On Some Finsler Metrics of Non-positive Constant Flag Curvature

(α, β)-norms on ${\mathbb{R}^N}$ induce Minkowski metrics, and the construction of related homothetic vector fields gives a family of new Finsler metrics of non-positive constant flag curvature for each non-trivial (α, β)-norm. The dimension of this family is at least ${\tfrac{1}{2}(N^2 - N + 4)}$ . In particular, we generalize the Funk metric on the unit ball via navigation representation of the standard Euclidean norm and the radial vector field. Finally, we describe the geodesics of these new Finsler metrics with constant flag curvature.     

Projectively Flat Singular Square Metrics with Constant Flag Curvature

In this paper, we study and characterize locally projectively flat singular square metrics with constant flag curvature. First, we obtain the sufficient and necessary conditions that singular square metrics are locally projectively flat. Furthermore, we classify locally projectively flat singular square metrics with constant flag curvature completely.     

Contact Pseudo-Metric Manifolds of Constant Curvature and CR Geometry

In this paper, we show that if an integrable contact pseudo-metric manifold of dimension 2n + 1, n ≥ 2, has constant sectional curvature \({\kappa}\) , then the structure is Sasakian and \({\kappa=\varepsilon=g(\xi,\xi)}\) , where \({\xi}\) is the Reeb vector field. We note that the notion of contact pseudo-metric structure is equivalent to the notion of non-degenerate almost CR manifold, then an equivalent statement of this result holds in terms of CR geometry. Moreover, we study the pseudohermitian torsion \({\tau}\) of a non-degenerate almost CR manifold.     

The Vectorial Ribaucour Transformation for Submanifolds of Constant Sectional Curvature

We obtain a reduction of the vectorial Ribaucour transformation that preserves the class of submanifolds of constant sectional curvature of space forms, which we call the L-transformation. It allows to construct a family of such submanifolds starting with a given one and a vector-valued solution of a system of linear partial differential equations. We prove a decomposition theorem for the L-transformation, which is a far-reaching generalization of the classical permutability formula for the Ribaucour transformation of surfaces of constant curvature in Euclidean three space. As a consequence, we derive a Bianchi-cube theorem, which allows to produce, from k initial scalar L-transforms of a given submanifold of constant curvature, a whole k-dimensional cube all of whose remaining \(2^k-(k+1)\) vertices are submanifolds with the same constant sectional curvature given by explicit algebraic formulae. We also obtain further reductions, as well as corresponding decomposition and Bianchi-cube theorems, for the classes of n-dimensional flat Lagrangian submanifolds of \({\mathbb {C}}^n\) and n-dimensional Lagrangian submanifolds with constant curvature c of the complex projective space \({\mathbb {C}}{\mathbb {P}}^n(4c)\) or the complex hyperbolic space \({\mathbb {C}}{\mathbb {H}}^n(4c)\) of complex dimension n and constant holomorphic curvature 4c.     

Lie Superalgebras Associated with Constant Sectional Curvature

This note describes an observation connecting Riemannian manifolds of constant sectional curvature with a particular class of Lie superalgebras. Specifically, it is shown that the structural equations of a space M with constant sectional curvature, of one variety or another, nearly coincide with some identities satisfied by tensors which can be used to construct some specific families of Lie superalgebras. In particular, one obtains either osp(n,2), spl(n,2), or osp(4,2n) if the Riemannian manifold has constant curvature, constant holomorphic curvature or constant quaternion-holomorphic curvature, respectively.Mathematics Subject Classiffications (2000). 17A70, 53C29, 53C99, 57Rxx     

Finsler Metrics of Constant Positive Curvature on the Lie Group S3

Guided by the Hopf fibration, a family (indexed by a positiveconstant K) of right invariant Riemannian metrics on the Liegroup S³ is singled out. Using the Yasuda–Shimada paperas an inspiration, a privileged right invariant Killing fieldof constant length is determined for each K > 1. Each suchRiemannian metric couples with the corresponding Killing fieldto produce a y-global and explicit Randers metric on S³. Employingthe machinery of spray curvature and Berwald's formula, it isproved directly that the said Randers metric has constant positiveflag curvature K, as predicted by Yasuda–Shimada. It isexplained why this family of Finslerian space forms is not projectivelyflat.     

Projective spherically symmetric Finsler metrics with constant flag curvature in R n

We investigate projective spherically symmetric Finsler metrics with constant flag curvature in R ⁿ and give the complete classification theorems. Furthermore, a new class of Finsler metrics with two parameters on n-dimensional disk is found to have constant negative flag curvature.     

Radial Graphs with Constant Mean Curvature in the Hyperbolic Space

The existence is proved of radial graphs with constant mean curvature in the hyperbolic space H ⁿ⁺¹ defined over domains in geodesic spheres of H ⁿ⁺¹ whose boundary has positive mean curvature with respect to the inward orientation.     

On Complete Hypersurfaces with Constant Mean Curvature and Finite L^P-norm Curvature in R^n＋1

By using curvature estimates, we prove that a complete non-compact hypersurface M with constant mean curvature and finite L^n-norm curvature in R^1＋1 must be minimal, so that M is a hyperplane if it is strongly stable. This is a generalization of the result on stable complete minimal hypersurfaces of R^n＋1. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite L^1-norm curvature in R^1＋1 are considered.     

Finsler Manifolds with Positive Constant Flag Curvature

It is shown that a Finsler metric with positive constant flag curvature and vanishing mean tangent curvature must be Riemannian. As applications, we also discuss the case of Cheng's maximal diameter theorem and Green's maximal conjugate radius theorem in Finsler manifolds.     

On the Essential Spectrum of the Laplacian and Vague Convergence of the Curvature at Infinity

We shall prove that under some volume growth condition, the essential spectrum of the Laplacian contains the interval [(n ? 1)² K/4, ∞) if an n-dimensional Rieman-nian manifold has an end and the average of the part of the Ricci curvature on the end which lies below a nonpositive constant (n ? 1)K converges to zero at infinity.     

Ricci Deformation of the Metric on Riemannian Orbifolds

In this note we generalize the Huisken’s (J Diff Geom 21:47–62, 1985) result to Riemannian orbifolds. We show that on any n-dimensional (n ≥ 4) orbifold of positive scalar curvature the metric can be deformed into a metric of constant positive curvature, provided the norm of the Weyl conformal curvature tensor and the norm of the traceless Ricci tensor are not large compared to the scalar curvature at each point, and therefore generalize 3-orbifolds result proved by Hamilton [Three- orbifolds with positive Ricci curvature. In: Cao HD, Chow B, Chu SC, Yau ST (eds) Collected Papers on Ricci Flow, Internat. Press, Somerville, 2003] to n-orbifolds (n ≥ 4).     

A Maximum Principle at Infinity for Surfaces with Constant Mean Curvature in Euclidean Space

Maximum principles at infinity generalize Hopf's maximum principle for hypersurfaces with constant mean curvature in R ⁿ . We establish such a maximum principle for parabolic surfaces in R³ with nonzero constant mean curvature and bounded Gaussian curvature.     

Rotational Hypersurfaces with Constant Gauss-Kronecker Curvature

The authors study rotational hypersurfaces with constant Gauss-Kronecker curvature in Rⁿ. They solve the ODE associated with the generating curve of such hypersurface using integral expressions and obtain several geometric properties of such hypersurfaces. In particular, they discover a class of non-compact rotational hypersurfaces with constant and negative Gauss-Kronecker curvature and finite volume, which can be seen as the higher-dimensional generalization of the pseudo-sphere.     

Biharmonic Submanifolds with Parallel Mean Curvature in \mathbb{S}^{n}\times\mathbb{R}

We find a Simons type formula for submanifolds with parallel mean curvature vector (pmc submanifolds) in product spaces M ⁿ (c)×?, where M ⁿ (c) is a space form with constant sectional curvature c, and then we use it to prove a gap theorem for the mean curvature of certain complete proper-biharmonic pmc submanifolds, and classify proper-biharmonic pmc surfaces in $\mathbb{S}^{n}(c)\times\mathbb{R}$ .     

Concordance and Isotopy of Metrics with Positive Scalar Curvature

Two positive scalar curvature metrics g ₀, g ₁ on a manifold M are psc-isotopic if they are homotopic through metrics of positive scalar curvature. It is well known that if two metrics g ₀, g ₁ of positive scalar curvature on a closed compact manifold M are psc-isotopic, then they are psc-concordant: i.e., there exists a metric ${\bar{g}}$ of positive scalar curvature on the cylinder ${M \times I}$ which extends the metrics g ₀ on ${M \times \{0\}}$ and g ₁ on ${M \times \{1\}}$ and is a product metric near the boundary. The main result of the paper is that if psc-metrics g ₀, g ₁ on M are psc-concordant, then there exists a diffeomorphism ${\Phi : M \times I \rightarrow M \times I}$ with ${\Phi|_{M \times \{0\}} = Id}$ (a pseudo-isotopy) such that the metrics g ₀ and ${(\Phi|_{M \times \{1\}})^{*}g_{1}}$ are psc-isotopic. In particular, for a simply connected manifold M with dim M ≥  5, psc-metrics g ₀, g ₁ are psc-isotopic if and only if they are psc-concordant. To prove these results, we employ a combination of relevant methods: surgery tools related to the Gromov–Lawson construction, classic results on isotopy and pseudo-isotopy of diffeomorphisms, standard geometric analysis related to the conformal Laplacian, and the Ricci flow.     

A Remark on Kahler Metrics of Constant Scalar Curvature on Ruled Complex Surfaces

In this paper we point out how some recent developments in thetheory of constant scalar curvature Kähler metrics canbe used to clarify the existence issue for such metrics in thespecial case of (geometrically) ruled complex surfaces. 2000Mathematics Subject Classification 53C55, 58E11.     

The First Eigenvalue of p-Laplace Operator Under Powers of the mth Mean Curvature Flow

In this paper, we derive the evolution equation for the eigenvalues of p-Laplace operator. Moreover, we show the following main results. Let ( ${M^{n}, g(t)), t\in [0,T),}$ be a solution of the unnormalized powers of the mth mean curvature flow on a closed manifold and λ_1,p (t) be the first eigenvalue of the p-Laplace operator (p ≥ n). At the initial time t = 0, if H > 0, and $$h_{ij}\geq \varepsilon Hg_{ij}\quad \left(\frac{1}{p}\leq\epsilon\leq \frac{1}{n}\right),$$ then λ_1,p (t) is nondecreasing and differentiable almost everywhere along the unnormalized powers of the mth mean curvature flow on [0,T). At last, we discuss some interesting monotonic quantities under unnormalized powers of the mth mean curvature flow.     

Complete Hypersurfaces with Constant Mean Curvature and Finite Total Curvature

The main result of this paper states that the traceless second fundamental tensor A⁰ of an n-dimensional complete hypersurface M, with constant mean curvature H and finite total curvature, _M |A⁰|ⁿ dv_M < , in a simply-connected space form (c), with non-positive curvature c, goes to zero uniformly at infinity. Several corollaries of this result are considered: any such hypersurface has finite index and, in dimension 2, if H ² + c > 0, any such surface must be compact.     

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 .