Hypersurfaces of the hyperbolic space with constant scalar curvature

We classify hypersurfaces of the hyperbolic space ?ⁿ⁺¹(c) with constant scalar curvature and with two distinct principal curvatures. Moreover, we prove that if Mⁿ is a complete hypersurfaces with constant scalar curvature n(n ? 1) R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n? 1, then R ≥ c. Additionally, we prove two rigidity theorems for such hypersurfaces.     

Gaussian mean curvature flow

In the Euclidean Space \mathbb Rⁿ⁺¹{\mathbb {R}^{n+1}} with a density e^{e\frac12 n m2 |x|2}, (e = ±1){e^{\varepsilon \frac12 n \mu^2 |x|^2},} {(\varepsilon =\pm1}), we consider the flow of a hypersurface driven by its mean curvature associated to this density. We give a detailed account of the evolution of a convex hypersurface under this flow. In particular, when e = -1{ \varepsilon=-1} (Gaussian density), the hypersurface can expand to infinity or contract to a convex hypersurface (not necessarily a sphere) depending on the relation between the bound of its principal curvatures and μ.     

Prescribed Mean Curvature Hypersurfaces in Hn+1(-1) with Convex Planar Boundary, I

We study immersed prescribed mean curvature compact hypersurfaces with boundary in Hⁿ⁺¹(-1). When the boundary is a convex planar smooth manifold with all principal curvatures greater than 1, we solve a nonparametric Dirichlet problem and use this, together with a general flux formula, to prove a parametric uniqueness result, in the class of all immersed compact hypersurfaces with the same boundary. We specialize this result to a constant mean curvature, obtaining a characterization of totally umbilic hypersurface caps.     

Über direkt integrierbare Kurven und strenge Böschungskurven im ℝn sowie geschlossene verallgemeinerte Hyperkreise

A curve in Euclidean space ?ⁿ is called “directly integrable”, if it can be explicitly calculated from the curvatures in a specified way. A necessary and sufficient condition for a curve to be directly integrable is that all its curvatures are real multiples of a single real function. Directly integrable curves in an odd-dimensional space ?ⁿ (n=2q+1) can be interpreted as generalized helices. In the case of even-dimensional space ?ⁿ (n=2p), we give a simple necessary and sufficient condition for a directly integrable curve to be closed.     

与高阶抛物型Schrodinger算子相关的Riesz变换的L^p估计——献给陆善镇教授75华诞

令δ/δt＋（-△）^2＋V^2为R^n＋1（n≥5）上的高阶抛物型Schrodinger算子,其中非负位势V与时间t无关且属于逆Holder类Bq1（Rn）（q1〉n/2）.本文得到与高阶抛物型Schrodinger算子相关的Riesz变换▽^2（δ/δt＋（-△）^2＋V^2）^-1/2的L^p（R^n＋1）估计.     

On Biconservative Lorentz Hypersurface with Non-diagonalizable Shape Operator

In this paper, we obtain some properties of biconservative Lorentz hypersurface \(M_{1}^{n}\) in \(E_{1}^{n+1}\) having shape operator with complex eigenvalues. We prove that every biconservative Lorentz hypersurface \(M_{1}^{n}\) in \(E_{1}^{n+1}\) whose shape operator has complex eigenvalues with at most five distinct principal curvatures has constant mean curvature. In addition, we investigate such a type of hypersurface with constant length of second fundamental form having six distinct principal curvatures.     

A finiteness theorem for isoparametric hypersurfaces

In this note we prove that for eachn there are only finitely many diffeomorphism classes of compact isoparametric hypersurfaces ofS ⁿ⁺¹ with four distinct principal curvatures.     

Anisotropic Local Hardy Spaces

In this paper we introduce and study the anisotropic local Hardy spaces h_A^p(\mathbbRⁿ)h_{A}^{p}(\mathbb{R}^{n}) 0<p≤1, associated with the expansive matrix A. We obtain an atomic characterization of the distributions in h_A^p(\mathbbRⁿ)h_{A}^{p}(\mathbb{R}^{n}). Also we describe the dual spaces of our local Hardy anisotropic spaces as anisotropic Campanato type spaces.     

Nonlinear partial differential systems on Riemannian manifolds with their geometric applications

We make the first study of how the existence of (essential) positive supersolutions of nonlinear degenerate partial differential equations on a manifold affects the topology, geometry, and analysis of the manifold. For example, for surfaces in R³ we prove a Bernstein-type theorem that generalizes and unifies three distinct theorems. In higher dimensions, we provide topological obstructions for a minimal hypersurface in Rⁿ⁺¹ to admit an essential positive supersolution. This immediately yields information about the Gauss map of complete minimal hypersurfaces in Rⁿ⁺¹. By coping with a wider class of nonlinear partial differential equations that are involved with (p)-harmonic maps and (p)-superstrongly unstable manifolds, we derive information on the regularity of minimizers, homotopy groups, and solutions to Dirichlet problems, from the existence of essential positive supersolutions.     

Anisotropic curvature-driven flow of convex sets

We study in this paper the mean curvature evolution, and in particular the anisotropic mean curvature evolution, of convex sets in R^N${\mathbb{R}}^N$ (without driving forces). If the anisotropy is smooth, we show that the evolution remains convex. If the anisotropy is crystalline, we build a convex evolution which satisfies an equation which is a weak form of the crystalline curvature motion equation.     

Weakly Convex Biharmonic Hypersurfaces in Nonpositive Curvature Space Forms are Minimal

A submanifold M ^m of a Euclidean space R ^m+p is said to have harmonic mean curvature vector field if ${\Delta \vec{H}=0}$ , where ${\vec{H}}$ is the mean curvature vector field of ${M\hookrightarrow R^{m+p}}$ and Δ is the rough Laplacian on M. There is a famous conjecture named after Bangyen Chen which states that submanifolds of Euclidean spaces with harmonic mean curvature vector fields are minimal. In this paper we prove that weakly convex hypersurfaces (i.e. hypersurfaces whose principle curvatures are nonnegative) with harmonic mean curvature vector fields in Euclidean spaces are minimal. Furthermore we prove that weakly convex biharmonic hypersurfaces in nonpositively curved space forms are minimal.     

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) × , where . We extend recent results of Hasanis et al. [5] and Otsuki [10].     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-harmonic forms and stable hypersurfaces in space forms

We prove that a complete noncompact orientable stable minimal hypersurface in \mathbbSⁿ⁺¹{\mathbb{S}^{n+1}} (n ≤ 4) admits no nontrivial L ²-harmonic forms. We also obtain that a complete noncompact strongly stable hypersurface with constant mean curvature in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} or \mathbbSⁿ⁺¹{\mathbb{S}^{n+1}} (n ≤ 4) admits no nontrivial L ²-harmonic forms. These results are generalized versions of Tanno’s result on stable minimal hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}}.     

Singular sets of convex bodies and surfaces with generalized curvatures

In this paper we consider generalized surfaces with curvature measures and we study the properties of those k-dimensional subsets Σ ^k of such surfaces where the curvatures have positive density with respect to k-dimensional Hausdorff measure. Special attention is given to boundaries of convex bodies inR ³. We introduce a class of convex sets whose curvatures live only on integer dimension sets. For such convex sets we consider integral functionals depending on the curvature and the area ofK and on the curvature andH ^k of Σ ^k .     

On submanifolds whose tubular hypersurfaces have constant higher order mean curvatures

Motivated by the theory of isoparametric hypersurfaces,we study submanifolds whose tubular hypersurfaces have some constant higher order mean curvatures.Here a k-th order mean curvature Q_k~v(k ≥ 1) of a submanifold M~n-is defined as the k-th power sum of the principal curvatures,or equivalently,of the shape operator with respect to the unit normal vector v.We show that if all nearby tubular hypersurfaces of M have some constant higher order mean curvatures,then the submanifold M itself has some constant higher order mean curvatures Q_k~v independent of the choice of v.Many identities involving higher order mean curvatures and Jacobi operators on such submanifolds are also obtained.In particular,we generalize several classical results in isoparametric theory given by E.Cartan,K.Nomizu,H.F.Miinzner,Q.M.Wang,et al.As an application,we finally get a geometrical filtration for the focal submanifolds of isoparametric functions on a complete Riemannian manifold.     

Biharmonic submanifolds of {\mathbb{C}P^n}

We give some general results on proper-biharmonic submanifolds of a complex space form and, in particular, of the complex projective space. These results are mainly concerned with submanifolds with constant mean curvature or parallel mean curvature vector field. We find the relation between the bitension field of the inclusion of a submanifold [`(M)]{\bar{M}} in \mathbbCPⁿ{\mathbb{C}P^n} and the bitension field of the inclusion of the corresponding Hopf-tube in \mathbbS²ⁿ⁺¹{\mathbb{S}^{2n+1}}. Using this relation we produce new families of proper-biharmonic submanifolds of \mathbbCPⁿ{\mathbb{C}P^n}. We study the geometry of biharmonic curves of \mathbbCPⁿ{\mathbb{C}P^n} and we characterize the proper-biharmonic curves in terms of their curvatures and complex torsions.     

Stable hypersurfaces via the first eigenvalue of the anisotropic Laplacian operator

In this paper we provide a characterization for stable hypersurfaces with constant anisotropic mean curvature immersed in the Euclidean space \(\mathbb {R}^{n+1}\) through the analysis of the first eigenvalue of the anisotropic Laplacian operator.     

Rigidity theorem for harmonic maps from Riemannian manifold to Grassmannian manifold

We first study the Grassmannian manifoldG _n (R^n+p)as a submanifold in Euclidean space ⁿ (R ^n+p). Then we give a local expression for each map from Riemannian manifoldM toG ⁿ (R ^n+p) ⁿ (R ^n+p), and use the local expression to establish a formula which is satisfied by any harmonic map fromM toG _n (R ^n+p). As a consequence of this formula we get a rigidity theorem.     

Linear Weingarten spacelike submanifolds in de Sitter space

Let M ⁿ be a spacelike linear Weingarten submanifold in a de Sitter space ${S^{n+p}_{p}(1)}$ with R?=?aH?+?b, where R and H are the normalized scalar curvature and the length of the mean curvature vector respectively. In this paper, we give intrinsic and extrinsic conditions for M ⁿ to be totally umbilical, respectively.     

The rigidity of Clifford torusS^1 \left( {\sqrt {\frac{1}{n}} } \right) \times S^{n - 1} \left( {\sqrt {\frac{{n - 1}}{n}} } \right)

In this paper, we prove that ifM is ann-dimensional closed minimal hypersurface with two distinct principal curvatures of a unit sphereS ⁿ⁺¹ (1), thenS=n andM is a Clifford torus ifn≤S≤n+[2n ²(n+4)/3(n(n+4)+4)], whereS is the squared norm of the second fundamental form ofM.