Rigidity for Nonnegatively Curved Metrics on S 2 × R3

We address the question: how large is the family of complete metricswith nonnegative sectional curvature on S ² × R³? We classify theconnection metrics, and give several examples of nonconnection metrics.We provide evidence that the family is small by proving some rigidityresults for metrics more general than connection metrics.

     

Isospectral deformations of metrics on spheres

We construct non-trivial continuous isospectral deformations of Riemannian metrics on the ball and on the sphere in R ⁿ for every n≥9. The metrics on the sphere can be chosen arbitrarily close to the round metric; in particular, they can be chosen to be positively curved. The metrics on the ball are both Dirichlet and Neumann isospectral and can be chosen arbitrarily close to the flat metric. Oblatum 19-VI-2000 & 21-II-2001?Published online: 4 May 2001     

Optimal pinching constants of odd dimensional homogeneous spaces

In this article we compute the pinching constants of all invariant Riemannian metrics on the Berger space B ¹³=SU(5)/(Sp(2)×_ℤ2S¹) and of all invariant U(2)-biinvariant Riemannian metrics on the Aloff–Wallach space W ⁷ _1,1=SU(3)/S¹ _1,1. We prove that the optimal pinching constants are precisely in both cases. So far B ¹³ and W ⁷ _1,1 were only known to admit Riemannian metrics with pinching constants .?We also investigate the optimal pinching constants for the invariant metrics on the other Aloff–Wallach spaces W ⁷ _k,l =SU(3)/S¹ _k,l . Our computations cover the cone of invariant T²-biinvariant Riemannian metrics. This cone contains all invariant Riemannian metrics unless k/l=1. It turns out that the optimal pinching constants are given by a strictly increasing function in k/l∈[0,1]. Thus all the optimal pinching constants are ≤.?In order to determine the extremal values of the sectional curvature of an invariant Riemannian metric on W ⁷ _k,l we employ a systematic technique, which can be applied to other spaces as well. The computation of the pinching constants for B ¹³ is reduced to the curvature computation for two proper totally geodesic submanifolds. One of them is diffeomorphic to ℂℙ³/ℤ₂ and inherits an Sp(2)-invariant Riemannian metric, and the other is W ⁷ _1,1 embedded as recently found by Taimanov. This approach explains in particular the coincidence of the optimal pinching constants for W ⁷ _1,1 and the Berger space B ¹³. Oblatum 9-XI-1998 & 3-VI-1999 / Published online: 20 August 1999     

Stability and Boundary Behavior of the Kobayashi Metrics

Uniform stability and localization results for the higher order and singular Kobayashi metrics are established. As an application we obtain the non-tangential weighted limits of these metrics in an h-extendible boundary point of a bounded domain in C ⁿ. This revised version was published online in August 2006 with corrections to the Cover Date.     

The Riemannian <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> topology on the manifold of Riemannian metrics

We study the manifold of all Riemannian metrics over a closed, finite-dimensional manifold. In particular, we investigate the topology on the manifold of metrics induced by the distance function of the L ² Riemannian metric—so-called because it induces an L ² topology on each tangent space. It turns out that this topology on the tangent spaces gives rise to an L ¹-type topology on the manifold of metrics itself. We study this new topology and its completion, which agrees homeomorphically with the completion of the L ² metric. We also give a user-friendly criterion for convergence (with respect to the L ² metric) in the manifold of metrics.     

Strongly Extremal Kähler Metrics

On a compact complex manifold (M, J) of the Kähler type, we consider the functional defined by the L²-norm of the scalar curvature with its domain the space of Kähler metrics of fixed total volume. We calculate its critical points, and derive a formula that relates the Kähler and Ricci forms of such metrics on surfaces. If these metrics have a nonzero constant scalar curvature, then they must be Einstein. For surfaces, if the scalar curvature is nonconstant, these critical metrics are conformally equivalent to non-Kähler Einstein metrics on an open dense subset of the manifold. We also calculate the Hessian of the lower bound of the functional at a critical extremal class, and show that, in low dimensions, these classes are weakly stable minima for the said bound. We use this result to discuss some applications concerning the two-points blow-up of CP².     

Einstein Finsler metrics and killing vector fields on Riemannian manifolds

We use a Killing form on a Riemannian manifold to construct a class of Finsler metrics. We find equations that characterize Einstein metrics among this class. In particular, we construct a family of Einstein metrics on S ³ with Ric = 2F ², Ric = 0 and Ric = -2F ², respectively. This family of metrics provides an important class of Finsler metrics in dimension three, whose Ricci curvature is a constant, but the flag curvature is not.     

Uniqueness of complex geodesics and characterization of circular domains

We study complex geodesics for complex Finsler metrics and prove a uniqueness theorem for them. The results obtained are applied to the case of the Kobayashi metric for which, under suitable hypotheses, we describe the exponential map and the relationship between the indicatrix and small geodesic balls. Finally, exploiting the connection between intrinsic metrics and the complex Monge-Ampère equation, we give characterizations for circular domains in ℂ ⁿ .     

On the new examples of complete noncompact Spin(7)-holonomy metrics

We construct some complete Spin(7)-holonomy Riemannian metrics on the noncompact orbifolds that are ?⁴-bundles with an arbitrary 3-Sasakian spherical fiber M. We prove the existence of the smooth metrics for M = S ⁷ and M = SU(3)/U(1) which were found earlier only numerically.     

On Douglas general (<Emphasis Type="Italic">α</Emphasis>, <Emphasis Type="Italic">β</Emphasis>)-metrics

Douglas metrics are metrics with vanishing Douglas curvature which is an important projective invariant in Finsler geometry. To find more Douglas metrics, in this paper we consider a class of Finsler metrics called general (α, β)-metrics, which are defined by a Riemannian metric \(\alpha = \sqrt {{a_{ij}}\left( x \right){y^i}{y^j}} \) and a 1-form β = b _i (x)y ⁱ . We obtain the differential equations that characterizes these metrics with vanishing Douglas curvature. By solving the equivalent PDEs, the metrics in this class are totally determined. Then many new Douglas metrics are constructed.     

A note on extremal metrics of non-constant scalar curvature

By working in ? ⁿ with potentials of the forma logu + s(u), u the square of the distant to the origin, we obtain extremal Kähler metrics of nonconstant scalar curvature on the blow-up of ? ⁿ at \(\vec 0\) . We then show that these metrics can be completed at ∞ by adding a ?? ^n?1, and reobtain the extremal Kähler metrics of non-constant scalar curvature constructed by Calabi on the blow-up of ?? ⁿ at one point. A similar construction produces this type of metrics on other bundles over ?? ^{n ? 1}.     

On certain geodesic conjugacies of flat cylinders

We prove C ⁰-conjugacy rigidity of any flat cylinder among two different classes of metrics on the cylinder, namely among the class of rotationally symmetric metrics and among the class of metrics without conjugate points.     

The complexity of computing metric distances between partitions

Quantitative measurement of the similarity of partitions is a problem of particular relevance to the social and behavioral sciences, where experimental procedures necessitate the analysis and comparison of partitions of objects. Since metrics used for this purpose vary considerably in computational complexity. I describe two related metric models that permit methodical enumeration of metrics which may be useful and computationally tractable. Twelve metrics on partitions are identified in this way. Five of them have appeared in the literature, while seven appear to be new. Four of them seem difficult to compute, but efficient algorithms for the remaining eight exist and exhibit time complexities ranging from O(n) to O(n³), where n is the number of objects in the partitions. These algorithms are all based on lattice- and graph-theoretic representations of the computational problems.     

Hypersurfaces in Hn and the space of its horospheres

A classical theorem, mainly due to Aleksandrov [Al2] and Pogorelov [P], states that any Riemannian metric on S ² with curvature K > —1 is induced on a unique convex surface in H ³ . A similar result holds with the induced metric replaced by the third fundamental form. We show that the same phenomenon happens with yet another metric on immersed surfaces, which we call the horospherical metric.?This result extends in higher dimensions, the metrics obtained are then conformally flat. One can also study equivariant immersions of surfaces or the metrics obtained on the boundaries of hyperbolic 3-manifolds. Some statements which are difficult or only conjectured for the induced metric or the third fundamental form become fairly easy when one considers the horospherical metric, which thus provides a good boundary condition for the construction of hyperbolic metrics on a manifold with boundary.?The results concerning the third fundamental form are obtained using a duality between H ³ and the de Sitter space . In the same way, the results concerning the horospherical metric are proved through a duality between H ⁿ and the space of its horospheres, which is naturally endowed with a fairly rich geometrical structure. Submitted: March 2001, Revised: November 2001.     

On Birkhoff Theorems for Lagrangian invariant tori with closed orbits

We show that a C¹ torus that is homologous to the zero section, invariant by the geodesic flow of a symmetric Finsler metric in T², and possesses closed orbits is a graph of the canonical projection. This result, together with the result obtained by Bialy in 1989 for continuous invariant tori without closed orbits of symmetric Finsler metrics in T², shows that the second Birkhoff Theorem holds for C¹ Lagrangian invariant tori of symmetric Finsler metrics in the two torus. We also study the first Birkhoff Theorem for continuous invariant tori of Finsler metrics in T² and give some sufficient conditions for a continuous minimizing torus with closed orbits to be a graph of the canonical projection. Partially supported by CNPq, FAPERJ, TWAS     

Filtrations for which All ℋ<Superscript>2</Superscript> Martingales Are of Integrable Variation; Distances between <Emphasis Type="Italic">σ</Emphasis>-Algebras

We consider filtrations for which all ℋ² martingales are of integrable variation. We find a sufficient condition and a necessary condition for this property. Both these conditions are stated in terms the volume of a filtration. The volume of a filtration is defined using a metric on a space of σ-algebras. To obtain the aforementioned conditions we use two equivalent metrics introduced by Boylan and Rogge. We also prove that the original definitions of these metrics can be simplified.      

Critical metrics of the Schouten functional

Given a Riemannian metric on a compact smooth manifold, we consider its Schouten tensor, which is a tensor field of type (0, 2) arising in the remainder of the Weyl part in the standard decomposition of the curvature tensor of the metric. We study extremal properties of the Schouten functional, defined to be the scaling-invariant L ²-norm of the Schouten tensor. It is proved, for instance, that space form metrics are characterized as critical points of the Schouten functional among conformally flat metrics.     

Clairaut relation for geodesics of Hopf tubes

Summary In this note we use the Hopf map π: S³ → S² to construct an interesting family of Riemannian metrics h^fon the 3-sphere, which are parametrized on the space of positive smooth functions f on the 2-sphere. All these metrics make the Hopf map a Riemannian submersion. The Hopf tube over an immersed curve γ in S² is the complete lift π^-1(γ) of γ, and we prove that any geodesic of this Hopf tube satisfies a Clairaut relation. A Hopf tube plays the role in S³ of the surfaces of revolution in R^3. Furthermore, we show a geometric integration method of the Frenet equations for curves in those non-standard S³. Finally, if we consider the sphere S³ equipped with a family h^f of Lorentzian metrics, then a new Clairautrelation is also obtained for timelike geodesics of the Lorentzian Hopf tube, and a geometric integration method for curves is still possible.     

Complex Surfaces with Cat(0) Metrics

We study complex surfaces with locally CAT(0) polyhedral K?hler metrics and construct such metrics on \mathbbCP²{\mathbb{C}P^{2}} with various orbifold structures. In particular, in relation to questions of Gromov and Davis–Moussong we construct such metrics on a compact quotient of the two-dimensional unit complex ball. In the course of the proof of these results we give criteria for Sasakian 3-manifolds to be globally CAT(1). We show further that for certain Kummer coverings of \mathbbCP²{\mathbb{C}P^{2}} of sufficiently high degree their desingularizations are of type K(π, 1).