Metrics Of Positive Curvature Research Articles

Convexity of complements of limit sets for holomorphic foliations on surfaces

Let $${\mathcal {F}}$$ be a holomorphic foliation on a compact Kähler surface with hyperbolic singularities and no foliation cycle. We prove that if the limit set of $${\mathcal {F}}$$ has zero Lebesgue measure, then its complement is a modification of a Stein domain. This applies for the case of suspensions of Kleinian representations, answering a question asked by Brunella. The proof consists in building, in several steps, a metric of positive curvature for the normal bundle of $${\mathcal {F}}$$ near the limit set. Then we construct a proper strictly plurisubharmonic exhaustion function for the complement of the limit set by extending Brunella’s method to our singular context. The arguments hold more generally when the limit set is thin, a property relying on Brownian motion.

A new explicit complete Kähler metric on ℂ n

In 1977, Klembeck presented an explicit Kähler metric of positive holomorphic curvature on [Klembeck P. A complete Kähler metric of positive curvature on . Proc Amer Math Soc. 1977;64(2):313–316]. Later, in 1991, To used Klembeck's metric and proved a non-compact version of the Kodaira embedding theorem [To W-K. Quasi-projective embeddings of noncompact complete Kähler manifolds of positive Ricci curvature and satisfying certain topological conditions. Duke Math J. 1991;63(3):745–789]. Wu H-H and Zheng F. [Examples of positively curved complete Kähler manifolds. In: Geometry and Analysis. No. 1. Somerville (MA): Int. Press; 2011. p. 517–542. (Adv Lect Math (ALM); 17)] discovered another explicit Kähler metric which is a natural perturbation of Klembeck's example. More examples (implicit) are studied by solutions of certain ODEs by Cao H-D. [Existence of gradient Kähler-Ricci solitons. In: Elliptic and parabolic methods in geometry. Minnesota: 1994. p. 1–16; Limits of solutions to the Kähler-Ricci flow. J Differ Geom. 1997;45:257–272] and Wu-Zheng [Examples of positively curved complete Kähler manifolds. In: Geometry and Analysis. No. 1. Somerville (MA): Int. Press; 2011. p. 517–542. (Adv Lect Math (ALM); 17)] respectively. Motivated by these, we investigate other complete Kähler metrics induced from certain weighted Fock spaces, following a similar scheme. In particular, we prove that these Bergman metrics induced by weighted Fock spaces possess positive holomorphic sectional curvature.

Geodesic nets with three boundary vertices

We prove that a geodesic net with three boundary (= unbalanced) vertices on a non-positively curved plane has at most one balanced vertex. We do not assume any a priori bound for the degrees of unbalanced vertices. The result seems to be new even in the Euclidean case. We demonstrate by examples that the result is not true for metrics of positive curvature on the plane, and that there are no immediate generalizations of this result for geodesic nets with four unbalanced vertices.

Local and global homogeneity for three obstinate spheres

In this note we complete a study of globally homogeneous Riemannian quotients Γ\\(M,ds2) in positive curvature. Specifically, M is a homogeneous space G/H that admits a G–invariant Riemannian metric of strictly positive sectional curvature, and ds2 is a G–invariant Riemannian metric on M, not necessarily normal and not necessarily positively curved. The Homogeneity Conjecture is that Γ\\(M,ds2) is (globally) homogeneous if and only if (M,ds2) is homogeneous and every γ∈Γ is of constant displacement on (M,ds2). In an earlier paper we verified that conjecture for all homogeneous spaces that admit an invariant Riemannian metric of positive curvature — with three exceptions, all of them odd dimensional spheres, which surprisingly did not yield to the earlier approaches. In this note we deal with those exceptions: we develop some methods that let us verify the Homogeneity Conjecture for those three obstinate spheres. That completes verification of the Homogeneity Conjecture in positive curvature.

Harmonic spinors and metrics of positive curvature via the Gromoll filtration and Toda brackets

We construct non-trivial elements of order 2 in the homotopy groups $\pi_{8j+1+*} Diff(D^6,\partial)$, for * congruent 1 or 2 modulo 8, which are detected by the "assembling homomorphism" (giving rise to the Gromoll filtration), followed by the alpha-invariant in $KO_*=Z/2$. These elements are constructed by means of Morlet's homotopy equivalence between $Diff(D^6,\partial)$ and $\Omega^7(PL_6/O_6)$, and Toda brackets in $PL_6/O_6$. We also construct non-trivial elements of order 2 in $\pi_* PL_m$ for every m greater or equal to 6 and * congruent to 1 or 2 modulo 8, which are detected by the alpha-invariant. As consequences, we (a) obtain non-trivial elements of order 2 in $\pi_* Diff(D^m,\partial)$ for m greater or equal to 6, and * + m congruent 0 or 1 modulo 8; (b) these elements remain non-trivial in $\pi_* Diff(M)$ where M is a closed spin manifold of the same dimension m and * > 0; (c) they act non-trivially on the corresponding homotopy group of the space of metrics of positive scalar curvature of such an M; in particular these homotopy groups are all non-trivial. The same applies to all other diffeomorphism invariant metrics of positive curvature, like the space of metrics of positive sectional curvature, or the space of metrics of positive Ricci curvature, provided they are non-empty. Further consequences are: (d) any closed spin manifold of dimension m greater or equal to 6 admits a metric with harmonic spinors; (e) there is no analogue of the odd-primary splitting of $(PL/O)_{(p)}$ for the prime 2; (f) for any $bP_{8j+4}$-sphere (where j > 0) of order which divides 4, the corresponding element in $\pi_0 Diff(D^{8j+2},\partial)$ lifts to $\pi_{8j-4} Diff(D^6,\partial)$, i.e., lies correspondingly deep down in the Gromoll filtration.

The Einstein−Λ flow on product manifolds

We consider the vacuum Einstein flow with a positive cosmological constant on spatial manifolds of product form . In dimensions we show the existence of continuous families of recollapsing models whenever at least one of the factors M1 or M2 admits a Riemannian Einstein metric with positive Einstein constant. We moreover show that these families belong to larger continuous families with models that have two complete time directions, i.e. do not recollapse. Complementarily, we show that whenever no factor has positive curvature, then any model in the product class expands in one time direction and collapses in the other. In particular, positive curvature of one factor is a necessary criterion for recollapse within this class. Finally, we relate our results to the instability of the Nariai solution in three spatial dimensions and point out why a similar construction of recollapsing models in that dimension fails. The present results imply that there exist different classes of initial data which exhibit fundamentally different types of long-time behavior under the Einstein– flow whenever the spatial dimension is strictly larger than three. Moreover, this behavior is related to the spatial topology through the existence of Riemannian Einstein metrics of positive curvature.

Geometrically Formal Homogeneous Metrics of Positive Curvature

A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to be topologically formal. Nonetheless, we show that among the homogeneous Riemannian metrics of positive sectional curvature a geometrically formal metric is either symmetric, or a metric on a rational homology sphere.

Lower estimates for the expected Betti numbers of random real hypersurfaces

We estimate from below the expected Betti numbers of real hypersurfaces taken at random in a smooth real projective n-dimensional manifold. These random hypersurfaces are chosen in the linear system of a large dth power of a real ample line bundle equipped with a Hermitian metric of positive curvature. As for the upper bounds that we recently established, these lower bounds read as a product of a factor which only depends on the dimension n of the manifold with the Kähler volume of its real locus R X and d n . Actually, any closed affine real algebraic hypersurface appears with positive probability as part of such random real hypersurfaces in any ball of R X of radius O ( 1 / d ) .

Suspending the Cartan Embedding of $${{\mathbb H}P^n}$$ Through Spindles and Generators of Homotopy Groups

We provide an equivariant suspension of the Cartan embedding of the symmetric space $${S^{4n+3} \to \mathbb {H}P^n \hookrightarrow Sp(n+1)}$$ ; this construction furnishes geometric generators of the homotopy group of π 4n+6 Sp(n + 1). We study the topology and geometry of the image of this generator; in particular we show that it is a spindle, minimal with respect to the biinvariant metric from Sp(n + 1). This spindle also admits a different metric of positive curvature away from the cone singular point.

Three-manifolds of positive curvature and convex weakly umbilic boundary

In this paper we study a boundary value problem in manifolds with weakly umbilic boundary (the Second Fundamental form of the boundary is a constant multiple of the metric). We show that if we start with a metric of positive curvature and convex boundary (positive Second Fundamental form), the Ricci flow uniformizes the curvature. In the case of a metric with rotational symmetry, we indicate how to weaken the hypothesis to positive Ricci curvature and positive Second Fundamental form.

Positively curved cohomogeneity one manifolds and 3-Sasakian geometry

We provide an exhaustive description of all simply connected positively curved cohomogeneity one manifolds. The description yields a complete classification in all dimensions but 7, where in addition to known examples, our list contains one exceptional space and two infinite families not yet known to carry metrics of positive curvature. The infinite families also carry a 3-Sasakian metric of cohomogeneity one, which is associated to a family of selfdual Einstein orbifold metrics on the 4-sphere constructed by Hitchin.

Metrics of positive curvature with conic singularities on the sphere

A simple proof is given of the necessary and sufficient condition on a triple of positive numbers θ 1 , θ 2 , θ 3 for the existence of a conformal metric of constant positive curvature on the sphere, with three conic singularities of total angles 2πθ 1 , 2πθ 2 , 2πθ 3 . The same condition is necessary and sufficient for the triple πθ 1 , πθ 2 , πθ 3 to be interior angles of a spherical triangular membrane.

Curvature properties of the positively curved Eschenburg spaces

Using a Cheeger construction, we enlarge the known one parameter family of metrics of positive curvature on the Eschenburg spaces to a simple explicit four parameter family. Then we show in a smaller class of metrics that all of the Eschenburg spaces of positive curvature have their pinching bounded above by 1/37. Since the Aloff–Wallach spaces are the homogeneous Eschenburg spaces and since Püttmann has calculated the pinching of W 1,1= SU(3)/ S 1 1,1 in the U(2) biinvariant metric to be exactly 1/37 this upper bound is sharp. It is also shown that the only Eschenburg space with pinching exactly 1/37 is W 1,1.

Conditions for nonnegative curvature on vector bundles and sphere bundles

This paper addresses J. Cheeger and D. Gromoll's question about which vector bundles admit a complete metric of nonnegative curvature, and it relates their question to the issue of which sphere bundles admit a metric of positive curvature. We show that any vector bundle that admits a metric of nonnegative curvature must admit a connection, a tensor, and a metric on the base space, which together satisfy a certain differential inequality. On the other hand, a slight sharpening of this condition is sufficient for the associated sphere bundle to admit a metric of positive curvature. Our results sharpen and generalize M. Strake and G. Walschap's conditions under which a vector bundle admits a connection metric of nonnegative curvature.

A class of Riemannian manifolds that pinch when evolved by Ricci flow

The purpose of this paper is to construct a set of Riemannian metrics on a manifold X with the property that will develop a pinching singularity in finite time when evolved by Ricci flow. More specifically, let , where N n is an arbitrary closed manifold of dimension n≥ 2 which admits an Einstein metric of positive curvature. We construct a (non-empty) set of warped product metrics on the non-compact manifold X such that if , then a smooth solution , t∈[0,T) to the Ricci flow equation exists for some maximal constant T, 0<T<∞, with initial value , and where K is some compact set .

Some sufficient conditions for immersion in the large of metrics of positive curvature

It is proved that a regular Riemannian manifold diffeomorphic to a circle and having positive Gaussian curvature bounded from zero is immersible into a three-dimensional Euclidean space in the form of a regular surface if it has smallLp (the norm of the gradient of Gaussian curvature), p > 2, or if it has a sufficiently small area (with any behavior of the geodesic boundary curvature).

Cohomology of biquotients

Biquotients are non-homogeneous quotient spaces of Lie groups. Using the Serre spectral sequence and the method of Borel, we compute the cohomology algebra of these spaces in cases where the Lie group cohomology is not too complicated. Among these are the biquotients which are known to carry a metric of positive curvature.

Curvature and linear subbundles of holomorphic vector bundles

We study the behavior of holomorphic vector bundles with supplementary curvature conditions. Let L ~ M be a holomorphic line or bundle on the complex manifold M and g be a fibrewise Hermitian metric of positive curvature on L. Then (as follows easily from [i, p. 170]) the function g(7, ~) has no relative minima for any everywhere nonzero section ~ ~ H ° (M~ ~ (L)). Generalizing this fact, S. Kobayashi noted [2, p. 166] that a global holomorphic section of a holomorphic Hermitian bundle with positive curvature and compact base has a nonempty set of zeros (the outline of the corresponding argument is given in [2]). Unfortunately, the latter is not always true (cf. the example below). Kobayashi's remark is valid under an additional restriction.

ON Σ-REALIZATIONS OF METRICS OF POSITIVE CURVATURE

A metric ds2 admits a Σ-realization if there is a realization of it in E3 in the form of a surface whose boundary lies on a given surface Σ. This paper proves the existence of Σ-realizations of a certain class of metrics of positive curvature for surfaces of quite general form, and describes a number of possible Σ-realizations of the given metric. The proof is based on a consideration of a nonlinear boundary-value problem for immersion equations. Bibliography: 3 titles.

ON GENERIC PROPERTIES OF CLOSED GEODESICS

A complete proof is given of the theorem asserting that bumpy metrics are generic. This result was announced by Abraham (Global Analysis (Proc. Sympos. Pure Math., vol 14), Amer. Math. Soc., Providence, R. I., 1970, pp. 1-3.). Related results are used to rigourously carry out Poincar?'s outline of a bifurcation-theoretic proof of the existence of closed geodesics without self-intersections for any Riemannian metric of positive curvature on the two-dimensional sphere . To do this, it is essential that the lengths of all non-self-intersecting closed geodesics for the metrics on , considered in the course of the proof, be uniformly bounded from above. Examples are given of metrics on (where the sign of the curvature alternates) for which there exist arbitrarily long closed geodesics without self-intersections. Bibliography: 27 titles.