Search

In this paper, we investigate nonexistence of stable integral currents in a compact hypersurface with positive Ricci curvature in a Euclidean spaceR m+1 and a vanishing theorem concerning the homology group is obtained.

For an $n$-dimensional compact submanifold $M^n$ in the Euclidean space $\mathbf R^{N}$, we study estimates for eigenvalues of the Paneitz operator on $M^n$. Our estimates for eigenvalues are sharp., 16 pages

It is our purpose to study complete self-shrinkers in Euclidean space. By introducing a generalized maximum principle for $\mathcal{L}$-operator, we give estimates on supremum and infimum of the squared norm of the second fundamental form of self-shrinkers without assumption on \emph{polynomial volume growth}, which is assumed in Cao and Li. Thus, we can obtain the rigidity theorems on complete self-shrinkers without assumption on \emph{polynomial volume growth}. For complete proper self-shrinkers of dimension 2 and 3, we give a classification of them under assumption of constant squared norm of the second fundamental form., Comment: 10 pages, a revised version

In this paper, we prove that the hyperbolic cylinderH 1(c 1)×H 2(c 2) is the only complete maximal spacelike hypersurfaces inH 1 4 (c) with nonzero constant Gauss-Kronecker curvature and give a characterization of complete maximal spacelike hypersurfaces ofH 1 4 (c) with constant scalar curvature.

LetM n (n>3) be a closed minimal hypersurface with constant scalar curvature in the unit sphereS n+1 (1) andS the square of the length of its second fundamental form. In this paper we prove thatS>n implies estimates of the formS>n+cn−d withc≥1/4. For example, forn>17 andS>n we proveS>n+1/4n which is sharper than a recent result of the authors [5]

In this paper, we completely classify complete hypersurfaces inR 4 with constant mean curvature and constant scalar curvature.

For a bounded domain $\Omega$ with a piecewise smooth boundary in a complete Riemannian manifold $M$, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal basis of $L^2(\Omega)$ in place of the Rayleigh-Ritz formula, we obtain inequalities for eigenvalues of the Laplacian. In particular, for lower order eigenvalues, our results extend the results of Chen and Cheng \cite{CC}., Comment: 17 pages

Let Mn be an n-dimensional Riemannian manifold minimally immersed in the unit sphere Sn+p (1) of dimension n + p. When Mn is compact, Chern, do Carmo and Kobayashi [1] proved that if the square ‖h‖2 of length of the second fundamental form h in Mn is not more than , then either Mn is totallygeodesic, or Mn is the Veronese surface in S4 (1) or Mn is the Clifford torus .In this paper, we generalize the results due to Chern, do Carmo and Kobayashi [1] to complete Riemannian manifolds.

In this note, we show that the totally geodesic sphere, Clifford torus and Cartan hypersurface are the only compact minimal hypersurfaces in S4(1) with constant scalar curvature if the Ricci curvature is not less than −1.

In this paper we study eigenvalues of a clamped plate problem on a bounded domain in an n-dimensional complete Riemannian manifold. By making use of Nash's theorem and introducing k free constants, we derive a universal bound for eigenvalues, which solves a problem proposed by Wang and Xia [16].

In this paper, we study eigenvalues of the poly-Laplacian with arbitrary order on a bounded domain in an $n$-dimensional Euclidean space and obtain a lower bound for eigenvalues, which gives an important improvement of results due to Levine and Protter. In particular, the result of Melas is included here.

This paper studies eigenvalues of the buckling problem of arbitrary order on bounded domains in Euclidean spaces and spheres. We prove universal bounds for the k-th eigenvalue in terms of the lower ones independent of the domains. Our results strengthen the recent work in [28] and generalize Cheng-Yang's recent estimates [16] on the buckling eigenvalues of order two to arbitrary order., Comment: 20 pages

In this paper, we study eigenvalues of a clamped plate problem. We obtain a lower bound for eigenvalues, which gives an important improvement of results due to Levine and Protter.

For a compact spin manifold M isometrically embedded into Euclidean space, we derive the extrinsic estimates from above and below for eigenvalues of the square of the Dirac operator, which depend on the second fundamental form of the embedding. We also show the bounds of the ratio of the eigenvalues. Since the unit sphere and the projective spaces admit the standard embedding into Euclidean spaces, we also obtain the corresponding results for their compact spin submanifolds.

In this paper, we classify complete space-like hypersurfaces in a de Sitter space with non-negative sectional curvature and r = kH, where k is a nonnegative constant, and r and H are the scalar curvature and the mean curvature respectively.

In this paper, we study Möbius characterizations of submanifolds without umbilical points in a unit sphere $S^{n+p}(1)$ . First of all, we proved that, for an $n$ -dimensional $(n\geq 2)$ submanifold $\mathbf x:M\mapsto S^{n+p}(1)$ without umbilical points and with vanishing Möbius form $\Phi$ , if $(n-2)||\tilde{\mathbf{A}}|| \leq\sqrt{\frac{n-1}n} \left\{ nR-\frac{1}{n}[(n-1)\left( 2-\frac{1}{p} \right)-1] \right\}$ is satisfied, then, $\mathbf x$ is Möbius equivalent to an open part of either the Riemannian product $S^{n-1}(r)\times S^{1}\left(\sqrt{1-r^2}\right)$ in $S^{n+1}(1)$ , or the image of the conformal diffeomorphism $\sigma$ of the standard cylinder $S^{n-1}(1)\times \mathbf{R}$ in $\mathbf{R}^{n+1}$ , or the image of the conformal diffeomorphism $\tau$ of the Riemannian product $S^{n-1}(r)\times \mathbf{H}^{1}\left(\sqrt{1+r^2}\right)$ in $\mathbf{H}^{n+1}$ , or $\mathbf x$ is locally Möbius equivalent to the Veronese surface in $S^4(1)$ . When $p=1$ , our pinching condition is the same as in Main Theorem of Hu and Li [6], in which they assumed that $M$ is compact and the Möbius scalar curvature $n(n-1)R$ is constant. Secondly, we consider the Möbius sectional curvature of the immersion $\mathbf x$ . We obtained that, for an $n$ -dimensional compact submanifold $\mathbf x:M\mapsto S^{n+p}(1)$ without umbilical points and with vanishing form $\Phi$ , if the Möbius scalar curvature $n(n-1)R$ of the immersion $\mathbf x$ is constant and the Möbius sectional curvature $K$ of the immersion $\mathbf x$ satisfies $K\geq 0$ when $p=1$ and $K>0$ when $p>1$ . Then, $\mathbf x$ is Möbius equivalent to either the Riemannian product $S^k(r)\times S^{n-k}\left(\sqrt{1-r^2}\right)$ , for $k=1, 2, \cdots, n-1$ , in $S^{n+1}(1)$ ; or $\mathbf x$ is Möbius equivalent to a compact minimal submanifold with constant scalar curvature in $S^{n+p}(1)$ .

It is well known that the spectrum of Laplacian on a compact Riemannian manifold $M$ is an important analytic invariant and has important geometric meanings. There are many mathematicians to investigate properties of the spectrum of Laplacian and to estimate the spectrum in term of the other geometric quantities of $M$ . When $M$ is a bounded domain in Euclidean spaces, a compact homogeneous Riemannian manifold, a bounded domain in the standard unit sphere or a compact minimal submanifold in the standard unit sphere, the estimates of the $k+1$ -th eigenvalue were given by the first $k$ eigenvalues (see [9], [12], [19], [20], [22], [23], [24] and [25]). In this paper, we shall consider the eigenvalue problem of the Laplacian on compact Riemannian manifolds. First of all, we shall give a general inequality of eigenvalues. As its applications, we study the eigenvalue problem of the Laplacian on a bounded domain in the standard complex projective space $\mathbf{CP}^n(4)$ and on a compact complex hypersurface without boundary in $\mathbf{CP}^n(4)$ . We shall give an explicit estimate of the $k+1$ -th eigenvalue of Laplacian on such objects by its first $k$ eigenvalues.

In this paper, we study $n$ -dimensional complete immersed submanifolds in a Euclidean space $E^{n+p}$ . We prove that if $M^{n}$ is an $n$ -dimensional compact connected immersed submanifold with nonzero mean curvature $H$ in $E^{n+p}$ and satisfies either: ¶ (1) $S\displaystyle \leq\frac{n^{2}H^{2}}{n-1},$ or ¶ (2) $n^{2}H^{2}\displaystyle \leq\frac{(n-1)R}{n-2}$ , ¶ then $M^{n}$ is diffeomorphic to a standard $n$ -sphere, where $S$ and $R$ denote the squared norm of the second fundamental form of $M^{n}$ and the scalar curvature of $M^{n}$ , respectively. ¶ On the other hand, in the case of constant mean curvature, we generalized results of Klotz and Osserman [11] to arbitrary dimensions and codimensions; that is, we proved that the totally umbilical sphere $S^{n}(c)$ , the totally geodesic Euclidean space $E^{n}$ , and the generalized cylinder $S^{n-1}(c)\times E^{1}$ are only $n$ -dimensional $(n>2)$ complete connected submanifolds $M^{n}$ with constant mean curvature $H$ in $E^{n+p}$ if $S\leq n^{2}H^{2}/(n-1)$ holds.

In this note, we corrected the coefficients of formula (2.12) in Proposition 1 and the third formula in Proposition 2 in our paper [1]. We should remark that our theorems in [1] are not harmed.

We classify complete conformally flat three dimensional Riemannian manifolds with constant scalar curvature and constant squared norm of Ricci curvature tensor by applying the Generalized Maximum Principle due to H. Omori.

In this paper, we study eigenvalues of the closed eigenvalue problem of the differential operator $ L$, which is introduced by Colding and Minicozzi in [4], on an $n$-dimensional compact self-shrinker in ${R}^{n+p}$. Estimates for eigenvalues of the differential operator $ L$ are obtained. Our estimates for eigenvalues of the differential operator $ L$ are sharp. Furthermore, we also study the Dirichlet eigenvalue problem of the differential operator $ L$ on a bounded domain with a piecewise smooth boundary in an $n$-dimensional complete self-shrinker in $ {R}^{n+p}$. For Euclidean space $ {R}^{n}$, the differential operator $ L$ becomes the Ornstein-Uhlenbeck operator in stochastic analysis. Hence, we also give estimates for eigenvalues of the Ornstein-Uhlenbeck operator., 21 pages, a final version, accepted for publication in Communications in Contemporary Mathematics

In this paper, we investigate universal inequalities for eigenvalues of a clamped plate problem on a bounded domain in an -dimensional hyperbolic space. It is well known that, for a bounded domain in the -dimensional Euclidean space, Payne, Polya and Weinberger (1955), Hook (1990) and Chen and Qian (1990) studied universal inequalities for eigenvalues of the clamped plate problem. Recently, Cheng and Yang (2006) have derived the Yang-type universal inequality for eigenvalues of the clamped plate problem on a bounded domain in the -dimensional Euclidean space, which is sharper than the other ones. For a domain in a unit sphere, Wang and Xia (2007) have also given a universal inequality for eigenvalues. For a bounded domain in the -dimensional hyperbolic space, although many mathematicians want to obtain a universal inequality for eigenvalues of the clamped plate problem, there are no results on universal inequalities for eigenvalues. The main reason that one could not derive a universal inequality is that one cannot find appropriate trial functions. In this paper, by constructing ``nice'' trial functions, we obtain a universal inequality for eigenvalues of the clamped plate problem on a bounded domain in the hyperbolic space. Furthermore, we can prove that if the first eigenvalue of the clamped plate problem tends to when the domain tends to the hyperbolic space, then all of the eigenvalues tend to .

Let Mp+P(c) be an (n + />)-dimensional connected indefinite Riemannian manifold of index p and of constant curvature c, which is called an indefinite space form of index p. According to c>0, c―0 or c

Let Mf(c) be an m-dimensional connected semi-Riemannian manifold ofindex sand of constant curvature c, which is calledan indefinitespace form ofindex s or simply a space form according as s>0 or s=0. An m-dimensional space form of constant curvature cis only denoted by Mm(c). The study of hypersurfaces with constant mean curvature of Mn+＼c) was initiatedby Nomizu and Smyth [13], who proved some excellent results. Itis seen that a complete space-like hypersurface of a Minkowski space

Let Mn be an n-dimensional Riemannian manifold which is minimally immersed in a unit sphere Sn+P(l) of dimension n+p. Then the second fundamental form h of the immersion is given by h{X, Y)=1XY―1 xY and it satisfies h{X, Y)―h(Y, X), where 7 and 7 denote the covariant differentiationon Sn+P(l) and Mn respectively, X and Y are vector fields on Mn. We choose a local field of orthonormal frames eu ・・・,en, ■■■, en+p in Sn+P(l) such that, restricted to Mn, the vector eu ・■・,en are tangent to Mn. We use the following convention on the range of indices unless otherwise stated: A, B, C, ・・・=!, 2, ・・・, n+p; i, j, k, ■・・=!,2, ・・・, n; a, j8,■■・ = n+l, ・■・,n+p. And we agree that repeated indices under a summation sign without indication are summed over the respective range. With respect to the frame fieldof Sn+P(l) chosen above, let