Finite Type Non—Minimal Submanifolds

The notion of finite type xubmanifolds was introduced by B.Y.Chen.In this paper we consider the characteristics and the classifications of finite type non-minimal submanifolds.The characteristic theorems of 2-type Chen submanifolds、mass-symmetric hypersurfaces and Dupin hypersurfaces in Es^m are obtained.The classification theorems of 3-type hypersurfaces and null 2-type curves in Es^m are also proved.     

Complete Kaehler Submanifolds in CP~n

In this paper we mainly investigate projectively flat complete Kaehler sub-manifolds, in CPn. We give the pinching constants and the local structure.     

The topological structure of complete noncompact submanifolds in spheres

In this paper, we first show that δ-super stable complete noncompact minimal submanifolds in $S^{m+n}$ or $R^{m+n}$ with $\delta >{(\frac{m?1}{m})}^2$ admit no nontrivial $L^2$ harmonic 1-forms and have only one nonparabolic end, which generalizes Cao–Shen–Zhu's result in [2] on stable minimal hypersurface in $R^{m+1}$ and Lin's result in [13] on $\frac{m?1}{m}$-super stable minimal submanifolds in $R^{m+n}$. Second, we prove that the dimension of the space of $L^2$ harmonic p-forms on $M^m$ is zero or finite and there is only one nonparabolic end or finitely many nonparabolic ends of M under the assumptions on the Schrödinger operators involving the squared norm of the traceless second fundamental form.     

GEOMETRIC INEQUALITIES FOR CERTAIN SUBMANIFOLDS IN A PINCHED RIEMANNIAN MANIFOLD

This article gives some geometric inequalities for a submanifold with parallel second fundamental form in a pinched Riemannian manifold and the distribution for the square norm of its second fundamental form.     

球面Sn+p(c)中子流形的拓扑

研究了球面Sn+p(c)子流形Mn的Pinching定理,证明了当s＜2√n-1c时,Mn(n＞3)与n维球面同胚.     

Vanishing theorems on Riemannian manifolds, and geometric applications

A general Liouville-type result and a corresponding vanishing theorem are proved under minimal regularity assumptions. The latter is then applied to conformal deformations of stable minimal hypersurfaces, to the L² cohomology of complete manifolds, to harmonic maps under various geometric assumptions, and to the topology of submanifolds of Cartan-Hadamard spaces with controlled extrinsic geometry.     

Sharp Reilly-type inequalities for a class of elliptic operators on submanifolds

Let M be an $n(>2)$-dimensional closed orientable submanifold in an $(n+p)$-dimensional space form ${\mathbb{R}}^{n+p}(c)$. We obtain an optimal upper bound for the second eigenvalue of a class of elliptic operators on M defined by $L_{T}f=-\text{div}(T\mathrm{\nabla }f)$, where T is a general symmetric, positive definite and divergence-free $(1,1)$-tensor on M. The upper bound is given in terms of an integration involving tr T and $|H_T{|}^2$, where tr T is the trace of the tensor T and $H_T={\sum }_{i=1}^{n}A(T{e}_i,e_i)$ is a normal vector field associated with T and the second fundamental form A of M. Furthermore, we give the sufficient and necessary conditions when the upper bound is attained. Our main theorem can be viewed as an extension of the famous “Reilly inequality”. The operator $L_T$ can be regarded as a natural generalization of the well-known operator $L_r$ which is the linearized operator of the first variation of the $(r+1)$-th mean curvature for hypersurfaces in a space form. As applications of our main theorem, we generalize the results of Grosjean [17] and Li–Wang [20] in codimension one to arbitrary codimension.     

‘Universal’ inequalities for the eigenvalues of the Hodge de Rham Laplacian

We obtain ‘universal’ inequalities for the eigenvalues of the Laplacian acting on differential forms of a Euclidean compact submanifold. These inequalities generalize the Yang inequality concerning the eigenvalues of the Dirichlet Laplacian of a bounded Euclidean domain.      

Submanifolds of positive Ricci curvature in a Euclidean space

In this paper we study the role of constant vector fields on a Euclidean space R ^n+p in shaping the geometry of its compact submanifolds. For an n-dimensional compact submanifold M of the Euclidean space R ^n+p with mean curvature vector field H and a constant vector field on R ^n+p , the smooth function is used to obtain a characterization of sphere among compact submanifolds of positive Ricci curvature (cf. main Theorem).      

The Outer Oblique Boundary Problem of Potential Theory

In this article, we prove existence and uniqueness results for solutions to the outer oblique boundary problem for the Poisson equation under very weak assumptions on boundary, coefficients, and inhomogeneities. The main tools are the Kelvin transformation and the solution operator for the regular inner problem, provided in Grothaus and Raskop (Stochastics 2006; 78(4):233–257). Moreover we prove regularization results for the weak solutions of both the inner and the outer problem. We investigate the nonadmissible direction for the oblique vector field, state results with stochastic inhomogeneities, and provide a Ritz–Galerkin approximation. The results are applicable to problems from Geomathematics (see, e.g., Freeden and Maier (Electronic Trans. Numer. Anal. 2002; 14:40–62 and Bauer, An Alternative Approach to the Oblique Derivative Problem in Potential Theory, Shaker Verlag, Aachen, 2004)).     

非紧对称空间中子流形焦点和形算子间的关系

首先详细地讨论了非紧Ｌｉｅ群的度量和Ｃａｒｔａｎ分解,然后由Ｌｉｅ群和对称空间的关系得到了非紧对称空间中的子流形焦点存在的充要条件,同时还给出了焦点重数的计算方法．     

Mean Curvature Flow of Arbitrary Codimension in Complex Projective Spaces*

Recently, Pipoli and Sinestrari [Pipoli, G. and Sinestrari, C., Mean curvature flow of pinched submanifolds of CPⁿ, Comm. Anal. Geom., 25, 2017, 799–846] initiated the study of convergence problem for the mean curvature flow of small codimension in the complex projective space CP^m. The purpose of this paper is to develop the work due to Pipoli and Sinestrari, and verify a new convergence theorem for the mean curvature flow of arbitrary codimension in the complex projective space. Namely, the authors prove that if the initial submanifold in CP^m satisfies a suitable pinching condition, then the mean curvature flow converges to a round point in finite time, or converges to a totally geodesic submanifold as t → ∞. Consequently, they obtain a differentiable sphere theorem for submanifolds in the complex projective space.     

A gap theorem on submanifolds with finite total curvature in spheres

We study a complete noncompact submanifold Mⁿ$M^n$ in a sphere S^n+p${\mathbb{S}}^{n+p}$. We prove that there admit no nontrivial L²$L^2$-harmonic 1-forms on M if the total curvature is bounded from above by a constant depending only on n. The gap theorem is a generalized version of Carron?s, Yun?s, Cavalcante?s and the first author?s results on submanifolds in Euclidean spaces and Seo?s result on submanifolds in hyperbolic space without the condition of minimality.     

A REPRESENTATION FORMULA RELATED TO SCHR(O)DINGER OPERATORS

Schr(o)dinger operator is a central subject in the mathematical study of quantum mechanics.Consider the Schrodinger operator H = -△ V on R, where △ = d2/dx2 and the potential function V is real valued. In Fourier analysis, it is well-known that a square integrable function admits an expansion with exponentials as eigenfunctions of -△. A natural conjecture is that an L2 function admits a similar expansion in terms of "eigenfunctions" of H, a perturbation of the Laplacian (see [7], Ch. Ⅺ and the notes), under certain condition on V.     

On a class of self-similar processes with stationary increments in higher order Wiener chaoses

We study a class of self-similar processes with stationary increments belonging to higher order Wiener chaoses which are similar to Hermite processes. We obtain an almost sure wavelet-like expansion of these processes. This allows us to compute the pointwise and local Hölder regularity of sample paths and to analyse their behaviour at infinity. We also provide some results on the Hausdorff dimension of the range and graphs of multidimensional anisotropic self-similar processes with stationary increments defined by multiple Wiener–Itô integrals.     

On a class of Riemann surfaces

It is considered the class of Riemann surfaces with dimT1 = 0, where T1 is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the spaceΩH of harmonic forms of the surface, namely The surfaces in the class OHD and the class of planar surfaces satisfy dimT1 = 0. A.Pfluger posed the question whether there might exist other surfaces outside those two classes. Here it is shown that in the case of finite genus g, we should look for a surface S with dimT1 = 0 among the surfaces of the form Sg\K , where Sg is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary.     

CONTENTS OF VOLUME 30

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Instructions for contributions

正Applied Mathematics-A Journal of Chinese Universities,Series B(Appl.Math.J.Chinese Univ.,Ser.B)is a comprehensive applied mathematics journal jointly sponsored by Zhejiang University,China Society for Industrial and Applied Mathematics,and Springer-Verlag.It is a quarterly journal with     

Journal of Mathematical Research with Applications Guide for Authors

正Journal overview:Journal of Mathematical Research with Applications(JMRA),formerly Journal of Mathematical Research and Exposition(JMRE)created in 1981,one of the transactions of China Society for Industrial and Applied Mathematics,is a home for original research papers of the highest quality in all areas of mathematics with applications.The target audience comprises:pure and applied mathematicians,graduate students in broad fields of sciences and technology,scientists and engineers interested in mathematics.