On the generalized Chern conjecture for hypersurfaces with constant mean curvature in a sphere

Let M be a compact hypersurface with constant mean curvature in Denote by H and S the mean curvature and the squared norm of the second fundamental form of M,respectively.We verify that there exists a positive constant γ(n) depending only on n such that if |H| ≤γ(n) and β(n,H)≤S ≤β(n,H)+n/18,then S≡β(n,H) and M is a Clifford torus.Here,β(n,H)=n+n~3/2(n-1)H~2+n(n-2)/2(n-1)(1/2)n~2H~4+4(n-1)H~2.     

Rigidity theorems for complete $$\lambda $$ λ -hypersurfaces

Archiv der Mathematik - In this article, we study hypersurfaces $$\Sigma \subset {\mathbb {R}}^{n+1}$$ with constant weighted mean curvature, also known as $$\lambda $$ -hypersurfaces. Recently,...     

Lipschitz constants for the real part and modulus of analytic mappings on a negatively curved surface

Archiv der Mathematik - We prove that if f, $$|\mathfrak {R}f|<1$$ , is an analytic mapping on a surface $$\Sigma $$ with curvature bounded from below by a constant $$k<0$$ , and if...     

p-Kähler and balanced structures on nilmanifolds with nilpotent complex structures

Annals of Global Analysis and Geometry - Let $$(X, T^{1,0}X)$$ be a compact connected orientable CR manifold of dimension $$2n+1$$ with non-degenerate Levi curvature. Assume that X admits a...     

Height estimates for constant mean curvature graphs in $$\mathrm {Nil}_3$$Nil3 and $${\widetilde{PSL_2}}({\mathbb {R}})$$PSL2~(R)

Archiv der Mathematik - In this paper we obtain height estimates for compact, constant mean curvature vertical graphs in the homogeneous spaces $$\mathrm {Nil}_3$$ and...     

Dark $$f(\mathcal{R},\varphi,\chi)$$ universe with Noether symmetry

Theoretical and Mathematical Physics - Using the Noether symmetry approach, we investigate $$f( \mathcal{R} , \varphi ,\chi)$$ theories of gravity, where $$ \mathcal{R} $$ is the scalar curvature,...     

First integrals for Finsler metrics with vanishing $$\chi $$ -curvature

Annals of Global Analysis and Geometry - We prove that in a Finsler manifold with vanishing $$\chi $$ -curvature (in particular with constant flag curvature) some non-Riemannian geometric...     

Free boundary constant mean curvature surfaces in a strictly convex three-manifold

Annals of Global Analysis and Geometry - Let C be a strictly convex domain in a three-dimensional Riemannian manifold with sectional curvature bounded above by a constant, and let $$\Sigma $$ be a...     

Manifolds of nonnegative Ricci curvature and infinite topological type

Geometriae Dedicata - We construct a family of complete n-dimensional ( $$n\ge 4$$ ) manifolds with nonnegative Ricci curvature and infinite topological type. Moreover, the curvature decay,...     

Linear Weingarten Spacelike Submanifolds in Semi-Riemannian Space Form

Mathematical Notes - We consider the linear Weingarten spacelike submanifold immersed in semi-Riemannian space form $$\mathbb {N}^{n+p} _q (c)$$ of constant sectional curvature $$c$$ and index...     

Integral points on convex curves

The Ramanujan Journal - We estimate the maximal number of integral points which can be on a convex arc in $${\mathbb {R}}^2$$ with given length, minimal radius of curvature and initial slope.     

Nonexistence of the NNSC-cobordism of Bartnik data

In this paper,we consider the problem of the nonnegative scalar curvature(NNSC)-cobordism of Bartnik data(∑_1~(n-1),γ_1,H_1) and(∑_2~(n-1),γ_2,H_2).We prove that given two metrics γ_1 and γ_2 on S~(n-1)(3≤n ≤ 7)with H_1 fixed,then(S~(n-1),γ_1,H_1) and(S~(n-1),γ_2,H_2) admit no NNSC-cobordism provided the prescribed mean curvature H2 is large enough(see Theorem 1.3).Moreover,we show that for n=3,a much weaker condition that the total mean curvature ∫_(s~2) H_2 dpγ_2 is large enough rules out NNSC-cobordisms(see Theorem 1.2);if we require the Gaussian curvature of γ_2 to be positive,we get a criterion for nonexistence of the trivial NNSCcobordism by using the Hawking mass and the Brown-York mass(see Theorem 1.1).For the general topology case,we prove that(∑_1~(n-1),γ_1,0) and(∑_2~(n-1),γ_2,H_2) admit no NNSC-cobordism provided the prescribed mean curvature H_2 is large enough(see Theorem 1.5).     

Eigenvalues of $$(-\triangle + \frac{R}{2})$$ on manifolds with nonnegative curvature operator

In this paper, we show that the eigenvalues of are nondecreasing under the Ricci flow for manifolds with nonnegative curvature operator. Then we show that the only steady Ricci breather with nonnegative curvature operator is the trivial one which is Ricci-flat.     

Area-stationary surfaces inside the sub-Riemannian three-sphere

We consider the sub-Riemannian metric g _h on given by the restriction of the Riemannian metric of curvature 1 to the plane distribution orthogonal to the Hopf vector field. We compute the geodesics associated to the Carnot–Carathéodory distance and we show that, depending on their curvature, they are closed or dense subsets of a Clifford torus. We study area-stationary surfaces with or without a volume constraint in (). By following the ideas and techniques by Ritoré and Rosales (Area-stationary surfaces in the Heisenberg group , arXiv:math.DG/0512547) we introduce a variational notion of mean curvature, characterize stationary surfaces, and prove classification results for complete volume-preserving area-stationary surfaces with non-empty singular set. We also use the behaviour of the Carnot–Carathéodory geodesics and the ruling property of constant mean curvature surfaces to show that the only C ² compact, connected, embedded surfaces in () with empty singular set and constant mean curvature H such that is an irrational number, are Clifford tori. Finally we describe which are the complete rotationally invariant surfaces with constant mean curvature in (). A. Hurtado has been partially supported by MCyT-Feder research project MTM2004-06015-C02-01. C. Rosales has been supported by MCyT-Feder research project MTM2004-01387.     

On constant mean curvature hypersurfaces with finite index

We discuss the non-existence of complete noncompact constant mean curvature hypersurfaces with finite index in a 4- or 5-dimensional manifold. As a consequence, we obtain that a complete noncompact constant mean curvature hypersurface in with finite index must be minimal. Received: 30 May 2005     

MINIMAL SUBMANIFOLDS IN A RIEMANNIAN MANIFOLD OF QUASI CONSTANT CURVATURE

A Riemannian manifold V~m which admits isometric imbedding into two spaces V~(m+p)ofdifferent constant curvatures is called a manifold of quasi constant curvature.TheRiemannian curvature of V~m is expressible in the formand conversely.In this paper it is proved that if M~n is any compact minimal submanifoldwithout boundary in a Riemannian manifold V~(n+p)of quasi constant curvature,then∫_(M~u)(2-1/p)σ~2-[na+1/2(b-丨b丨)(n+1)]σ+n(n-1)b~2*丨≥0,where σ is the square of the norm of the second fundamental form of M~n When V~(n+p)is amanifold of constant curvature,b=0,the above inequality reduces to that of Simons.     

Complete surfaces of constant curvature in H<Superscript>2</Superscript> × R and S<Superscript>2</Superscript> × R

We study isometric immersions of surfaces of constant curvature into the homogeneous spaces and . In particular, we prove that there exists a unique isometric immersion from the standard 2-sphere of constant curvature c > 0 into and a unique one into when c > 1, up to isometries of the ambient space. Moreover, we show that the hyperbolic plane of constant curvature c < −1 cannot be isometrically immersed into or . J.A. Aledo was partially supported by Ministerio de Education y Ciencia Grant No. MTM2004-02746 and Junta de Comunidades de Castilla-La Mancha, grant no. PAI-05-034. J.M. Espinar and J.A. Gálvez were partially supported by Ministerio de Education y Ciencia grant no. MTM2004-02746 and Junta de Andalucía Grant No. FQM325.     

P-separation of variables in Laplace's equation

The P-separation of variables in Laplace's equation Δ₂u = 0 in flat n-dimensional space S_n is proved to be equivalent to the complete separation of variables in the invariant Laplace equation $$\Delta u \equiv \left\{ {\Delta _2 + \frac{{n - 2}}{{4(n - 1)}}R} \right\}u = 0$$ , in a space V_n of constant curvature K ≠ 0 (Δ is the invariant Laplacian, and R is the scalar curvature, all in V_n).     

Pinching estimates for negatively curved manifolds with nilpotent fundamental groups

Let M be a complete Riemannian metric of sectional curvature within [−a²,−1] whose fundamental group contains a k-step nilpotent subgroup of finite index. We prove that a ≥ k answering a question of M. Gromov. Furthermore, we show that for any the manifold M admits a complete Riemannian metric of sectional curvature within Received: May 2004 Revision: July 2004 Accepted: July 2004     

Curvature homogeneous Lorentzian three-manifolds

We study three-dimensional curvature homogeneous Lorentzian manifolds. We prove that for all Segre types of the Ricci operator, there exist examples of nonhomogeneous curvature homogeneous Lorentzian metrics in .