlet Laplacian acting on differential forms of a Euclidean compact submanifold were shown to be universal.
Abstract: We obtain ‘universal’ inequalities for the eigenvalues of the Laplacian acting on differential forms of a Euclidean compact submanifold. These inequalities generalize the Yang inequality concerning the eigenvalues of the Dirichlet Laplacian of a bounded Euclidean domain.