Energy quantization for triholomorphic maps

Let be weakly convergent stationary triholomorphic maps from a hyperkähler manifold M to another hyperkähler manifold N. We establish an energy quantization for the density function of the defect measure on the concentration set.Received: 10 July 2002, Accepted: 30 September 2002, Published online: 17 December 2002     

Kähler hyperbolicity and the Euler number of a compact manifold with geodesic flow of Anosov type

 Let M be a 2m-dimensional compact Riemannian manifold with Anosov geodesic flow. We prove that every closed bounded k form, k≥2, on the universal covering of M is d(bounded). Further, if M is homotopy equivalent to a compact K?hler manifold, then its Euler number χ(M) satisfies (−1) ^m χ(M)>0. Received: 25 September 2001 / Published Online: 16 October 2002     

Conormal bundles with vanishing Maslov form

IfM is a Riemannian manifold, andL is a Lagrangian submanifold ofT ^* M, the Maslov class ofL has a canonical representative 1-form which we call theMaslov form ofL. We prove that ifL =v ^* N = conormal bundle of a submanifoldN ofM, its Maslov form vanishes iffN is a minimal submanifold. Particularly, ifM is locally flatv ^* N is a minimal Lagrangian submanifold ofT ^* M iffN is a minimal submanifold ofM. This strengthens a result of Harvey and Lawson [H L].     

Topological restrictions for circle actions and harmonic morphisms

 Let M ^m be a compact oriented smooth manifold admitting a smooth circle action with isolated fixed points which are isolated as singularities as well. Then all the Pontryagin numbers of M ^m are zero and its Euler number is nonnegative and even. In particular, M ^m has signature zero. We apply this to obtain non-existence of harmonic morphisms with one-dimensional fibres from various domains, and a classification of harmonic morphisms from certain 4-manifolds. Received: 16 May 2002 Published online: 14 February 2003 Mathematics Subject Classification (2000): 58E20, 53C43, 57R20.     

Minimal entropy and collapsing with curvature bounded from below

We show that if a closed manifold M admits an ℱ-structure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a non-trivial S ¹-action. As a corollary we obtain that the simplicial volume of a manifold admitting an ℱ-structure is zero.?We also show that if M admits an ℱ-structure then it collapses with curvature bounded from below. This in turn implies that M collapses with bounded scalar curvature or, equivalently, its Yamabe invariant is non-negative.?We show that ℱ-structures of rank zero appear rather frequently: every compact complex elliptic surface admits one as well as any simply connected closed 5-manifold.?We use these results to study the minimal entropy problem. We show the following two theorems: suppose that M is a closed manifold obtained by taking connected sums of copies of S ⁴, ℂP ², ²,S ²×S ²and the K3 surface. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S ⁴,ℂP ²,S ²×S ²,ℂP ²#  ² or ℂP ²# ℂP ². Finally, suppose that M is a closed simply connected 5-manifold. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S ⁵,S ³×S ², then on trivial S ³-bundle over S ² or the Wu-manifold SU(3)/SO(3). Oblatum 13-III-2002 & 12-VIII-2002?Published online: 8 November 2002 G.P. Paternain was partially supported by CIMAT, Guanajuato, México.?J. Petean is supported by grant 37558-E of CONACYT.     

Laminations and groups of homeomorphisms of the circle

If M is an atoroidal 3-manifold with a taut foliation, Thurston showed that π₁(M) acts on a circle. Here, we show that some other classes of essential laminations also give rise to actions on circles. In particular, we show this for tight essential laminations with solid torus guts. We also show that pseudo-Anosov flows induce actions on circles. In all cases, these actions can be made into faithful ones, so π₁(M) is isomorphic to a subgroup of Homeo(S ¹). In addition, we show that the fundamental group of the Weeks manifold has no faithful action on S ¹. As a corollary, the Weeks manifold does not admit a tight essential lamination with solid torus guts, a pseudo-Anosov flow, or a taut foliation. Finally, we give a proof of Thurston’s universal circle theorem for taut foliations based on a new, purely topological, proof of the Leaf Pocket Theorem. Oblatum 20-III-2002 & 30-IX-2002?Published online: 18 December 2002 RID="*" ID="*"Both authors partially supported by the U.S. National Science Foundation.     

On 4-manifolds and span-related numbers for cat manifolds

LetM ⁿ denote any closed connected CAT manifold of positive dimensionn. We define CATs(Mⁿ) to be the smallest positive dimension of all closed connected CAT manifoldsN for which the CAT span ofM×N is strictly greater than the CAT span ofN. We determine a formula for this characteristic number which involves only the Kirby-Siebenmann numberks(M) ofM and a Stiefel-Whitney number. Several results on splitting the tangent bundles of closed 4-manifolds are obtained. For example, both the Euler number ofM ⁴ andks(M⁴) represent the total obstruction to positive CAT span for a non-smoothable closed connected 4-manifold. Dedicated to the memory of Professor Otto Endler     

Codimension one or two immersions and submersions of low dimensional manifolds

LetM be a manifold satisfying certain conditions which are weaker than those of E. Thomas^[12], andf:M→N be a map with codimension one or two. We give necessary and sufficient conditions forf to be homotopic to a map with maximal rank. As an application, we completely determine the codimension one or two immersions of Dold manifolds in real projective spaces.     

Random Projections of Smooth Manifolds

We propose a new approach for nonadaptive dimensionality reduction of manifold-modeled data, demonstrating that a small number of random linear projections can preserve key information about a manifold-modeled signal. We center our analysis on the effect of a random linear projection operator Φ:ℝ ^N →ℝ ^M , M<N, on a smooth well-conditioned K-dimensional submanifold ℳ⊂ℝ ^N . As our main theoretical contribution, we establish a sufficient number M of random projections to guarantee that, with high probability, all pairwise Euclidean and geodesic distances between points on ℳ are well preserved under the mapping Φ. Our results bear strong resemblance to the emerging theory of Compressed Sensing (CS), in which sparse signals can be recovered from small numbers of random linear measurements. As in CS, the random measurements we propose can be used to recover the original data in ℝ ^N . Moreover, like the fundamental bound in CS, our requisite M is linear in the “information level” K and logarithmic in the ambient dimension N; we also identify a logarithmic dependence on the volume and conditioning of the manifold. In addition to recovering faithful approximations to manifold-modeled signals, however, the random projections we propose can also be used to discern key properties about the manifold. We discuss connections and contrasts with existing techniques in manifold learning, a setting where dimensionality reducing mappings are typically nonlinear and constructed adaptively from a set of sampled training data. This research was supported by ONR grants N00014-06-1-0769 and N00014-06-1-0829; AFOSR grant FA9550-04-0148; DARPA grants N66001-06-1-2011 and N00014-06-1-0610; NSF grants CCF-0431150, CNS-0435425, CNS-0520280, and DMS-0603606; and the Texas Instruments Leadership University Program. Web: dsp.rice.edu/cs.     

Berezin–Toeplitz quantization for lower energy forms

Let M be an arbitrary complex manifold and let L be a Hermitian holomorphic line bundle over M. We introduce the Berezin–Toeplitz quantization of the open set of M where the curvature on L is nondegenerate. In particular, we quantize any manifold admitting a positive line bundle. The quantum spaces are the spectral spaces corresponding to [0,k^?N], where N>1 is fixed, of the Kodaira Laplace operator acting on forms with values in tensor powers L^k. We establish the asymptotic expansion of associated Toeplitz operators and their composition in the semiclassical limit k→∞ and we define the corresponding star-product. If the Kodaira Laplace operator has a certain spectral gap this method yields quantization by means of harmonic forms. As applications, we obtain the Berezin–Toeplitz quantization for semi-positive and big line bundles.     

The second twisted Betti number and the convergence of collapsing Riemannian manifolds

Let M _i X denote a sequence of n-manifolds converging to a compact metric space, X, in the Gromov-Hausdorff topology such that the sectional curvature is bounded in absolute value and dim(X)<n. We prove the following stability result: If the fundamental groups of M _i are torsion groups of uniformly bounded exponents and the second twisted Betti numbers of M _i vanish, then there is a manifold, M, and a sequence of diffeomorphisms from M to a subsequence of {M _i } such that the distance functions of the pullback metrics converge to a pseudo-metric in C ⁰-norm. Furthermore, M admits a foliation with leaves diffeomorphic to flat manifolds (not necessarily compact) such that a vector is tangent to a leaf if and only if its norm converges to zero with respect to the pullback metrics. These results lead to a few interesting applications. Oblatum 17-I-2002 & 27-II-2002?Published online: 29 April 2002     

On wedge extendability of CR-meromorphic functions

In this article, we consider metrically thin singularities E of the solutions of the tangential Cauchy-Riemann operators on a -smooth embedded Cauchy-Riemann generic manifold M (CR functions on ) and more generally, we consider holomorphic functions defined in wedgelike domains attached to . Our main result establishes the wedge- and the L¹-removability of E under the hypothesis that the -dimensional Hausdorff volume of E is zero and that M and are globally minimal. As an application, we deduce that there exists a wedgelike domain attached to an everywhere locally minimal M to which every CR-meromorphic function on M extends meromorphically. Received: 7 September 2000; in final form: 20 December 2001 / Published online: 6 August 2002     

Uniqueness Theorems for CR Functions

Let M be a CR manifold embedded in ?^s of arbitrary codimension. M is called generic if the complex hull of the tangent space in all points of M is the whole ?^s. M is minimal (in sense of Tumanov) in p ? M if there does not exist any CR submanifold of M passing through p with the same CR dimension as M but of smaller dimension. Let M be generic and minimal in some point p ? M and N be a generic submanifold of M passing through p. We prove that a continuous CR function on M vanishes identically in some neigbourhood of p if its restriction to N either vanishes in p faster then some function with non-integrable logarithm or it vanishes on a subset of N of positive measure.     

Best Sobolev Constants and Manifolds with Positive Scalar Curvature Metrics

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 ofS ^{n + m} . We will find some estimates for the constant K in the case N =S ⁿ .     

Geometry of the Normal Bundle of a Submanifold*

 We study the geometric behavior of the normal bundle T ^⊥ M of a submanifold M of a Riemannian manifold . We compute explicitely the second fundamental form of T ^⊥ M and look at the relation between the minimality of T ^⊥ M and M. Finally we show that the Maslov forms with respect to a suitable connection of the pair (T ^⊥ M, are null. Received March 14, 2001; in revised form February 11, 2002     

On the regularity of spherically symmetric wave maps

Wave maps are critical points U: M → N of the Lagrangian ??[U] = ∞_M ‖dU‖², where M is an Einsteinian manifold and N a Riemannian one. For the case M = ?^2,1 and U a spherically symmetric map, it is shown that the solution to the Cauchy problem for U with smooth initial data of arbitrary size is smooth for all time, provided the target manifold N satisfies the two conditions that: (1) it is either compact or there exists an orthonormal frame of smooth vectorfields on N whose structure functions are bounded; and (2) there are two constants c and C such that the smallest eigenvalue λ and the largest eigenvalue λ of the second fundamental form k_AB of any geodesic sphere Σ(p, s) of radius s centered at p ? N satisfy sλ ≧ c and s A ≦ C(1 + s). This is proved by first analyzing the energy-momentum tensor and using the second condition to show that near the first possible singularity, the energy of the solution cannot concentrate, and hence is small. One then proves that for targets satisfying the first condition, initial data of small energy imply global regularity of the solution. © 1993 John Wiley & Sons, Inc.     

Nonexistence Results of Minimal Immersions

Let M be a Riemannian m-dimensional manifold with m ≥ 3, endowed with non zero parallel p-form. We prove that there is no minimal isometric immersions of M in a Riemannian manifold N with constant strictly negative sectional curvature. Next we show that, under the conform flatness of the manifold N and some assumptions on the Ricci curvature of N, there is no α-pluriharmonic isometric immersion.     

On the homotopy class of maps with finite p-energy into non-positively curved manifolds

We prove that a map f : M → N with finite p-energy, p > 2, from a complete manifold (M, á , ñ ){\left(M,\left\langle ,\right\rangle \right)} into a non-positively curved, compact manifold N is homotopic to a constant, provided the negative part of the Ricci curvature of the domain manifold is small in a suitable spectral sense. The result relies on a Liouville-type theorem for finite q-energy, p-harmonic maps under spectral assumptions.     

Sard’s theorem for mappings between Fréchet manifolds

We prove an infinite-dimensional version of Sard’s theorem for Fréchet manifolds. Let M (respectively, N) be a bounded Fréchet manifold with compatible metric d _M (respectively, d _N ) modeled on Fréchet spaces E (respectively, F) with standard metrics. Let f : M → N be an MC ^k -Lipschitz–Fredholm map with k > max{Ind f, 0}: Then the set of regular values of f is residual in N.     

Fiberwise volume non-increasing diffeomorphisms on product manifolds

Given a closed connected Riemannian manifold M and a connected Riemannian manifold N, we consider fiberwise, i.e. M×{z}, z ? N{M\times \{z\}, z\in N}, volume non-increasing diffeomorphisms on the product M × N. Our main theorems show that in the presence of a certain cohomological condition on M and N such diffeomorphisms must map a fiber diffeomorphically onto another fiber and are therefore fiberwise volume preserving. As a first corollary, we show that the isometries of M × N split. As a second corollary, we prove a special case of an extension of Talelli’s conjecture.