Holomorphically planar conformal vector fields on contact metric manifolds

We study holomorphically planar conformal vector fields (HPCV) on contact metric manifolds under some curvature conditions. In particular, we have studied HPCV fields on (i) contact metric manifolds with pointwise constant ξ-sectional curvature (under this condition M is either K-contact or V is homothetic), (ii) Einstein contact metric manifolds (in this case M becomes K contact), (iii) contact metric manifolds with parallel Ricci tensor (under this condition M is either K-contact Einstein or is locally isometric to E ⁿ⁺¹×S ⁿ (4)).     

Systolic volume of hyperbolic manifolds and connected sums of manifolds

The systolic volume of a closed n-manifold M is defined as the optimal constant σ(M) satisfying the inequality vol(M, g) ≥ σ(M) sys(M, g) ⁿ between the volume and the systole of every metric g on M. First, we show that the systolic volume of connected sums of closed oriented essential manifolds is unbounded. Then, we prove that the systolic volume of every sequence of closed hyperbolic (three-dimensional) manifolds is also unbounded. These results generalize systolic inequalities on surfaces in two different directions.      

Isoperimetric regions in spherical cones and Yamabe constants of M ×?S1

We study isoperimetric regions on Riemannian manifolds of the form (M ⁿ × (0, π), sin²(t)g + dt ²) where g is a metric of positive Ricci curvature ≥ n − 1. When g is an Einstein metric we use this to compute the Yamabe constant of (M ×\mathbbR, g+ dt² ){(M \times \mathbb{R}, g+ dt^2 )} and so to obtain lower bounds for the Yamabe invariant of M × S ¹.     

On Harnack inequalities and singularities of admissible metrics in the Yamabe problem

In this paper we study the local behaviour of admissible metrics in the k-Yamabe problem on compact Riemannian manifolds (M, g ₀) of dimension n ≥ 3. For n/2 < k < n, we prove a sharp Harnack inequality for admissible metrics when (M, g ₀) is not conformally equivalent to the unit sphere S ⁿ and that the set of all such metrics is compact. When (M, g ₀) is the unit sphere we prove there is a unique admissible metric with singularity. As a consequence we prove an existence theorem for equations of Yamabe type, thereby recovering as a special case, a recent result of Gursky and Viaclovsky on the solvability of the k-Yamabe problem for k > n/2. This work was supported by the Australian Research Council.     

The scalar curvature flow on <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>—perturbation theorem revisited

For the problem of finding a geometry on S ⁿ , for n≥3, with a prescribed scalar curvature, there is a well-known result which is called the perturbation theorem; it is due to Chang and Yang (Duke Math. J. 64, 27–69, 1991). Their key assumption is that the candidate f for the prescribed scalar curvature is sufficiently near the scalar curvature of the standard metric in the sup norm. It is important to know how large that difference in sup norm can possibly be. Here we consider prescribing scalar curvature problem using the scalar curvature flow.     

Complex<Emphasis Type="BoldItalic">n</Emphasis>-dimensional manifolds with a real<Emphasis Type="BoldItalic">n</Emphasis><Superscript>2</Superscript>-dimensional automorphism group

The main theorem of this article is a characterization of non compact simply connected complete Kobayashi hyperbolic complex manifold of dimension n≽ 2 with real n ²-dimensional holomorphic automorphism group. Together with the earlier work [11, 12] and [13] of Isaev and Krantz, this yields a complete classification of the simply-connected, complete Kobayashi hyperbolic manifolds with dim_ℝ Aut (M) ≽ (dim_ℂ M)².     

Optimal control under persistent disturbances

Let f be an orientation-preserving Morse-Smale diffeomorphism of an n-dimensional (n ≥ 3) closed orientable manifold M ⁿ . We show the possibility of representing the dynamics of f in a “source-sink” form. The roles of the “source” and “sink” are played by invariant closed sets one of which, A _f , is an attractor, and the other, R _f , is a repeller. Such a representation reveals new topological invariants that describe the embedding (possibly, wild) of stable and unstable manifolds of saddle periodic points in the ambient manifold. These invariants have allowed us to obtain a classification of substantial classes of Morse-Smale diffeomorphisms on 3-manifolds. In this paper, for any n ≥ 3, we describe the topological structure of the sets A _f and R _f and of the space of orbits that belong to the set M ⁿ \ (A _f ∪ R _f ).     

On classification of immersions ofn-manifolds in (2n−1)-manifolds

Under the assumption of (f, M ⁿ ,N ²ⁿ⁻¹) being trivial, the classification of immersions homotopic tof: M ⁿ →N ²ⁿ⁻¹ is obtained in many cases. The triviality of (f, M ⁿ ,P ²ⁿ⁻¹) is proved for anyM ⁿ andf. LetM, N be differentiable manifolds of dimensionn and2n−1 respectively. For a mapf: M → N, denote byI[M, N] _f the set of regular homotopy classes of immersions homotopic tof. It has been proved in [1] that, whenn>1,I[M, N] _f is nonempty for anyf. In this paper we will determine the setI[M, N] _f in some cases. For example, ifN=P ²ⁿ⁻¹ or more generally, the lens spacesS _m ²ⁿ⁻¹ =Z _m /S ²ⁿ⁻¹,M is any orientablen-manifold or nonorientable butn≡0, 1, 3 mod 4, then, for anyf, theI[M, N] _f is determined completely. WhenN=R ²ⁿ⁻¹, the setI[M, N] of regular homotopy classes of all immersions has been enumerated by James and Thomas in [2] and McClendon in [3] forn>3. Applying our results toN=R ²ⁿ⁻¹ we obtain their results again, except for the casen≡2 mod 4 andM nonorientable. Whenn=3, McClendon's results cannot be used. Our results include the casesn=3,M orientable or not (for orientableM, I[M, R⁵] is known by Wu [4]).     

Teichmüller Spaces and Negatively Curved Fiber Bundles

In the first part of the paper we introduce the theory of bundles with negatively curved fibers. For a space X there is a forgetful map F _X between bundle theories over X, which assigns to a bundle with negatively curved fibers over X its subjacent smooth bundle. Our main result states that, for certain k-spheres ${\mathbb{S}^k}In the first part of the paper we introduce the theory of bundles with negatively curved fibers. For a space X there is a forgetful map F _X between bundle theories over X, which assigns to a bundle with negatively curved fibers over X its subjacent smooth bundle. Our main result states that, for certain k-spheres \mathbbS^k{\mathbb{S}^k}, the forgetful map F_\mathbbS^k{F_{\mathbb{S}^k}} is not one-to-one. This result follows from Theorem A, which proves that the quotient map MET  ^{sec < 0} (M)?T  ^{sec < 0} (M){\mathcal{MET}^{\,\,sec <0 }(M)\rightarrow\mathcal{T}^{\,\,sec <0 }(M)} is not trivial at some homotopy levels, provided the hyperbolic manifold M satisfies certain conditions. Here MET  ^{sec < 0} (M){\mathcal{MET}^{\,\,sec <0 }(M)} is the space of negatively curved metrics on M and T  ^{sec < 0} (M) = MET  ^{sec < 0} (M)/ DIFF₀(M){\mathcal{T}^{\,\,sec <0 }(M) = \mathcal{MET}^{\,\,sec <0 }(M)/ {\rm DIFF}_0(M)} is, as defined in [FO2], the Teichmüller space of negatively curved metrics on M. In particular we conclude that T  ^{sec < 0} (M){\mathcal{T}^{\,\,sec <0 }(M)} is, in general, not connected. Two remarks: (1) the nontrivial elements in p_kMET  ^{sec < 0} (M){\pi_{k}\mathcal{MET}^{\,\,sec <0 }(M)} constructed in [FO3] have trivial image by the map induced by MET  ^{sec < 0} (M)?T  ^{sec < 0} (M){\mathcal{MET}^{\,\,sec <0 }(M)\rightarrow\mathcal{T}^{\,\,sec <0 }(M)} ; (2) the nonzero classes in p_kT  ^{sec < 0} (M){\pi_{k}\mathcal{T}^{\,\,sec <0 }(M)} constructed in [FO2] are not in the image of the map induced by MET  ^{sec < 0} (M)?T  ^{sec < 0} (M){\mathcal{MET}^{\,\,sec <0 }(M)\rightarrow\mathcal{T}^{\,\,sec <0 }(M)} ; the nontrivial classes in p_kT  ^{sec < 0} (M){\pi_{k}\mathcal{T}^{\,\,sec <0 }(M)} given here, besides coming from MET  ^{sec < 0} (M){\mathcal{MET}^{\,\,sec <0 }(M)} and being harder to construct, have a different nature and genesis: the former classes – given in [FO2] – come from the existence of exotic spheres, while the latter classes – given here – arise from the non-triviality and structure of certain homotopy groups of the space of pseudo-isotopies of the circle \mathbbS¹{\mathbb{S}^1}. The strength of the new techniques used here allowed us to prove also a homology version of Theorem A, which is given in Theorem B.     

Complex hyperbolic hyperplane complements

We study spaces obtained from a complete finite volume complex hyperbolic n-manifold M by removing a compact totally geodesic complex (n − 1)-submanifold S. The main result is that the fundamental group of M\ S{M{\setminus} S} is relatively hyperbolic, relative to fundamental groups of the ends of M\ S{M{\setminus} S} , and M\ S{M{\setminus} S} admits a complete finite volume A-regular Riemannian metric of negative sectional curvature. It follows that for n > 1 the fundamental group of M\ S{M{\setminus} S} satisfies Mostow-type Rigidity, has solvable word and conjugacy problems, has finite asymptotic dimension and rapid decay property, satisfies Borel and Baum-Connes conjectures, is co-Hopf and residually hyperbolic, has no nontrivial subgroups with property (T), and has finite outer automorphism group. Furthermore, if M is compact, then the fundamental group of M\ S{M{\setminus} S} is biautomatic and satisfies Strong Tits Alternative.     

Splittings and Cr-structures for manifolds with nonpositive sectional curvature

Let denote the universal covering space of a compact Riemannian manifold, M ⁿ , with sectional curvature, −1≤K _Mn ≤0. We show that a collection of deck transformations of , satisfying certain (metric dependent) conditions, determines an open dense subset of M ⁿ , at every point of which, there exists a local isometric splitting with nontrivial flat factor. Such a collection, which we call an abelian structure, also gives rise to an essentially canonical Cr-structure in the sense of Buyalo, i.e an atlas for an injective F-structure, for which additional conditions hold. It follows in particular that the minimal volume of M ⁿ vanishes. We show that an abelian structure exists if the injectivity radius at all points of M ⁿ is less than ε(n)>0. This yields a conjecture of Buyalo as well as a strengthened version of the conclusion of Gromov’s “gap conjecture” in our special situation. In addition, we observe that abelian structures on nonpositively curved manifolds have certain stability properties under suitably controlled changes of metric. Oblatum 26-III-1999 & 14-IX-2000?Published online: 8 December 2000     

A variational problem for pullback metrics

Let (M, g) and (N, h) be Riemannian manifolds without boundary and let f : M → N be a smooth map. Let ||f^*h||{\|f^*h\|} denote the norm of the pullback metric of h by f. In this paper, we consider the functional F(f) = ò_M ||f^*h||² dv_g{{\Phi (f) = \int_M \|f^*h\|^2 dv_g}}. We prove the existence of minimizers of the functional Φ in each 3-homotopy class of maps, where maps f ₁ and f ₂ are 3-homotopic if they are homotopic on the three dimensional skeltons of a triangulation of M. Furthermore, we give a monotonicity formula and a Bochner type formula.     

Some basic inequalities in higher dimensional non-Euclid space

In this paper, the concept of a finite mass-points system∑N(H(A))(N>n) being in a sphere in an n-dimensional hyperbolic space Hn and a finite mass-points system∑N(S(A))(N>n) being in a hyperplane in an n-dimensional spherical space Sn is introduced, then, the rank of the Cayley-Menger matrix AN(H)(or a AN(S)) of the finite mass-points system∑∑N(S(A))(or∑N(S(A))) in an n-dimensional hyperbolic space Hn (or spherical space Sn) is no more than n 2 when∑N(H(A))(N>n) (or∑N(S(A))(N>n)) are in a sphere (or hyperplane). On the one hand, the Yang-Zhang's inequalities, the Neuberg-Pedoe's inequalities and the inequality of the metric addition in an n-dimensional hyperbolic space Hn and in an n-dimensional spherical space Sn are established by the method of characteristic roots. These are basic inequalities in hyperbolic geometry and spherical geometry. On the other hand, some relative problems and conjectures are brought.     

About the Blowup of Quasimodes on Riemannian Manifolds

On any compact Riemannian manifold (M,g) of dimension n, the L ²-normalized eigenfunctions φ _λ satisfy ||f_l||_￥ ￡ Cl^\fracn-12\|\phi_{\lambda}\|_{\infty}\leq C\lambda^{\frac{n-1}{2}} where −Δφ _λ =λ ² φ _λ . The bound is sharp in the class of all (M,g) since it is obtained by zonal spherical harmonics on the standard n-sphere S ⁿ . But of course, it is not sharp for many Riemannian manifolds, e.g., flat tori ℝ ⁿ /Γ. We say that S ⁿ , but not ℝ ⁿ /Γ, is a Riemannian manifold with maximal eigenfunction growth. The problem which motivates this paper is to determine the (M,g) with maximal eigenfunction growth. In an earlier work, two of us showed that such an (M,g) must have a point x where the set ℒ _x of geodesic loops at x has positive measure in S^*_xMS^{*}_{x}M. We strengthen this result here by showing that such a manifold must have a point where the set ℛ _x of recurrent directions for the geodesic flow through x satisfies |{ℛ} _x |>0. We also show that if there are no such points, L ²-normalized quasimodes have sup-norms that are o(λ ^(n−1)/2), and, in the other extreme, we show that if there is a point blow-down x at which the first return map for the flow is the identity, then there is a sequence of quasimodes with L ^∞-norms that are Ω(λ ^(n−1)/2).     

On the pontryagin-steenrod-wu theorem

We present a short and direct proof (based on the Pontryagin-Thom construction) of the following Pontryagin-Steenrod-Wu theorem: (a) LetM be a connected orientable closed smooth (n + 1)-manifold,n≥3. Define the degree map deg: π ⁿ (M) →H ⁿ (M; ℤ) by the formula degf =f*[S ⁿ ], where [S ⁿ ] εH ⁿ (M; ℤ) is the fundamental class. The degree map is bijective, if there existsβ εH ₂(M, ℤ/2ℤ) such thatβ ·w ₂(M) ε 0. If suchβ does not exist, then deg is a 2-1 map; and (b) LetM be an orientable closed smooth (n+2)-manifold,n≥3. An elementα lies in the image of the degree map if and only ifρ ₂ α ·w ₂(M)=0, whereρ ₂: ℤ → ℤ/2ℤ is reduction modulo 2.     

Homotopy Types of Stabilizers and Orbits of Morse Functions on Surfaces

Let M be a smooth compact surface, orientable or not, with boundary or without it, P either the real line ℝ ¹ or the circle S ¹, and D(M) the group of diffeomorphisms of M acting on C^∞(M, P) by the rule h⋅ f = f ∘ h ⁻¹ for h ∊ D(M) and f ∊ C^∞ (M,P). Let f: M → P be an arbitrary Morse mapping, Σ _f the set of critical points of f, D(M,Σ _f ) the subgroup of D(M) preserving Σ _f , and S(f), S (f,Σ _f ), O(f), and O(f,Σ _f ) the stabilizers and the orbits of f with respect to D(M) and D(M,Σ _f ). In fact S(f) = S(f,Σ _f ).In this paper we calculate the homotopy types of S(f), O(f) and O(f,Σ _f ). It is proved that except for few cases the connected components of S(f) and O(f,Σ _f ) are contractible, π _k O(f) = π _k M for k ≥ 3, π₂ O(f) = 0, and π₁ O(f) is an extension of π₁ D(M) ⊕ Z ^k (for some k ≥ 0) with a (finite) subgroup of the group of automorphisms of the Kronrod-Reeb graph of f.We also generalize the methods of F. Sergeraert to give conditions for a finite codimension orbit of a tame smooth action of a tame Lie group on a tame Fréchet manifold to be a tame Fréchet manifold itself. In particular, we obtain that O(f) and O(f, Σ _f ) are tame Fréchet manifolds. Communicated by Peter Michor Vienna Mathematics Subject Classifications (2000): 37C05, 57S05, 57R45.     

The number of exceptional orbits of a pseudofree circle action on S 5

Let M be a closed Riemannian manifold of dimension 5 which admits a Riemannian metric of nonnegative sectional curvature. The aim of this short paper is to show that under certain lower bound of the orders of isotropy subgroups, every pseudofree and isometric S ¹-action on M cannot have more than five exceptional circle orbits. As a consequence, we conclude that a pseudofree and isometric S ¹-action on a 5-sphere S ⁵ with a Riemannian metric of nonnegative sectional curvature cannot have more than five exceptional circle orbits. This gives a result related to the Montgomery–Yang problem. In addition, we also give some further related result about nonnegatively curved manifolds of dimension 5 with an isometric but not necessarily pseudofree circle action.     

n-dimensional totally real minimal submanifolds of CP n

Let CP ⁿ be the n-dimensional complex projective space with the Study-Fubini metric of constant holomorphic sectional curvature 4 and let M be a compact, orientable, n-dimensional totally real minimal submanifold of CP ⁿ . In this paper we prove the following results.

TT-eigentensors for the Lichnerowicz Laplacian on some asymptotically hyperbolic manifolds with warped products metrics

Let (M =]0, ∞[×N, g) be an asymptotically hyperbolic manifold of dimension n + 1 ≥ 3, equipped with a warped product metric. We show that there exist no TT L ²-eigentensors with eigenvalue in the essential spectrum of the Lichnerowicz Laplacian Δ _L . If (M, g) is the real hyperbolic space, there is no symmetric L ²-eigentensors of Δ _L .     

Commensurability of Hyperbolic Manifolds with Geodesic Boundary

Suppose , let M ₁, M ₂ be n-dimensional connected complete finite-volume hyperbolic manifolds with nonempty geodesic boundary, and suppose that π₁ (M ₁) is quasi-isometric to π₁ (M ₂) (with respect to the word metric). Also suppose that if n=3, then ∂M ₁ and ∂M ₂ are compact. We show that M ₁ is commensurable with M ₂. Moreover, we show that there exist homotopically equivalent hyperbolic 3-manifolds with non-compact geodesic boundary which are not commensurable with each other. We also prove that if M is as M ₁ above and G is a finitely generated group which is quasi-isometric to π₁ (M), then there exists a hyperbolic manifold with geodesic boundary M′ with the following properties: M′ is commensurable with M, and G is a finite extension of a group which contains π₁ (M′) as a finite-index subgroupMathematics Subject Classification (2000). Primary: 20F65; secondary: 30C65, 57N16