The Structure of a Class of K-contact Manifolds

In this paper we study a class of K-contact manifolds, namely -conformally flat K-contact manifolds and we show that a compact -conformally flat K-contact manifold with regular contact vector field is a principal S¹-bundle over an almost Kaehler space of constant holomorphic sectional curvature.     

On the Existence and Regularity of Mass-Minimizing Currents with an Elastic Boundary

We study the following variational problem. For a compact manifold S₀ embedded in the Euclidean space we consider deformations of S₀. They are represented by Lipschitz continuous homeomorphisms of S₀ whose images are embedded manifolds. We introduce an energy of a deformation which depends on the first derivative of the curvature of (S₀) and the mass of a mass minimizing current which is bounded by (S₀). In this paper it is shown that an energy minimizing deformation of (S₀) exists. Moreover, in the case that S₀ has codimension 1, (S₀) is an embedded C^3a -submanifold, if is of the class C^2,1.     

Torsion tensor and critical metrics on contact (2n+1)-manifolds

Contact Riemannian manifolds (M, ,g) satisfying the condition (1) =0, where is the torsion introduced byChern andHamilton [6] and is the characteristic vector field, have interesting geometric properties (see [6], [9], [11]). In this paper we give a variational characterization of compact contact Riemannian manifolds which satisfy (1). Moreover we study the tangent sphere bundles (T ₁ M, , g), where (,g) is the standard contact Riemannian structure, which satisfy the condition (1); in particular in the 3-dimensional case we find a surprising result (see Corollary 5.3).Supported by funds of the M.U.R.S.T.     

Standard pseudo-Hermitian structure on manifolds and seifert fibration

A strictly pseudoconvex pseudo-Hermitian manifoldM admits a canonical Lorentz metric as well as a canonical Riemannian metric. Using these metrics, we can define a curvaturelike function onM. AsM supports a contact form, there exists a characteristic vector field dual to the contact structure. If induces a local one-parameter group ofCR transformations, then a strictly pseudoconvex pseudo-Hermitian manifoldM is said to be a standard pseudo-Hermitian manifold. We study topological and geometric properties of standard pseudo-Hermitian manifolds of positive curvature or of nonpositive curvature . By the definition, standard pseudo-Hermitian manifolds are calledK-contact manifolds by Sasaki. In particular, standard pseudo-Hermitian manifolds of constant curvature turn out to be Sasakian space forms. It is well known that a conformally flat manifold contains a class of Riemannian manifolds of constant curvature. A sphericalCR manifold is aCR manifold whose Chern-Moser curvature form vanishes (equivalently, Weyl pseudo-conformal curvature tensor vanishes). In contrast, it is emphasized that a sphericalCR manifold contains a class of standard pseudo-Hermitian manifolds of constant curvature (i.e., Sasakian space forms). We shall classify those compact Sasakian space forms. When 0, standard pseudo-Hermitian closed aspherical manifolds are shown to be Seifert fiber spaces. We consider a deformation of standard pseudo-Hermitian structure preserving a sphericalCR structure.Dedicated to Professor Sasao Seiya for his sixtieth birthday     

Disconnectedness of sublevel sets of some Riemannian functionals

LetM be a compact manifold of dimension greater than four. Denote byRiem(M) the space of Riemannian structures onM (i.e. of isometry classes of Riemannian metrics onM) endowed with the Gromov-Hausdorff metric. LetRiem (M) Riem(M) be its subset formed by all Riemannian structures such that vol()=1 andinj() , whereinj() denotes the injectivity radius of.We prove that for all sufficiently small positive the spaceRiem (M) is disconnected. Moreover, if is sufficiently small, thenRiem (M) is representable as the union of two non-empty subsetsA andB such that the Gromov-Hausdorff distance between any element ofA and any element ofB is greater than/9. We also prove a more general result with the following informal meaning: There exist two Riemannian structures of volume one and arbitrarily small injectivity radius onM such that any continuous path (and even any sequence of sufficiently small jumps) in the space of Riemannian structures of volume one onM connecting these Riemannian structures must pass through Riemannian structures of injectivity radius uncontrollably smaller than the injectivity radii of these two Riemannian structures.These results can be generalized for at least some four-dimensional manifolds. The technique used in this paper can also be used to prove the disconnectedness of many other subsets of the space of Riemannian structures onM formed by imposing various constraints on curvatures, volume, diameter, etc.This work was partially supported by the New York University Research Challenge Fund grant, by NSF grant DMS 9114456 and by the NSERC operating grant OGP0155879.     

Bounds for the first Dirichlet eigenvalue attained at an infinite family of Riemannian manifolds

LetM be a compact Riemannian manifold with smooth boundary M. We get bounds for the first eigenvalue of the Dirichlet eigenvalue problem onM in terms of bounds of the sectional curvature ofM and the normal curvatures of M. We discuss the equality, which is attained precisely on certain model spaces defined by J. H. Eschenburg. We also get analog results for Kähler manifolds. We show how the same technique gives comparison theorems for the quotient volume(P)/volume(M),M being a compact Riemannian or Kähler manifold andP being a compact real hypersurface ofM.Work partially supported by a DGICYT Grant No. PB94-0972 and by the E.C. Contract CHRX-CT92-0050 GADGET II.     

Three dimensional expansive diffeomorphisms with homoclinic points

LetM be a compact connected oriented three dimensional manifold andf:MM an expansive diffeomorphism such that (f)=M. Let us also assume that there is a hyperbolic periodic point with a homoclinic intersection. Thenf is conjugate to an Anosov isomorphism ofT ³. Moreover, we show that at a homoclinic point the stable and unstable manifolds of the hyperbolic periodic point are topologically transverse.     

A global pinching theorem for compact surfaces inS 3 with constant mean curvature

LetM be a compact minimal surface inS ³. Y. J. Hsu^[5] proved that if S₂22, thenM is either the equatorial sphere or the Clifford torus, whereS is the square of the length of the second fundamental form ofM, ·₂ denotes theL ²-norm onM. In this paper, we generalize Hsu's result to any compact surfaces inS ³ with constant mean curvature.Supported by NSFH.     

Uniqueness for the harmonic map flow in two dimensions

LetM be a two-dimensional Riemannian manifold with smooth (possibly empty) boundary. Ifu andv are weak solutions of the harmonic map flow inH ¹(M×[0,T]; S^N) whose energy is non-increasing in time and having the same initial data u₀ H¹(M,S^N) (and same boundary values H ^3/2(M; S^N) if M; S^N Ø) thenu=v.     

Flächen im isotropen s2×ℝ

In [4] the author published the theory of curves in isotropic S² × . New results of Pottmann [1] show, that isotropic geometry has a meaning in CAGD, especially in questions on scattered data and visualisation. These are not only considered in euclidean space, but also on manifolds. So it may be interesting to look at the theory of surfaces in isotropic manifolds. This will be done in this paper for the manifold S² × by embedding it in I₄. Special surfaces on isotropic S² × will be geometrically interpreted.

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Herrn Prof. Dr. Oswald Giering zum 60. Geburtstag gewidmet     

A Liouville-Type Theorem for Strongly Subharmonic Functions on Complete Non-compact Riemannian Manifolds and Some Applications

We prove that if Mis a complete non-compact Riemannian manifold and ₁(M)=0, then any C ²solution of uk> 0 is unbounded. We apply this result to obtain an estimate for the size of the image set of some types of maps between Riemannian manifolds.     

Surgery, Curvature, and Minimal Volume

On a Riemannian manifold X, we consider aK+s, where a is a nonnegative constant, K is the sectional curvature and s is the scalar curvature. It is shown that if X admits a metric with aK+s > 0, then so does any manifold obtained from X by surgeries of codimension 3. This implies the existence of such metrics on certain compact simply connected manifolds of dimension 5 by using the cobordism argument. We also study the corresponding minimal volume problem. As a corollary, we derive that every compact simply connected manifold of dimension 5 and every compact complex surface of Kodaira dimension 1 whose minimal model is not of Class VII collapse with aK+s bounded below.     

Gelfand theory of certainC *-algebras on nonasymptotically flat manifolds

Let be a Riemannian manifold with finitely many conical ends. Under certain conditions which do not require to be asymptotically flat, we study aC ^*-algebra containing pseudodifferential operators on . Results on compact commutators and the Gelfand space are presented. Criteria for differential operators or systems within reach to be Fredholm are just simple consequences of Atkinson's theorem and our results.     

Some topological properties of cohomogeneity one manifolds with negative curvature

This paper is aimed at studying negatively curved Riemannian manifolds acted on by a Lie group of isometries with principal orbits of codimension one. The orbit space of such a manifold M is proved to be always homeomorphic to or ⁺ and this second case may occur only when either the singular orbit is a geodesic of M or when the space is simply connected. Several corollaries are given.     

Three-dimensional hyperelliptic manifolds

Let X³ = H³, E³, S³, H² × E¹, S² × E¹, T₁(H²), Nil of Solv be one of the eight 3-dimensional geometrics of Thurston [10] and G be a discrete group of isometrics of X³ acting without fixed points. A manifold M³ = X³/G is said to be hyperelliptic if there is an isometric involution on it such that the factor space M³/<> is diffeomorphic to the 3-sphere S³. In analogy with the theory of Riemann surfaces we call involution.In the present paper the existence of hyperelliptic manifolds in each light of the eight 3-dimensional geometrics will be obtained. All the proofs given there will be written in the language of orbifolds whose basic facts can be found in [9].     

Foliated control theory, II

The main result is a control theorem for the structure space of E with control near the leaves F in M, where : E M is a fiber bundle over the Riemannian manifold M having a compact closed manifold for fiber and F is a smooth foliation of M, each leaf of which inherits a flat Riemannian geometry from M. A similar result has been proved by the authors under the assumption that each leaf of F is one-dimensional and the fiber of : E M is homotopy stable.Both authors were supported in part by the National Science Foundations.     

A topological obstruction to the geodesibility of a foliation of odd dimension

Let M be a compact Riemannian manifold of dimension n, and let be a smooth foliation on M. A topological obstruction is obtained, similar to results of R. Bott and J. Pasternack, to the existence of a metric on M for which is totally geodesic. In this case, necessarily that portion of the Pontryagin algebra of the subbundle must vanish in degree n if is odd-dimensional. Using the same methods simple proofs of the theorems of Bott and Pasternack are given.     

Homogeneous Submanifolds of Codimension Two

We study codimension 2 homogeneous submanifolds of Euclidean space for which the index of minimum relative nullity is small. We prove that if min_xMf(x)n-5, where (x) denotes the nullity of the second fundamental form of the immersion f at the point x, then the manifold M ⁿ is either isometric to a sphere or to a product of two spheres S²×S ^n–2 or covered by the Riemannian product S ^n–1 ×R. As a consequence, we obtain a classification of compact codimension 2 homogeneous submanifolds of dimension at least 5.     

Spinc Structures and Scalar Curvature Estimates

In this note, we look at estimates for the scalar curvature of a compact, connected Riemannian manifold Mwhich are related to spin ^c Dirac operators.We show that one may not enlarge a Kähler metric with positiveRicci curvature without making smaller somewhere on M.More generally, if f: N M is an area-nonincreasing map of a certain topological type,then the scalar curvature k of Ncannot be everywhere larger than f.If k f, then N is isometric to M × F, where F possesses a parallel untwisted spinor.We also give explicit upper bounds for min for arbitrary Riemannian metrics on certainsubmanifolds of complex projective space.In certain cases, these estimates are sharp:we give examples where equality is obtained.     

Manifolds of Maps in Riemannian Foliations

Let (M,F) and (M,F) be two (compact or not) foliated manifolds, C _F (M, M) the space of smooth maps which send leaves into leaves. In this paper we prove that C _F (M, M) admits a structure of an infinite-dimensional manifold modeled on LF-spaces, provided that F is a Riemannian foliation or, more generally, when it admits an adapted local addition.