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In this paper, we first construct compact embedded λ-hypersurfaces with the topology of torus which are called λ-torus in Euclidean spaces ℝn+1. Then, we give many compact immersed λ-hypersurfaces in Euclidean spaces ℝn+1.

The purpose of this paper is to study complete $$\lambda $$ -surfaces in Euclidean space $${\mathbb {R}}^3$$ . A complete classification for 2-dimensional complete $$\lambda $$ -surfaces in Euclidean space $$\mathbb R^3$$ with constant squared norm of the second fundamental form is given, which confirms a conjecture of Guang (Self-shrinkers and translating solitons of mean curvature flow, 2016, p 74).

In this paper, we classify $3$-dimensional complete self-shrinkers in Euclidean space $\mathbb R^{4}$ with constant squared norm of the second fundamental form $S$ and constant $f_{4}$., arXiv admin note: text overlap with arXiv:1807.06760

It is our purpose to study complete self-shrinkers in Euclidean space. By making use of the generalized maximum principle for \(\mathcal {L}\)-operator, we give a complete classification for 2-dimensional complete self-shrinkers with constant squared norm of the second fundamental form in \(\mathbb R^3\). Ding and Xin (Trans Am Math Soc 366:5067–5085, 2014) have proved this result under the assumption of polynomial volume growth, which is removed in our theorem.

Since $n$-dimensional $\lambda$-hypersurfaces in the Euclidean space $\mathbb {R}^{n+1}$ are critical points of the weighted area functional for the weighted volume-preserving variations, in this paper, we study the rigidity properties of complete $\lambda$-hypersurfaces. We give a gap theorem of complete $\lambda$-hypersurfaces with polynomial area growth. By making use of the generalized maximum principle for $\mathcal L$ of $\lambda$-hypersurfaces, we prove a rigidity theorem of complete $\lambda$-hypersurfaces., Comment: Comments are welcome

In this paper, we study complete self-shrinkers in Euclidean space and prove that an $n$-dimensional complete self-shrinker with polynomial volume growth in Euclidean space $\mathbb{R}^{n+1}$ is isometric to either $\mathbb{R}^{n}$, $S^{n}(\sqrt{n})$, or $\mathbb{R}^{n-m}\times S^m (\sqrt{m})$, $1\leq m\leq n-1$, if the squared norm $S$ of the second fundamental form is constant and satisfies $S, Comment: All comments are welcome

In this paper, we investigate the Dirichlet eigenvalue problems of the poly-Laplacian with any order and the quadratic polynomial operator of the Laplacian. We give some estimates for lower bounds of the sums of their first k eigenvalues.

In this paper, we investigate an eigenvalue problem for a biharmonic operator on a bounded domain in an n-dimensional Euclidean space, which is also called a buckling problem. We introduce a new method to construct ``nice'' trial functions and we derive a universal inequality for higher eigenvalues of the buckling problem by making use of the trial functions. Thus, we give an affirmative answer for the problem on universal bounds for eigenvalues of the buckling problem, which was proposed by Payne, Polya and Weinberger in [14] and this problem has been mentioned again by Ashbaugh in [1].

Let Ω be a bounded domain in an n-dimensional Euclidean space R n . We study eigenvalues of an eigenvalue problem of a system of elliptic equations: { Δ u + α grad ( div u ) = − σ u , in Ω , u | ∂ Ω = 0 . Estimates for eigenvalues of the above eigenvalue problem are obtained. Furthermore, we obtain an upper bound on the ( k + 1 ) th eigenvalue σ k + 1 . We also obtain sharp lower bound for the first eigenvalue of two kinds of eigenvalue problems of the biharmonic operator on compact manifolds with boundary and positive Ricci curvature.

For a bounded domain Ω in a complete Riemannian manifold M n , we study estimates for lower order eigenvalues of a clamped plate problem. We obtain universal inequalities for lower order eigenvalues. We would like to remark that our results are sharp.

In this paper, we investigate eigenvalues of the Dirichlet eigenvalue problem of Laplacian on a bounded domain Ω in an n-dimensional complete Riemannian manifold M. When M is an n-dimensional Euclidean space R n , the conjecture of Polya is well known: the kth eigenvalue λ k of the Dirichlet eigenvalue problem of Laplacian satisfies λ k ⩾ 4 π 2 ( ω n vol Ω ) 2 n k 2 n , for k = 1 , 2 , … . Li and Yau [P. Li, S.T. Yau, On the Schrodinger equation and the eigenvalue problem, Comm. Math. Phys. 88 (1983) 309–318] (cf. Lieb [E. Lieb, The number of bound states of one-body Schrodinger operators and the Weyl problem, in: Proc. Sympos. Pure Math., vol. 36, 1980, pp. 241–252]) have given a partial solution for the conjecture of Polya, that is, they have proved 1 k ∑ i = 1 k λ i ⩾ n n + 2 4 π 2 ( ω n vol Ω ) 2 n k 2 n , for k = 1 , 2 , … , which is sharp in the sense of average. In this paper, we consider a general setting for complete Riemannian manifolds. We establish an analog of the Li and Yau's inequality for eigenvalues of the Dirichlet eigenvalue problem of Laplacian on a bounded domain in a complete Riemannian manifold. Furthermore, we obtain a universal inequality for eigenvalues of the Dirichlet eigenvalue problem of Laplacian on a bounded domain in a hyperbolic space H n ( − 1 ) . From it, we prove that when the bounded domain Ω tends to H n ( − 1 ) , all eigenvalues tend to ( n − 1 ) 2 4 .

Let M be an n-dimensional closed hypersurface with constant mean curvature H satisfying |H| ≤ ϵ(n) in a unit sphere Sn+1, n ≤ 7, and S the square of the length of the second fundamental form of M. There exists a constant δ(n, H) > 0, which depends only on n and H, such that if S0 ≤ S ≤ S0 + δ(n, H), then S ≡ S0 and M is isometric to a Clifford hypersurface, where ϵ(n) is a sufficiently small constant depending on n and $S_0=n+\frac{n^3}{2(n-1)}H^2+\frac{n(n-2)}{2(n-1)}\sqrt{n^2H^4+4(n-1)H^2}$.

Let M M be an n n -dimensional compact hypersurface with constant scalar curvature n ( n − 1 ) r n(n-1)r , r > 1 r> 1 , in a unit sphere S n + 1 ( 1 ) S^{n+1}(1) . We know that such hypersurfaces can be characterized as critical points for a variational problem of the integral ∫ M H d M \int _MHdM of the mean curvature H H . In this paper, we first study the eigenvalue of the Jacobi operator J s J_s of M M . We derive an optimal upper bound for the first eigenvalue of J s J_s , and this bound is attained if and only if M M is a totally umbilical and non-totally geodesic hypersurface or M M is a Riemannian product S m ( c ) × S n − m ( 1 − c 2 ) S^m(c)\times S^{n-m}(\sqrt {1-c^2}) , 1 ≤ m ≤ n − 1 1\leq m\leq n-1 .

In this paper, we study the first eigenvalue of Jacobi operator on an $n$-dimensional non-totally umbilical compact hypersurface with constant mean curvature $H$ in the unit sphere $S^{n+1}(1)$. We give an optimal upper bound for the first eigenvalue of Jacobi operator, which only depends on the mean curvature $H$ and the dimension $n$., Comment: the title is changed. to appear in Calculus of Variations and PDE's

In this paper, we investigate an eigenvalue problem for a biharmonic operator on a bounded domain in an n-dimensional Euclidean space, which is also called a buckling problem. We introduce a new method to construct ``nice'' trial functions and we derive a universal inequality for higher eigenvalues of the buckling problem by making use of the trial functions. Thus, we give an affirmative answer for the problem on universal bounds for eigenvalues of the buckling problem, which was proposed by Payne, Polya and Weinberger in [14] and this problem has been mentioned again by Ashbaugh in [1].

Let D be a connected bounded domain in an n-dimensional Euclidean space R n . Assume that 0 < λ 1 < λ 2 ≤ ··· ≤λ k ≤··· are eigenvalues of a clamped plate problem or an eigenvalue problem for the Dirichlet biharmonic operator: Then, we give an upper bound of the (k+1)-th eigenvalue λ k+1 in terms of the first k eigenvalues, which is independent of the domain D, that is, we prove the following: Further, a more explicit inequality of eigenvalues is also obtained.

It is our purpose to study curvature structures of compact hypersurfaces in the unit sphere S n+1 (1). We proved that the Riemannian product S 1 (√1 - c 2 ) × S n-1 (c) is the only compact hypersurfaces in S n+1 (1) with infinite fundamental group, which satisfy r > n-2/n-1 and S < (n-1)n(r-1)+2/n-2 + n-2/n(r-1)+2, where n(n-1)r is the scalar curvature of hypersurfaces and c 2 = n-2/nr. In particular, we obtained that the Riemannian product S 1 (√1 - c2) × S n-1 (c) is the only compact hypersurfaces with infinite fundamental group in S n+1 (1) if the sectional curvatures are nonnegative.

In this paper, we prove that if M^2 is a complete maximal spacelike surface of an anti-de Sitter space {\bf H}^{4}_{2}(c) with constant scalar curvature, then S=0, S={-10c\over 11}, S={-4c\over 3} or S=-2c, where S is the squared norm of the second fundamental form of M^{2}. Also(1) S=0 if and only if M^2 is the totally geodesic surface {\bf H}^2(c);(2) S={-4c\over 3} if and only if M^2 is the hyperbolic Veronese surface;(3) S=-2c if and only if M^2 is the hyperbolic cylinder of the totally geodesicsurface {\bf H}^{3}_{1}(c) of {\bf H}^{4}_{2}(c).1991 Mathematics Subject Classifaction 53C40, 53C42.

In this paper, we prove that an n n -dimensional closed minimal hypersurface M M with Ricci curvature R i c ( M ) ≥ n 2 Ric(M) \geq \dfrac {n}{2} of a unit sphere S n + 1 ( 1 ) S^{n+1}(1) is isometric to a Clifford torus if n ≤ S ≤ n + 14 ( n + 4 ) 9 n + 30 n\leq S\leq n+\frac {14(n+4)}{9n+30} , where S S is the squared norm of the second fundamental form of M M .

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.

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.