Hypersurfaces in a Unit Sphere Sn+1(1) with Constant Scalar Curvature

The paper considers n-dimensional hypersurfaces with constantscalar curvature of a unit sphere S^n–1(1). The hypersurfaceS^k(c₁)xS^n–k(c₂) in a unit sphere Sⁿ⁺¹(1) is characterized,and it is shown that there exist many compact hypersurfaceswith constant scalar curvature in a unit sphere Sⁿ⁺¹(1) whichare not congruent to each other in it. In particular, it isproved that if M is an n-dimensional (n > 3) complete locallyconformally flat hypersurface with constant scalar curvaturen(n–1)r in a unit sphere Sⁿ⁺¹(1), then r > 1–2/n,and (1) when r (n–2)/(n–1), if then M is isometric to S¹xS^n–1(c),where S is the squared norm of the second fundamental form ofM; (2) there are no complete hypersurfaces in Sⁿ⁺¹(1) with constantscalar curvature n(n–1)r and with two distinct principalcurvatures, one of which is simple, such that r = (n–2)/(n–1)and      

An Euler-Type Version of the Local Steiner Formula for Convex Bodies

The purpose of this note is to establish a new version of thelocal Steiner formula and to give an application to convex bodiesof constant width. This variant of the Steiner formula generalizesresults of Hann [3] and Hug [6], who use much less elementarytechniques than the methods of this paper. In fact, Hann askedfor a simpler proof of these results [4, Problem 2, p. 900].We remark that our formula can be considered as a Euclideananalogue of a spherical result proved in [2, p. 46], and thatour method can also be applied in hyperbolic space. For some remarks on related formulas in certain two-dimensionalMinkowski spaces, see Hann [5, p. 363]. For further information about the notions used below, we referto Schneider's book [9]. Let Kⁿ be the set of all convex bodiesin Euclidean space Rⁿ, that is, the set of all compact, convex,non-empty subsets of Rⁿ. Let S^n–1 be the unit sphere.For KKⁿ, let NorK be the set of all support elements of K, thatis, the pairs (x, u)RⁿxS^n–1 such that x is a boundarypoint of K and u is an outer unit normal vector of K at thepoint x. The support measures (or generalized curvature measures)of K, denoted by ₀(K.), ..., _n–1(K.), are the unique Borelmeasures on RⁿxS^n–1 that are concentrated on NorK andsatisfy [formula] for all integrable functions f:RⁿR; here denotes the Lebesguemeasure on Rⁿ. Equation (1), which is a consequence and a slightgeneralization of Theorem 4.2.1 in Schneider [9], is calledthe local Steiner formula. Our main result is the following.1991 Mathematics Subject Classification 52A20, 52A38, 52A55.     

Volume Estimate Via Total Curvature in Hyperbolic Spaces

Let D Hⁿ(–k²) be a convex compact subset of the hyperbolicspace Hⁿ(–k²) with non-empty interior and smooth boundary.It is shown that the volume of D can be estimated by the totalcurvature of D. More precisely, , where K denotes the Gauss–Kronecker curvature of D andVol(S^n–1) denotes the Euclidean volume of the sphere.2000 Mathematics Subject Classification 53C21.     

On the Ricci curvature of a Randers metric of isotropic <Emphasis Type="Italic">S</Emphasis>-curvature

We derive the integral inequality of a Randers metric with isotropic S-curvature in terms of its navigation representation. Using the obtained inequality we give some rigidity results under the condition of Ricci curvature. In particular, we show the following result： Assume that an n-dimensional compact Randers manifold （M, F） has constant S-curvature c. Then （M, F） must be Riemannian if its Ricci curvature satisfies that Ric 〈 -（n - 1）c^2.     

Complete Manifolds with Sectional Curvature Bounded Below and Large Volume Growth

This paper provides a proof that an n-dimensional complete openRiemannian manifold M with sectional curvature K_M –1is diffeomorphic to a Euclidean n-space Rⁿ if the volume growthof geodesic balls in M is close to that of the balls in an n-dimensionalhyperbolic space Hⁿ(–1) of sectional curvature –1.     

Totally Real Minimal Surfaces with Non-Circular Ellipse of Curvature in the Nearly Kahler S6

In [2] we discussed almost complex curves in the nearly KählerS⁶. These are surfaces with constant Kähler angle 0 or and, as a consequence of this, are also minimal and have circularellipse of curvature. We also considered minimal immersionswith constant Kähler angle not equal to 0 or , but withellipse of curvature a circle. We showed that these are linearlyfull in a totally geodesic S⁵ in S⁶ and that (in the simplyconnected case) each belongs to the S¹-family of horizontallifts of a totally real (non-totally geodesic) minimal surfacein CP². Indeed, every element of such an S¹-family has constantKähler angle and in each family all constant Kählerangles occur. In particular, every minimal immersion with constantKähler angle and ellipse of curvature a circle is obtainedby rotating an almost complex curve which is linearly full ina totally geodesic S⁵.     

Circle Actions and Morse Theory on Quaternion-Kahler Manifolds

In this paper we study quaternion-Kähler manifolds endowedwith an isometric S¹-action. We consider the corresponding momentmap µ and prove that the only compact quaternion-Kählermanifold with positive scalar curvature which admits an isometriccircle action free on µ^–1(0) is the quaternionicprojective space HPⁿ.     

Shapiro's Cyclic Sum

Shapiro's cyclic sum is defined by , If K is the cone in Rⁿ of points withnon-negative coordinates, it is shown that the minimum of Ein K is a fixed point of T², where T is the non-linear operatordefined by (Tx)_i = x_n–i+1/(x_n–i+2 + x_n–i+3)²for i = 1,2,...,n. It is conjectured that Tx = S^kx, where Sis the shift operator in Rⁿ, and a proof is given under someadditional hypotheses. One of the consequences is a simple proofthat at the minimum point, a_i(x) = a_n–i+1–k(x) fori = 1,2,...,n.     

Non-Complex Symplectic 4-Manifolds with Formula

Simply connected closed symplectic 4-manifolds with and K² = 0 are investigated. As aresult, it is confirmed that most of homotopy elliptic surfaces{E(1)_k|K is a fibred knot in S³} constructed by R. Fintusheland R. Stern in Invent. Math. 134 (1998) 363–400 are simplyconnected closed minimal symplectic 4-manifolds that do notadmit a complex structure. 2000 Mathematics Subject Classification57R17, 57R57 (primary), 14J26 (secondary).     

A Multilinear Extension Inequality In Rn

Sharp decay estimates are provided in this paper for sphericalaverages of a certain multilinear extension operator on L² (S^n–1)x ... x L² (S^n–1). 2000 Mathematics Subject Classification42B10.     

Curvatures of homogeneous Randers spaces

We study curvatures of homogeneous Randers spaces. After deducing the coordinate-free formulas of the flag curvature and Ricci scalar of homogeneous Randers spaces, we give several applications. We first present a direct proof of the fact that a homogeneous Randers space is Ricci quadratic if and only if it is a Berwald space. We then prove that any left invariant Randers metric on a non-commutative nilpotent Lie group must have three flags whose flag curvature is positive, negative and zero, respectively. This generalizes a result of J.A. Wolf on Riemannian metrics. We prove a conjecture of J. Milnor on the characterization of central elements of a real Lie algebra, in a more generalized sense. Finally, we study homogeneous Finsler spaces of positive flag curvature and particularly prove that the only compact connected simply connected Lie group admitting a left invariant Finsler metric with positive flag curvature is SU(2)$\text{SU}\left(2\right)$.     

On 4-manifolds with zero-entropy metrics

We sharpen a recent result of Paternain and Petean by showingthat a closed 4-manifold which admits a Riemannian metric withzero topological entropy and infinite fundamental group eitheris aspherical or has a finite covering space homeomorphic toS³ x S¹ or S² x S¹ x S¹.     

Finsler空间上的Weyl曲率

The Weyl curvature of a Finsler metric is investigated. This curvature constructed from Riemannain curvature. It is an important projective invariant of Finsler metrics. The author gives the necessary conditions on Weyl curvature for a Finsler metric to be Randers metric and presents examples of Randers metrics with non-scalar curvature. A global rigidity theorem for compact Finsler manifolds satisfying such conditions is proved. It is showed that for such a Finsler manifold,if Ricci scalar is negative,then Finsler metric is of Randers type.     

The Normal Holonomy Group of Kahler Submanifolds

We study the (restricted) holonomy group Hol() of the normalconnection (shortened to normal holonomy group) of a Kählersubmanifold of a complex space form. We prove that if the normalholonomy group acts irreducibly on the normal space then itis linear isomorphic to the holonomy group of an irreducibleHermitian symmetric space. In particular, it is a compact groupand the complex structure J belongs to its Lie algebra. We prove that the normal holonomy group acts irreducibly ifthe submanifold is full (that is, it is not contained in a totallygeodesic proper Kähler submanifold) and the second fundamentalform at some point has no kernel. For example, a Kähler–Einsteinsubmanifold of CPⁿ has this property. We define a new invariant µ of a Kähler submanifoldof a complex space form. For non-full submanifolds, the invariantµ measures the deviation of J from belonging to the normalholonomy algebra. For a Kähler–Einstein submanifold,the invariant µ is a rational function of the Einsteinconstant. By using the invariant µ, we prove that thenormal holonomy group of a not necessarily full Kähler–Einsteinsubmanifold of CPⁿ is compact, and we give a list of possibleholonomy groups. The approach is based on a definition of the holonomy algebrahol(P) of an arbitrary curvature tensor field P on a vectorbundle with a connection and on a De Rham type decompositiontheorem for hol(P). 2000 Mathematics Subject Classification53C40 (primary), 53B25 (secondary).     

The Kirwan Map for Singular Symplectic Quotients

Let M be a Hamiltonian K-space with proper moment map µ.The symplectic quotient X = µ^–1(0)/K is a singularstratified space with a symplectic structure on the strata.In this paper we generalise the Kirwan map, which maps the Kequivariant cohomology of µ^–1(0) to the middle perversityintersection cohomology of X, to this symplectic setting. The key technical results which allow us to do this are Meinrenken'sand Sjamaar's partial desingularisation of singular symplecticquotients and a decomposition theorem, proved in Section 2 ofthis paper, exhibiting the intersection cohomology of a ‘symplecticblowup’ of the singular quotient X along a maximal depthstratum as a direct sum of terms including the intersectioncohomology of X.     

On （α,β）-Metrics of Scalar Flag Curvature with Constant S-curvature

In this paper, we study （α,β）-metrics of scalar flag curvature on a manifold M of dimension n （n 〉 3）. Suppose that an （α,β）-metric F is not a Finsler metric of Randers type, that is, F ≠k1 V√α^2 ＋ k2β^2 ＋ k3β, where k1 〉 0, k2 and k3 are scalar functions on M. We prove that F is of scalar flag curvature and of vanishing S-curvature if metric. In this case, F is a locally Minkowski and only if the flag curvature K = 0 and F is a Berwald metric.     

Exotic smooth structures on Formula

Motivated by Stipsicz and Szabó's exotic 4-manifoldswith b₂⁺ = 3 and b₂^– = 8, we construct a family of simplyconnected smooth 4-manifolds with b₂⁺ = 3 and b₂^– = 8.As a corollary, we conclude that the topological 4-manifold     

Some Sharp Bounds for the Cone Multiplier of Negative Order in R3

This paper considers the cone multiplier operator which is definedby where and . For –3/2 < µ < –3/14, sharp L^p – L^q estimatesand endpoint estimates for S^µ are obtained. 2000 MathematicsSubject Classification 42B15 (primary).     

Waves in the Presence of an Infinite Dock with Gap

After extending some completeness results of an earlier paper,the two-dimensional problem of the infinite dock with gap isconsidered. With the frequency and 2 the non-dimensional lengthof the gap, eigenvalues of K = ²/g are first computed and thentheir apparent asymptotic form is established not only to orderK^–1 but also up to terms in K^–2.     

The Bicanonical Map of Surfaces with pg = 0 and K2 >= 7, II

Minimal complex surfaces of general type with p_g = 0 and K²= 7 or 8 whose bicanonical map is not birational are studied.It is shown that if S is such a surface, then the bicanonicalmap has degree 2 (see Bulletin of the London Mathematical Society33 (2001) 1–10) and there is a fibration f: S P¹ suchthat (i) the general fibre F of f is a genus 3 hyperellipticcurve; (ii) the involution induced by the bicanonical map ofS restricts to the hyperelliptic involution of F. Furthermore, if , then f isan isotrivial fibration with six double fibres, and if , then f has five double fibres andit has precisely one fibre with reducible support, consistingof two components. 2000 Mathematics Subject Classification 14J29.