Flat Riemannian manifolds are boundaries

In this paper we prove that each compact flat Riemannian manifold is the boundary of a compact manifold. Our method of proof is to construct a smooth action of (₂) ^k on the flat manifold. We are independently preceded in this approach by Marc W. Gordon who proved the flat Riemannian manifolds, whose holonomy groups are of a certain class of groups, bound. By analyzing the fixed point data of this group action we get the complete result. As corollaries to the main theorem it follows that those compact flat Riemannian manifolds which are oriented bound oriented manifolds; and, if we have an involution on a homotopy flat manifold, then the manifold together with the involution bounds. We also give an example of a nonbounding manifold which is finitely covered byS ³ ×S ³ ×S ³.     

Symmetries of Lagrangian fibrations

We construct fiber-preserving anti-symplectic involutions for a large class of symplectic manifolds with Lagrangian torus fibrations. In particular, we treat the K3 surface and the six-dimensional examples constructed by Castaño-Bernard and Matessi (2009) [8], which include a six-dimensional symplectic manifold homeomorphic to the quintic threefold. We interpret our results as corroboration of the view that in homological mirror symmetry, an anti-symplectic involution is the mirror of duality. In the same setting, we construct fiber-preserving symplectomorphisms that can be interpreted as the mirror to twisting by a holomorphic line bundle.     

Symplectic involutions on deformations of K3[2]

Let X be a hyperk?hler manifold deformation equivalent to the Hilbert square of a K3 surface and let φ be an involution preserving the symplectic form. We prove that the fixed locus of φ consists of 28 isolated points and one K3 surface, and moreover that the anti-invariant lattice of the induced involution on H ₂(X, ℤ) is isomorphic to E ₈(−2). Finally we show that any couple consisting of one such manifold and a symplectic involution on it can be deformed into a couple consisting of the Hilbert square of a K3 surface and the involution induced by a symplectic involution on the K3 surface.     

Nonexistence of minimal Lagrangian spheres in hyperKähler manifolds

We prove that for one cannot immerse as a minimal Lagrangian manifold into a hyperK?hler manifold. More generally we show that any minimal Lagrangian immersion of an orientable closed manifold into a hyperK?hler manifold must have nonvanishing second Betti number and that if , is a K?hler manifold and more precisely a K?hler submanifold in w.r.t. one of the complex structures on . In addition we derive a result for the other Betti numbers. Received February 10, 1999 / Accepted April 23, 1999     

The heat kernel weighted Hodge Laplacian on noncompact manifolds

On a compact orientable Riemannian manifold, the Hodge Laplacian has compact resolvent, therefore a spectral gap, and the dimension of the space of harmonic -forms is a topological invariant. By contrast, on complete noncompact Riemannian manifolds, is known to have various pathologies, among them the absence of a spectral gap and either ``too large' or ``too small' a space . In this article we use a heat kernel measure to determine the space of square-integrable forms and to construct the appropriate Laplacian . We recover in the noncompact case certain results of Hodge's theory of in the compact case. If the Ricci curvature of a noncompact connected Riemannian manifold is bounded below, then this ``heat kernel weighted Laplacian' acts on functions on in precisely the manner we would wish, that is, it has a spectral gap and a one-dimensional kernel. We prove that the kernel of on -forms is zero-dimensional on , as we expect from topology, if the Ricci curvature is nonnegative. On Euclidean space, there is a complete Hodge theory for . Weighted Laplacians also have a duality analogous to Poincaré duality on noncompact manifolds. Finally, we show that heat kernel-like measures give desirable spectral properties (compact resolvent) in certain general cases. In particular, we use measures with Gaussian decay to justify the statement that every topologically tame manifold has a strong Hodge decomposition.

     

The Geometry of Two Generator Groups: Hyperelliptic Handlebodies

A Kleinian group naturally stabilizes certain subdomains and closed subsets of the closure of hyperbolic three space and yields a number of different quotient surfaces and manifolds. Some of these quotients have conformal structures and others hyperbolic structures. For two generator free Fuchsian groups, the quotient three manifold is a genus two solid handlebody and its boundary is a hyperelliptic Riemann surface. The convex core is also a hyperelliptic Riemann surface. We find the Weierstrass points of both of these surfaces. We then generalize the notion of a hyperelliptic Riemann surface to a hyperelliptic three manifold. We show that the handlebody has a unique order two isometry fixing six unique geodesic line segments, which we call the Weierstrass lines of the handlebody. The Weierstrass lines are, of course, the analogue of the Weierstrass points on the boundary surface. Further, we show that the manifold is foliated by surfaces equidistant from the convex core, each fixed by the isometry of order two. The restriction of this involution to the equidistant surface fixes six generalized Weierstrass points on the surface. In addition, on each of these equidistant surfaces we find an orientation reversing involution that fixes curves through the generalized Weierstrass points.Mathematics Subject Classifications (2000). primary 30F10, 30F35, 30F40; secondary 14H30, 22E40.     

Compact leaves of structurally stable foliations

We prove that any compact manifold whose fundamental group contains an abelian normal subgroup of positive rank can be represented as a leaf of a structurally stable suspension foliation on a compact manifold. In this case, the role of a transversal manifold can be played by an arbitrary compact manifold. We construct examples of structurally stable foliations that have a compact leaf with infinite solvable fundamental group which is not nilpotent. We also distinguish a class of structurally stable foliations each of whose leaves is compact and locally stable in the sense of Ehresmann and Reeb.     

The visual core of a hyperbolic 3-manifold

In this note we introduce the notion of the visual core of a hyperbolic 3-manifold , and explore some of its basic properties. We investigate circumstances under which the visual core of a cover of N embeds in N, via the usual covering map . We go on to show that if the algebraic limit of a sequence of isomorphic Kleinian groups is a generalized web group, then the visual core of the algebraic limit manifold embeds in the geometric limit manifold. Finally, we discuss the relationship between the visual core and Klein-Maskit combination along component subgroups. Received: 16 March 1999 / Revised version: 14 May 2001 / Published online: 19 October 2001     

Twofold unbranched coverings of genus two 3-manifolds are hyperelliptic

A closed 3-dimensional manifold is hyperelliptic if it admits an involution such that the quotient space of the manifold by the action of the involution is homeomorphic to the 3-sphere. We prove that a twofold unbranched covering of a genus two 3-manifold is hyperelliptic. This result is reminiscent of a theorem, which seems to have first appeared in a paper by Enriques and which has been reproved more recently by Farkas and Accola, which states that a twofold unbranched covering of a Riemann surface of genus two is hyperelliptic.     

A new geometric construction of compact complex manifolds in any dimension

We consider holomorphic linear foliations of dimension m of (with ) fulfilling a so-called weak hyperbolicity condition and equip the projectivization of the leaf space (for the foliation restricted to an adequate open dense subset) with a structure of compact, complex manifold of dimension . We show that, except for the limit-case where we obtain any complex torus of any dimension, this construction gives non-symplectic manifolds, including the previous examples of Hopf, Calabi-Eckmann, Haefliger (linear case), Loeb-Nicolau (linear case) and López de Medrano-Verjovsky. We study some properties of these manifolds, that is to say meromorphic functions, holomorphic vector fields, forms and submanifolds. For each manifold, we construct an analytic space of deformations of dimension and show that, under some additional conditions, it is universal. Lastly, we give explicit examples of new compact, complex manifolds, in particular of connected sums of products of spheres and show the existence of a momentum-like map which classifies these manifolds, up to diffeomorphism. Received: 28 October 1998 / in final form: 7 September 1999     

Geometric cohomology frames on Hausmann-Holm-Puppe conjugation spaces

For certain manifolds with an involution the mod 2 cohomology ring of the set of fixed points is isomorphic to the cohomology ring of the manifold, up to dividing the degrees by two. Examples include complex projective spaces and Grassmannians with the standard antiholomorphic involution (with real projective spaces and Grassmannians as fixed point sets).

Hausmann, Holm and Puppe have put this observation in the framework of equivariant cohomology, and come up with the concept of conjugation spaces, where the ring homomorphisms arise naturally from the existence of what they call cohomology frames. Much earlier, Borel and Haefliger had studied the degree-halving isomorphism between the cohomology rings of complex and real projective spaces and Grassmannians using the theory of complex and real analytic cycles and cycle maps into cohomology.

The main result in the present note gives a (purely topological) connection between these two results and provides a geometric intuition into the concept of a cohomology frame. In particular, we see that if every cohomology class on a manifold with involution is the Thom class of an equivariant topological cycle of codimension twice the codimension of its fixed points (inside the fixed point set of ), these topological cycles will give rise to a cohomology frame.

     

Discriminant and Clifford algebras

The centralizer of a square-central skew-symmetric unit in a central simple algebra with orthogonal involution carries a unitary involution. The discriminant algebra of this unitary involution is shown to be an orthogonal summand in one of the components of the Clifford algebra of the orthogonal involution. As an application, structure theorems for orthogonal involutions on central simple algebras of degree 8 are obtained. Received: 30 January 2001; in final form: 28 May 2001 / Published online: 1 February 2002     

Geodesic equivalence of metrics as a particular case of integrability of geodesic flows

We consider the recently found connection between geodesically equivalent metrics and integrable geodesic flows. If two different metrics on a manifold have the same geodesics, then the geodesic flows of these metrics admit sufficiently many integrals (of a special form) in involution, and vice versa. The quantum version of this result is also true: if two metrics on one manifold have the same geodesics, then the Beltrami Laplace operator Δ for each metric admits sufficiently many linear differential operators communiting with Δ. This implies that the topology of a manifold with two different metrics with the same geodesics must be sufficiently simple. We also have that the nonproportionality of the metrics at a point implies the nonproportionality of the metrics at almost all points. In memory of Mikhail Vladimirovich Saveliev Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 2, pp. 285–293, May, 2000.     

Implicitly defined harmonic PHH submersions

The concept of harmonic pseudo-horizontally homothetic submersion, alias harmonic PHH submersion, expresses the more geometric notion of smooth family of minimal submanifolds indexed by a K?hler parameter manifold. We give results that allow to construct harmonic PHH submersions by solving some implicit equations and we apply this method to the case of sphere bundles. Received: 3 September 1998 / Accepted: 19 March 1999     

approximations of inertial manifolds via finite differences

We construct an inertial manifold for the evolution equation as a limit of the inertial manifolds for the difference approximations of the Trotter-Kato type and show that this limit is taken in a topology.

     

Geodesic knots in closed hyperbolic 3-manifolds

We consider the existence of simple closed geodesics or “geodesic knots” in finite volume orientable hyperbolic 3-manifolds. Every such manifold contains at least one geodesic knot by results of Adams, Hass and Scott in (Adams et al. Bull. London Math. Soc. 31: 81–86, 1999). In (Kuhlmann Algebr. Geom. Topol. 6: 2151–2162, 2006) we showed that every cusped orientable hyperbolic 3-manifold in fact contains infinitely many geodesic knots. In this paper we consider the closed manifold case, and show that if a closed orientable hyperbolic 3-manifold satisfies certain geometric and arithmetic conditions, then it contains infinitely many geodesic knots. The conditions on the manifold can be checked computationally, and have been verified for many manifolds in the Hodgson-Weeks census of closed hyperbolic 3-manifolds. Our proof is constructive, and the infinite family of geodesic knots spiral around a short simple closed geodesic in the manifold.      

INVOLUTIONS WITH FIXED POINT SET RP(2m)∪P(2m, 2n + 1)

Let （M, T） be a smooth closed manifold with a smooth involution T whose fixed point set is a disjoint union of an even-dimensional real projective space and a Dold manifold. In some cases, the equivariant bordism classes of （M, T） are determined.     

Fiber Bundles and Regular Approximation of Codimension-One Cycles

We derive an approximation of codimension-one integral cycles(and cycles modulo p) in a compact Riemannian manifold bymeans of piecewise regular cycles: we obtain both flat convergence andconvergence of the masses. The theorem is proved by using suitableprincipal bundles with a discrete group. As a byproduct, we give analternative proof of the main results, which does not use the regularitytheory for homology minimizers in a Riemannian manifold. This also givesa result of -convergence.     

Nonnegative solutions of an elliptic equation and Harnack ends of Riemannian manifolds

In this paper, we give a barrier argument at infinity for solutions of an elliptic equation on a complete Riemannian manifold. By using the barrier argument, we can construct a nonnegative (bounded, respectively) solution of the elliptic equation, which takes the given data at infinity of each end. In particular, we prove that if a complete Riemannian manifold has finitely many ends, each of which is Harnack and nonparabolic, then the set of bounded solutions of the elliptic equation is finite dimensional, in some sense. We also prove that if a complete Riemannian manifold is roughly isometric to a complete Riemannian manifold satisfying the volume doubling condition, the Poincaré inequality and the finite covering condition on each end, then there exists a nonnegative solution of an elliptic equation taking the given data at infinity of each end of the manifold. These results generalize those of Yau, of Donnelly, of Grigor'yan, of Li and Tam, of Holopainen, and of the present authors, but with the barrier argument at infinity that enables one to overcome the obstacle due to the nonlinearity of solutions. Received: 11 November 1999     

Quotient spaces of anti-holomorphic involutions

Let σ be an anti-holomorphic involution on an almost complex four manifold X, a necessary and sufficient condition is given to determine weather X/σ admits an almost complex structure.