Finite Topological Type and Volume Growth

We show that a complete noncompact n-dimensional Riemannian manifold Mwith Ricci curvature Ric _M –(n – 1) and conjugateradius conj _M c > 0 has finite topological type, provided that the volume growth of geodesic balls in M is not very far from that of the balls in an n-dimensional hyperbolic space H ⁿ (–1)of sectional curvature –1. We also show that a complete open Riemannian manifold M with nonnegative intermediate Ricci curvature and quadratic curvature decay has finite topological typeif the volume of geodesic balls of M around the base point grows slowly.     

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      

Complete Hypersurfaces with Constant Mean Curvature in a Unit Sphere

By investigating hypersurfaces M ⁿ in the unit sphere S ⁿ⁺¹(1) with constant mean curvature and with two distinct principal curvatures, we give a characterization of the torus S ¹(a) × S^n-1(?{1-a²})S^{n-1}(\sqrt{1-a^2}) , where a²=\frac2+nH²±?{n²H⁴+4(n-1)H²}2n(1+H²)a^2=\frac{2+nH^2\pm\sqrt{n^2H^4+4(n-1)H^2}}{2n(1+H^2)} . We extend recent results of Hasanis et al. [5] and Otsuki [10].     

The Extension of the H~k Mean Curvature Flow in Riemannian Manifolds

In this paper,the authors consider a family of smooth immersions Ft : Mn→Nn+1of closed hypersurfaces in Riemannian manifold Nn+1with bounded geometry,moving by the Hkmean curvature flow.The authors show that if the second fundamental form stays bounded from below,then the Hkmean curvature flow solution with finite total mean curvature on a finite time interval [0,Tmax)can be extended over Tmax.This result generalizes the extension theorems in the paper of Li(see "On an extension of the Hkmean curvature flow,Sci.China Math.,55,2012,99–118").     

Upper bounds on the length of a shortest closed geodesic and quantitative Hurewicz theorem

In this paper we present two upper bounds on the length of a shortest closed geodesic on compact Riemannian manifolds. The first upper bound depends on an upper bound on sectional curvature and an upper bound on the volume of the manifold. The second upper bound will be given in terms of a lower bound on sectional curvature, an upper bound on the diameter and a lower bound on the volume.The related questions that will also be studied are the following: given a contractible k-dimensional sphere in M ⁿ , how “fast” can this sphere be contracted to a point, if π _i (M ⁿ )={0} for 1≤i<k. That is, what is the maximal length of the trajectory described by a point of a sphere under an “optimal” homotopy? Also, what is the “size” of the smallest non-contractible k-dimensional sphere in a (k-1)-connected manifold M ⁿ providing that M ⁿ is not k-connected?     

On the pseudohermitian sectional curvature of a strictly pseudoconvex CR manifold

We show that the pseudohermitian sectional curvature Hθ(σ) of a contact form θ on a strictly pseudoconvex CR manifold M measures the difference between the lengths of a circle in a plane tangent at a point of M and its projection on M by the exponential map associated to the Tanaka-Webster connection of (M,θ). Any Sasakian manifold (M,θ) whose pseudohermitian sectional curvature Kθ(σ) is a point function is shown to be Tanaka-Webster flat, and hence a Sasakian space form of φ-sectional curvature c=−3. We show that the Lie algebra i(M,θ) of all infinitesimal pseudohermitian transformations on a strictly pseudoconvex CR manifold M of CR dimension n has dimension ?²(n+1) and if dimRi(M,θ)=²(n+1) then Hθ(σ)= constant.     

The Riemannian product structures of spacelike hypersurfaces with constant k-th mean curvature in the de Sitter spaces

In this paper, we investigate complete spacelike hypersurfaces in the de Sitter space with constant k-th mean curvature and two distinct principal curvatures one of which is simple. We obtain some characterizations of the Riemannian product H¹(c₁)×Sn−1(c₂) or Hn−1(c₁)×S¹(c₂) in the de Sitter space .     

A characterization of a certain real hypersurface of type (A<Subscript>2</Subscript>) in a complex projective space

In the class of real hypersurfaces M ^2n?1 isometrically immersed into a nonflat complex space form \(\widetilde {{M_n}}\left( c \right)\) of constant holomorphic sectional curvature c (≠ 0) which is either a complex projective space ?P ⁿ (c) or a complex hyperbolic space ?H ⁿ (c) according as c > 0 or c < 0, there are two typical examples. One is the class of all real hypersurfaces of type (A) and the other is the class of all ruled real hypersurfaces. Note that the former example are Hopf manifolds and the latter are non-Hopf manifolds. In this paper, inspired by a simple characterization of all ruled real hypersurfaces in \(\widetilde {{M_n}}\left( c \right)\), we consider a certain real hypersurface of type (A₂) in ?P ⁿ (c) and give a geometric characterization of this Hopf manifold.     

Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ?=?Aψ?+?b

We study hypersurfaces in the Lorentz-Minkowski space \mathbbLⁿ⁺¹{\mathbb{L}^{n+1}} whose position vector ψ satisfies the condition L _k ψ = Aψ + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k = 0, . . . , n − 1, A ? \mathbbR^(n+1)×(n+1){A\in\mathbb{R}^{(n+1)\times(n+1)}} is a constant matrix and b ? \mathbbLⁿ⁺¹{b\in\mathbb{L}^{n+1}} is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature, open pieces of totally umbilical hypersurfaces \mathbbSⁿ₁(r){\mathbb{S}^n_1(r)} or \mathbbHⁿ(-r){\mathbb{H}^n(-r)}, and open pieces of generalized cylinders \mathbbS^m₁(r)×\mathbbR^n-m{\mathbb{S}^m_1(r)\times\mathbb{R}^{n-m}}, \mathbbH^m(-r)×\mathbbR^n-m{\mathbb{H}^m(-r)\times\mathbb{R}^{n-m}}, with k + 1 ≤ m ≤ n − 1, or \mathbbL^m×\mathbbS^n-m(r){\mathbb{L}^m\times\mathbb{S}^{n-m}(r)}, with k + 1 ≤ n − m ≤ n − 1. This completely extends to the Lorentz-Minkowski space a previous classification for hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} given by Alías and Gürbüz (Geom. Dedicata 121:113–127, 2006).     

A note on K<Emphasis Type="Italic">ä</Emphasis>hler manifolds with almost nonnegative bisectional curvature

In this note, we prove the following result. There is a positive constant ε(n, Λ) such that if M ⁿ is a simply connected compact Kähler manifold with sectional curvature bounded from above by Λ, diameter bounded from above by 1, and with holomorphic bisectional curvature H ≥ ?ε(n, Λ), then M ⁿ is diffeomorphic to the product M ₁ × ? × M _k , where each M _i is either a complex projective space or an irreducible Kähler–Hermitian symmetric space of rank ≥ 2. This resolves a conjecture of Fang under the additional upper bound restrictions on sectional curvature and diameter.     

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.     

Geometric, topological and differentiable rigidity of submanifolds in space forms

Let M be an n-dimensional submanifold in the simply connected space form F ^n+p (c) with c + H ² > 0, where H is the mean curvature of M. We verify that if M ⁿ (n ≥ 3) is an oriented compact submanifold with parallel mean curvature and its Ricci curvature satisfies Ric _M ≥ (n ? 2)(c + H ²), then M is either a totally umbilic sphere, a Clifford hypersurface in an (n + 1)-sphere with n = even, or ${\mathbb{C}P^{2} \left(\frac{4}{3}(c + H^{2})\right) {\rm in} S^{7} \left(\frac{1}{\sqrt{c + H^{2}}}\right)}$ C P 2 4 3 ( c + H 2 ) in S 7 1 c + H 2 . In particular, if Ric _M > (n ? 2)(c + H ²), then M is a totally umbilic sphere. We then prove that if M ⁿ (n ≥ 4) is a compact submanifold in F ^n+p (c) with c ≥ 0, and if Ric _M > (n ? 2)(c + H ²), then M is homeomorphic to a sphere. It should be emphasized that our pinching conditions above are sharp. Finally, we obtain a differentiable sphere theorem for submanifolds with positive Ricci curvature.     

Hypersurfaces with constant mean curvature in a hyperbolic space

Let Mⁿ be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space Hⁿ⁺¹（c） with non-zero constant mean curvature H and two distinct principal curvatures. In this paper, we show that （1） if the multiplicities of the two distinct principal curvatures are greater than 1,then Mⁿ is isometric to the Riemannian product S^k（r）×H^n-k（-1/（r² + ρ²））, where r > 0 and 1 < k < n - 1;（2）if H² > -c and one of the two distinct principal curvatures is simple, then Mⁿ is isometric to the Riemannian product S^n-1（r） × H¹（-1/（r² +ρ²）） or S¹（r） × H^n-1（-1/（r² +ρ²））,r > 0, if one of the following conditions is satisfied （i） S≤（n-1）t²2+c²t^-22 on Mⁿ or （ii）S≥ （n-1）t²1+c²t^-21 on Mⁿ or（iii）（n-1）t²2+c²t^-22≤ S≤（n-1）t²1+c²t^-21 on Mⁿ, where t₁ and t₂ are the positive real roots of （1.5）.     

Spacelike hypersurfaces with constant scalar curvature

In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y. Cheng and S.T. Yau [7] to compact spacelike hypersurfaces which are immersed in de Sitter space S ^{n +1} ₁(c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant scalar curvature n(n-1)r is isometric to a sphere if r << c. Received: 18 December 1996 / Revised version: 26 November 1997     

Geometric and topological rigidity for compact submanifolds of odd dimension

We investigate rigidity problems for odd-dimensional compact submanifolds.We show that if Mn(n 5)is an odd-dimensional compact submanifold with parallel mean curvature in Sn+p,and if RicM(n-2-1n)(1+H2)and Hδn,whereδn is an explicit positive constant depending only on n,then M is a totally umbilical sphere.Here H is the mean curvature of M.Moreover,we prove that if Mn(n 5)is an odd-dimensional compact submanifold in the space form Fn+p(c)with c 0,and if RicM(n-2-εn)(c+H2),whereεn is an explicit positive constant depending only on n,then M is homeomorphic to a sphere.     

Uniqueness of horospheres and geodesic cylinders in hyperbolic space

A hyperbolic analogon to Hartman’s characterization of orthogonal sphere cylinders is proved: Let Mn ? Hⁿ⁺¹ be a noncompact closed hypersurface with sectional curvature K ≥ 0 which bounds a convex set. Assume further H_r ≡ c for one normalized mean curvature. Then Mⁿ is a horosphere or a geodesic cylinder if $r{\leq}\ {2\over 3}\ (n+1)$ . For $r >\ {2\over 3}\ (n+1)$ the same follows but only if c lies in a specified interval which however covers the case of a horosphere. The argumentation is based on results of S.B. Alexander and R.B. Currier on the infinity set of certain convex hypersurfaces, the comparison with interior spindle surfaces, first eigenvalue estimates for Voss operators and variational properties of relevant curvature expressions.     

Topological change in mean convex mean curvature flow

Consider the mean curvature flow of an (n+1)-dimensional compact, mean convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We prove that elements of the mth homotopy group of the complementary region can die only if there is a shrinking S ^k ×R ^n?k singularity for some k≤m. We also prove that for each m with 1≤m≤n, there is a nonempty open set of compact, mean convex regions K in R ⁿ⁺¹ with smooth boundary ?K for which the resulting mean curvature flow has a shrinking S ^m ×R ^n?m singularity.     

Rigidity theorems for hypersurfaces with constant mean curvature

Let M ⁿ be a compact oriented hypersurface of a unit sphere \(\mathbb{S}^{n + 1} \) (1) with constant mean curvature H. Given an integer k between 2 and n ? 1, we introduce a tensor ? related to H and to the second fundamental form A of M, and show that if |?|² ≤ B _H,k and tr(? ³) ≤ C _n,k |?|³, where B _H,k and C _n,k are numbers depending only on H, n and k, then either |?|² ≡ 0 or |?|² ≡ B _H,k . We characterize all M ⁿ with |?|² ≡ B _H,k . We also prove that if \(\left| A \right|^2 \leqslant 2\sqrt {k(n - k)}\) and tr(? ³) ≤ C _n,k |?|³ then |A|² is constant and characterize all M ⁿ with |A|² in the interval \(\left[ {0,2\sqrt {k\left( {n - k} \right)} } \right] \) . We also study the behavior of |?|², with the condition additional tr(? ³) ≤ C _n,k |?|³, for complete hypersurfaces with constant mean curvature immersed in space forms and show that if sup _M |?|² = B _H,k and this supremum is attained in M ⁿ then M ⁿ is an isoparametric hypersurface with two distinct principal curvatures of multiplicities k y n ? k. Finally, we use rotation hypersurfaces to show that the condition on the trace of ? ³ is necessary in our results; more precisely, for each integer k with 2 ≤ k ≤ n ? 1 and \(H \geqslant 1/\sqrt {2n - 1} \) there is a complete hypersurface M ⁿ in \(\mathbb{S}^{n + 1} \) (1) with constant mean curvature H such that sup _M |?|² = B _H,k , and this supremum is attained in M ⁿ , and which is not a product of spheres.     

New <Emphasis Type="Italic">r</Emphasis>-Minimal Hypersurfaces via Perturbative Methods

A hypersurface in a Riemannian manifold is r-minimal if its (r+1)-curvature, the (r+1)th elementary symmetric function of its principal curvatures, vanishes identically. If n>2(r+1) we show that the rotationally invariant r-minimal hypersurfaces in ? ⁿ⁺¹ are nondegenerate in the sense that they carry no nontrivial Jacobi fields decaying rapidly enough at infinity. Combining this with a computation of the (r+1)-curvature of normal graphs and the deformation theory in weighted Hölder spaces developed by Mazzeo, Pacard, Pollack, Uhlenbeck and others, we produce new infinite dimensional families of r-minimal hypersurfaces in ? ⁿ⁺¹ obtained by perturbing noncompact portions of the catenoids. We also consider the moduli space \({\mathcal{M}}_{r,k}(g)\) of elliptic r-minimal hypersurfaces with k≥2 ends of planar type in ? ⁿ⁺¹ endowed with an ALE metric g, and show that \({\mathcal{M}}_{r,k}(g)\) is an analytic manifold of formal dimension k(n+1), with \({\mathcal{M}}_{r,k}(g)\) being smooth for a generic g in a neighborhood of the standard Euclidean metric.     

Characterizations of Some Product Manifolds by Their Spectrum

Let Sⁿ(c) denote the n-dimensional Euclidean sphere of constant sectional curvature c and denote by CPⁿ(c) the complex projective space of complex dimension n and of holomorphic sectional curvature c. In this paper, we obtain some characterizations of the manifolds S²(c) × S²(c′), S⁴(c) × S⁴(c′), CP²(c) × CP²(c′) by their spectrum.