Showing papers in "Annals of Global Analysis and Geometry in 2005"

Classification des variété approximativement kähleriennes homogénes

Abstract: A Riemannian homogeneous manifold admitting a strict nearly-Kahler structure is 3-symmetric. We actually classify them in dimension 6 and use previous results of Swann, Cleyton and Nagy to prove the conjecture in higher dimensions. The six-dimensional homogeneous spaces, S3 × S3, S6, CP(3) and the flag manifold F(1, 2) have a unique (after a change of scale) nearly-Kahler, invariant structure. For the first one we solve a differential equation on the SU(3)-structure given by Reyes Carrion. For the last two it is obtained by canonical variation of the Kahler structure of the twistor space over a four-dimensional manifold. Finally, from Bar, a nearly-Kahler structure on the sphere S6 corresponds to a constant 3-form on the Riemannian cone R7.

Weyl Transforms, the Heat Kernel and Green Function of a Degenerate Elliptic Operator

Abstract: We give a formula for the heat kernel of a degenerate elliptic partial differential operator L on ℝ2 related to the Heisenberg group. The formula is derived by means of pseudo-differential operators of the Weyl type, {i.e.}, Weyl transforms, and the Fourier–Wigner transforms of Hermite functions, which form an orthonormal basis for L2(ℝ2). Using the heat kernel, we give a formula for the Green function of L. Applications to the global hypoellipticity of L in the sense of tempered distributions, the ultracontractivity and hypercontractivity of the strongly continuous one-parameter semigroup e−tL, t > 0, are given.

Centers of convex subsets of buildings

Abstract: We prove that two-dimensional convex subsets of spherical buildings are either buildings or have a center.

Spectral Bounds on Orbifold Isotropy

Abstract: We first show that a Laplace isospectral family of Riemannian orbifolds, satisfying a lower Ricci curvature bound, contains orbifolds with points of only finitely many isotropy types. If we restrict our attention to orbifolds with only isolated singularities, and assume a lower sectional curvature bound, then the number of singular points in an orbifold in such an isospectral family is universally bounded above. These proofs employ spectral theory methods of Brooks, Perry and Petersen, as well as comparison geometry techniques developed by Grove and Petersen.

General Normal Cycles and Lipschitz Manifolds of Bounded Curvature

Abstract: Closed Legendrian (d − 1)-dimensional locally rectifiable currents on the sphere bundle in \(\mathbb{R}\)d are considered and the associated index functions are studied. A topological condition assuring the validity of a local version of the Gauss–Bonnet formula is established. The case of lower-dimensional Lipschitz submanifolds in \(\mathbb{R}\)d and their associated normal cycles is examined in detail.

Discretization of Compact Riemannian Manifolds Applied to the Spectrum of Laplacian

Abstract: For κ ⩾ 0 and r0 > 0 let ℳ(n, κ, r0) be the set of all connected, compact n-dimensional Riemannian manifolds (M n , g) with Ricci (M, g) ⩾ −(n−1) κ g and Inj (M) ⩾ r0. We study the relation between the kth eigenvalue λ k (M) of the Laplacian associated to (M n ,g), Δ = −div(grad), and the kth eigenvalue λ k (X) of a combinatorial Laplacian associated to a discretization X of M. We show that there exist constants c, C > 0 (depending only on n, κ and r0) such that for all M ∈ ℳ(n, κ, r0) and X a discretization of $${M, c \leqslant \frac{\lambda_{k}(M)}{\lambda_{k}(X)} \leqslant C}$$ for all k 0. We show that there exist constants c, C > 0 depending on η, n, κ and r0 such that $${c \leqslant \frac{\lambda_{k}(M)}{\lambda_{k}(N)} \leqslant C}$$ for all $${k \in \mathbb{N}}$$ .

Hamiltonian Stationary Lagrangian Surfaces in ℂP2

Abstract: In this paper we use a new equivalent condition of Hamiltonian stationary Lagrangian surfaces in ℂP2 to show that any Hamiltonian stationary Lagrangian torus in ℂP2 can be constructed from a pair of commuting Hamiltonian ODEs on a finite dimensional subspace of a certain loop Lie algebra, i.e., is of finite type.

Initial Value Problems of the Sine-Gordon Equation and Geometric Solutions

Abstract: Recent results using inverse scattering techniques interpret every solution φ(x, y) of the sine-Gordon equation as a nonlinear superposition of solutions along the axes x=0 and y=0. This has a well-known geometric interpretation, namely that every weakly regular surface of Gauss curvature K=−1, in arc length asymptotic line parametrization, is uniquely determined by the values φ(x, 0) and φ(0, y) of its coordinate angle along the axes. We introduce a generalized Weierstrass representation of pseudospherical surfaces that depends only on these values, and we explicitely construct the associated family of pseudospherical immersions corresponding to it.

O(m) × O(n)-Invariant Minimal Hypersurfaces in \(\mathbb{R}\)m+n

Abstract: We classify the nonextendable immersed O(m) × O(n)-invariant minimal hypersurfaces in the Euclidean space \(\mathbb{R}\)m+n, m, n ≥ 3, analyzing also whether they are embedded or stable. We show also the existence of embedded, complete, stable minimal hypersurfaces in \(\mathbb{R}\)m+n, m + n ≥ 8, m, n ≥ 3 not homeomorphic to \(\mathbb{R}\)m+n−1 that are O(m) × O(n)-invariant.

Calibrated Subbundles in Noncompact Manifolds of Special Holonomy

Abstract: This paper is a continuation of Math. Res. Lett. 12 (2005), 493–512. We first construct special Lagrangian submanifolds of the Ricci-flat Stenzel metric (of holonomy SU(n)) on the cotangent bundle of S n by looking at the conormal bundle of appropriate submanifolds of S n . We find that the condition for the conormal bundle to be special Lagrangian is the same as that discovered by Harvey–Lawson for submanifolds in R n in their pioneering paper, Acta Math. 148 (1982), 47–157. We also construct calibrated submanifolds in complete metrics with special holonomy G2 and Spin(7) discovered by Bryant and Salamon (Duke Math. J. 58 (1989), 829–850) on the total spaces of appropriate bundles over self-dual Einstein four manifolds. The submanifolds are constructed as certain subbundles over immersed surfaces. We show that this construction requires the surface to be minimal in the associative and Cayley cases, and to be (properly oriented) real isotropic in the coassociative case. We also make some remarks about using these constructions as a possible local model for the intersection of compact calibrated submanifolds in a compact manifold with special holonomy.

Killing forms on quaternion-Kähler manifolds

Abstract: We show that every Killing p-form on a compact quaternion manifold has to be parallel for p≥ 2.

tt * -Geometry and Pluriharmonic Maps

Abstract: In this paper we use the real differential geometric definition of a metric (a unimodular oriented metric) tt*-bundle of Cortes and the author (Topological-anti-topological fusion equations, pluriharmonic maps and special Kahler manifolds) to define a map Φ from the space of metric (unimodular oriented metric) tt*-bundles of rank r over a complex manifold M to the space of pluriharmonic maps from M to {GL}(r)/O(p,q) (respectively {SL}(r)/SO(p,q)), where (p,q) is the signature of the metric. In the sequel the image of the map Φ is characterized. It follows, that in signature (r,0) the image of Φ is the whole space of pluriharmonic maps. This generalizes a result of Dubrovin (Comm. Math. Phys. 152 (1992; S539–S564).

Complete k-Curvature Homogeneous Pseudo-Riemannian Manifolds

Abstract: . For k ≥ 2, we exhibit complete k-curvature homogeneous neutral signature pseudo-Riemannian manifolds which are not locally affine homogeneous (and hence not locally homogeneous). All the local scalarWeyl invariants of these manifolds vanish. These manifolds are Ricci flat, Osserman, and Ivanov-Petrova.

Marginally Trapped Surfaces in L4 and an Extended Weierstrass–Bryant Representation

Abstract: Let E be the total space of a locally trivial torus bundle over a surface Σ g of genus g > 1. Using Seiberg–Witten theory and spectral sequences, we prove that E carries a symplectic structure if and only if the homology class of the fiber [T 2] is nonzero in H 2(E, R).

The Index of b-Pseudodifferential Operators on Manifolds with Corners

Abstract: On compact manifolds with corners of arbitrary codimension, we characterize the ‘multi-cylindrical end’ (or b-type) pseudodifferential operators that are Fredholm on weighted Sobolev spaces and we compute their indices. The index formula contains the usual interior term manufactured from the local symbols of the operator and also contains boundary correction terms corresponding to eta-type invariants of the induced operators on the boundary faces.

Moduli of Framed Parabolic Sheaves

Abstract: This paper gives a construction of the moduli space of framed parabolic sheaves on a Riemann surface This space serves as a universal, master, space for the well known moduli space of parabolic bundles, as well as moduli spaces of vector bundles, which can all be obtained from this space by torus quotients The construction is given for the structure group SL(N, C), and indeed is adapted to this case At the end of the paper, an approach is suggested for dealing with the case of arbitrary reductive groups, involving the associated loop group

Complex Product Structures and Affine Foliations

Abstract: A complex product structure on a manifold is an appropriate combination of a complex structure and a product structure. The existence of such a structure determines many interesting properties of the underlying manifold, notably that the manifold admits a pair of complementary foliations whose leaves carry affine structures. This is due to the existence of a unique torsion-free connection which preserves both the complex and the product structure; this connection is not necessarily flat. We study the existence of complex product structures on tangent bundles of smooth manifolds, and we investigate the structure of manifolds admitting a complex product structure and a compatible hypersymplectic metric, showing that the foliations mentioned earlier are either symplectic or Lagrangian, depending on the symplectic form under consideration.

p -Laplace Operator and Diameter of Manifolds

Abstract: Let $${(M^{n},g)}$$ be a compact Riemannian manifold without boundary. In this paper, we consider the first nonzero eigenvalue of the p-Laplacian $${\lambda_{1,p}(M)}$$ and we prove that the limit of $$p{\sqrt {\lambda_{1,p}(M)}}$$ when $$p\rightarrow\infty$$ is 2/d(M), where d(M) is the diameter of M. Moreover, if $${(M^{n},g)}$$ is an oriented compact hypersurface of the Euclidean space $${\mathbb{R}^{n+1}}$$ or $${\mathbb{S}^{n+1}}$$ , we prove an upper bound of $${\lambda_{1,p}(M)}$$ in terms of the largest principal curvature κ over M. As applications of these results, we obtain optimal lower bounds of d(M) in terms of the curvature. In particular, we prove that if M is a hypersurface of $${\mathbb{R}^{n+1}}$$ then: $$d(M)\ge\pi/\kappa$$ .

Existence of symplectic structures on torus bundles over surfaces

Abstract: Let E be the total space of a locally trivial torus bundle over a surface Σg of genus g > 1. Using Seiberg–Witten theory and spectral sequences, we prove that E carries a symplectic structure if and only if the homology class of the fiber [T2] is nonzero in H2(E, ℝ).

Integrability of Analytic Almost Complex Structures on Banach Manifolds

Abstract: We prove that the classical integrability condition for almost complex structures on finite-dimensional smooth manifolds also works in infinite dimensions in the case of almost complex structures that are real analytic on real analytic Banach manifolds. With this result at hand, we extend some known results concerning existence of invariant complex structures on homogeneous spaces of Banach–Lie groups. By way of illustration, we construct the complex flag manifolds associated with unital C*-algebras.

Moduli space of Fedosov structures

Abstract: We consider the space of germs of Fedosov structures at a point, under the action of origin-preserving diffeomorphisms. We calculate dimensions of moduli spaces of k-jets of generic structures and construct the Poincare series of the moduli space. It is shown to be a rational function.

The Moduli Space of Six-Dimensional Two-Step Nilpotent Lie Algebras

Abstract: We determine the moduli space of metric two-step nilpotent Lie algebras of dimension up to 6. This space is homeomorphic to a cone over a four-dimensional contractible simplicial complex. Moreover, we exhibit standard metric representatives of the seven isomorphism types of six-dimensional two-step nilpotent Lie algebras within our picture.

Schiffer’s Problem and an Isoperimetric Inequality for the First Buckling Eigenvalue of Domains on \(\mathbb{S}^{2}\)

Abstract: We consider the overdetermined eigenvalue problem on a sufficiently regular connected open domain Ω on the 2-sphere \(\mathbb{S}^2\): $$ \begin{array}{@{}l@{}} \Delta u+\alpha u = 0\quad {\rm in}\quad \Omega,\\ u = {\rm constant},\ \displaystyle\frac{\partial u}{\partial u} = {\rm constant\quad on}\quad \partial \Omega, \end{array}$$ where α ≠ 0. We show that if α = 2 and Ω is simply connected then the problem admits a (nonzero) solution if and only if Ω is a geodesic disk. We furthermore extend to domains on \(\mathbb{S}^{2}\) the isoperimetric inequality of Payne–Weinberger for the first buckling eigenvalue of compact planar domains. As a corollary we prove that Ω is a geodesic disk if the above overdetermined eigenvalue problem admits a (nonzero) solution with ∂u/∂ν = 0 on ∂Ω and α = λ2 the second eigenvalue of the Laplacian with Dirichlet boundary condition. This extends a result proved in the case of the Euclidean plane by C. Berenstein.

On Stability of Cones in Rn+1 with Zero Scalar Curvature

Abstract: In this work we generalize the case of scalar curvature zero the results of Simmons (Ann. Math. 88 (1968), 62–105) for minimal cones in Rn+1. If Mn−1 is a compact hypersurface of the sphere S n (1) we represent by C(M)e the truncated cone based on M with center at the origin. It is easy to see that M has zero scalar curvature if and only if the cone base on M also has zero scalar curvature. Hounie and Leite (J. Differential Geom. 41 (1995), 247–258) recently gave the conditions for the ellipticity of the partial differential equation of the scalar curvature. To show that, we have to assume n ⩾ 4 and the three-curvature of M to be different from zero. For such cones, we prove that, for n ≤slant 7 there is an e for which the truncate cone C(M)e is not stable. We also show that for n ⩾ 8 there exist compact, orientable hypersurfaces Mn−1 of the sphere with zero scalar curvature and S3 different from zero, for which all truncated cones based on M are stable.

Positively Curved Self-Dual Einstein Metrics on Weighted Projective Planes

Abstract: We classify positively curved self-dual Einstein Hermitian orbifold metrics of Galicki – Lawson on the weighted projective planes. We thus determine which of the 3-Sasakian S1-reductions of S11 possess canonical variation metrics of positive sectional curvature.

Einstein Metrics, Symplectic Minimality, and Pseudo-Holomorphic Curves

Abstract: Let (M4, g,ω) be a compact, almost-Kahler–Einstein manifold of negative star-scalar curvature. Then (M,ω) is a minimal symplectic 4-manifold of general type. In particular, M cannot be differentiably decomposed as a connected sum $$N\,{\#}\,\overline{\mathbb{CP}}_2$$ .

Towards an Effectivisation of the Riemann Theorem

Abstract: Let Q be a connected and simply connected domain on the Riemann sphere, not coinciding with the Riemann sphere and with the whole complex plane ℂ. Then, according to the Riemann Theorem, there exists a conformal bijection between Q and the exterior of the unit disk. In this paper, we find an explicit form of this map for a broad class of domains with analytic boundaries.

The Rough Laplacian and Harmonicity of Hopf Vector Fields

Abstract: Let M be an orientable real hypersurface of a general Kahler manifold \(\bar{M}\). The characteristic vector field ξ of the induced almost contact metric structure (ξ,η, g,ϕ) is also called the Hopf vector field of M. In this paper, we compute the ‘rough’ Laplacian of ξ in terms of the shape operator A and also (as a natural generalization of the contact metric case) in terms of torsion τ = Lξg. Then we give some criteria of harmonicity of ξ. Moreover, we consider hypersurfaces M of contact type and give some criteria for M to admit an H-contact structure.

Conjugate Loci of Pseudo-Riemannian 2-Step Nilpotent Lie Groups with Nondegenerate Center

Abstract: We determine the complete conjugate locus along all geodesics parallel or perpendicular to the center (Theorem 2.3). When the center is one-dimensional we obtain formulas in all cases (Theorem 2.5), and when a certain operator is also diagonalizable these formulas become completely explicit (Corollary 2.7). These yield some new information about the smoothness of the pseudoriemannian conjugate locus. We also obtain the multiplicities of all conjugate points.

Volume and Projective Equivalence Between Riemannian Manifolds

Abstract: Let (M, g) and (M, $${\bar{g}}$$ ) be two Riemannian metrics which are pointwise projectively equivalent, i.e. they have the same geodesics as point sets. We prove that the pointwise projective equivalence is trivial, if (M, g) is a noncompact complete manifold which has at most quadratic volume growth and nonnegative total scalar curvature, and (M, $${\bar{g}}$$ ) has nonpositive Ricci curvature.