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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.

In this paper, we introduce a special class of hypersurfaces which are called $$\lambda $$ -hypersurfaces related to a weighted volume preserving mean curvature flow in the Euclidean space. We prove that $$\lambda $$ -hypersurfaces are critical points of the weighted area functional for the weighted volume-preserving variations. Furthermore, we classify complete $$\lambda $$ -hypersurfaces with polynomial area growth and $$H-\lambda \ge 0$$ . The classification result generalizes the results of Huisken (J Differ Geom 31:285–299, 1990) and Colding and Minicozzi (Ann Math 175:755–833, 2012).

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

The totally umbilical and non-totally geodesic hypersurfaces in the (n + 1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. In our 2010 paper we proved that the weak stability index of a compact hypersurface M with constant scalar curvature n(n − 1)r, r> 1, in an (n + 1)-dimensional sphere Sn + 1(1), which is not a totally umbilical hypersurface, is greater than or equal to n + 2 if the mean curvature H and H3 are constant. In this paper, we prove the same results, without the assumption that H3 is constant. In fact, we show that the weak stability index of a compact hypersurface M with constant scalar curvature n(n − 1)r, r> 1, in Sn + 1(1), which is not a totally umbilical hypersurface, is greater than or equal to n + 2 if the mean curvature H is constant.

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.

In this paper, we study n-dimensional hypersurfaces with constant mth mean curvature in a unit sphere Sn+1(1) and construct many compact nontrivial embedded hypersurfaces with constant mth mean curvature Hm > 0 in Sn+1(1), for 1 ≤ m ≤ n-1. Moreover, if the 2nd mean curvature H2 takes value between [Formula: see text] and [Formula: see text] for any integer k ≥ 2 and n ≥ 3, then there exists an n-dimensional compact nontrivial embedded hypersurface with constant H2 (i.e. constant scalar curvature) in Sn+1(1); If the 4th mean curvature H4 takes value between [Formula: see text] and [Formula: see text] for any integer k ≥ 3 and n ≥ 5, then there exists an n-dimensional compact nontrivial embedded hypersurface with constant H4 in Sn+1(1).

We study the weak stability index of an immersion ϕ: M → Sn+1 (1) ⊂ Rn+2 of an n-dimensional compact Riemannian manifold. We prove that the weak stability index of a compact hypersurface M with constant scalar curvature in Sn+1 (1), which is not totally umbilical, is greater than or equal to n + 2 if the mean curvature H1 and H3 are constant, and that the equality holds if and only if M is $\smash{S^m(c)\times S^{n-m}(\sqrt{1-c^2})}$. As an application, we show that the weak stability index of an n-dimensional compact hypersurface with constant scalar curvature in Sn+1 (1), which is neither totally umbilical nor a Clifford hypersurface, is greater than or equal to 2n + 4 if the mean curvature H1 and H3 are constant.

In this paper, we classify all immersed hypersurfaces in the unit sphere Sm+1 with parallel para-Blaschke tensor.

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 .

In this paper, we study eigenvalues of Laplacian with any order on a bounded domain in an n-dimensional Euclidean space and obtain estimates for eigenvalues, which are the Yang-type inequalities. In particular, the sharper result of Yang is included here. Furthermore, for lower order eigenvalues, we obtain two sharper inequalities. As a consequence, a proof of results announced by Ashbaugh [1] is also given.

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 $S_0=n+\frac{n^3}{2(n-1)}H^2+\frac{n(n-2)}{2(n-1)}\sqrt{n^2H^4+4(n-1)H^2}$.

In this paper we study eigenvalues of the poly-Laplacian with any order on a domain in an n-dimensional unit sphere and obtain estimates for eigenvalues. In particular, the optimal result of Cheng and Yang (Math Ann 331:445–460, 2005) is included in our ones. In order to prove our results, we introduce 2(l + 1) functions ai and bi, for i = 0, 1, . . . , l and two operators μ and η. First of all, we study properties of functions ai and bi and the operators μ and η. By making use of these properties and introducing k free constants, we obtain estimates for eigenvalues.

Let D be a bounded domain in an n-dimensional Euclidean space ℝn. Assume thatare eigenvalues of an eigenvalue problem of a system of n elliptic equations:In particular, when n=3, the eigenvalue problem describes the behaviour of the elastic vibration. We obtain universal inequalities for eigenvalues of the above eigenvalue problem by making use of a direct and explicit method; our results are sharper than one of Hook. Furthermore, a universal inequality for lower-order eigenvalues of the above eigenvalue problem is also derived.

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 survey, we discuss critical points of functionals by various aspects. We review properties of critical points of weighted area functional, that is, self-shrinkers of mean curvature flow in Euclidean spaces and examples of compact self-shrinkers are discussed. We also review properties of critical points for weighted area functional for weighted volume-preserving variations, that is, \(\lambda \)-hypersurfaces of weighted volume-preserving mean curvature flow in Euclidean spaces.

In this paper, we will introduce the notion of harmonic stability for complete minimal hypersurfaces in a complete Riemannian manifold. The first result we prove, is that a complete harmonic stable minimal surface in a Riemannian manifold with non-negative Ricci curvature is conformally equivalent to either a plane R2 or a cylinder R × S1, which generalizes a theorem due to Fischer-Colbrie and Schoen [12].

In this paper, we investigate an eigenvalue problem for the Dirichlet Laplacian on a domain in an n-dimensional compact Riemannian manifold. First we give a general inequality for eigenvalues. As one of its applications, we study eigenvalues of the Laplacian on a domain in an n-dimensional complex projective space, on a compact complex submanifold in complex projective space and on the unit sphere. By making use of the orthogonalization of Gram–Schmidt (QR-factorization theorem), we construct trial functions. By means of these trial functions, estimates for lower order eigenvalues are obtained.

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].

We study curvature structures of compact hypersurfaces in the unit sphere Sn+1(1) with two distinct principal curvatures. First of all, we prove that the Riemannian product is the only compact hypersurface in Sn+1(1) with two distinct principal curvatures, one of which is simple and satisfies where n(n − 1)r is the scalar curvature of hypersurfaces and c2 = (n − 2)/nr. This generalized the result of Cheng, where the scalar curvature is constant is assumed. Secondly, we prove that the Riemannian product is the only compact hypersurface with non-zero mean curvature in Sn+1(1) with two distinct principal curvatures, one of which is simple and satisfies This gives a partial answer for the problem proposed by Cheng.

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.

In this paper, we study eigenvalues of Laplacian on either a bounded connected domain in an n-dimensional unit sphere Sn(1), or a compact homogeneous Riemannian manifold, or an n-dimensional compact minimal submanifold in an N-dimensional unit sphere SN(1). We estimate the k+1-th eigenvalue by the first k eigenvalues. As a corollary, we obtain an estimate of difference between consecutive eigenvlaues. Our results are sharper than ones of P. C. Yang and Yau [25], Leung [19], Li [20] and Harrel II and Stubbe [12], respectively. From Weyl’s asymptotical formula, we know that our estimates are optimal in the sense of the order of k for eigenvalues of Laplacian on a bounded connected domain in an n-dimensional unit sphere Sn(1).

In this paper we investigate three-dimensional complete minimal hypersurfaces with constant Gauss-Kronecker curvature in a space form M4(c) (c ≤ 0). We prove that if the scalar curvature of a such hypersurface is bounded from below, then its Gauss-Kronecker curvature vanishes identically. Examples of complete minimal hypersurfaces which are not totally geodesic in the Euclidean space E4 and the hyperbolic space H4(c) with vanishing Gauss-Kronecker curvature are also presented. It is also proved that totally umbilical hypersurfaces are the only complete hypersurfaces with non-zero constant mean curvature and with zero quasi-Gauss-Kronecker curvature in a space form M4(c) (c ≤ 0) if the scalar curvature is bounded from below. In particular, we classify complete hypersurfaces with constant mean curvature and with constant quasi-Gauss-Kronecker curvature in a space form M4(c) (c ≤ 0) if the scalar curvature r satisfies r≥ ⅔c.

The purpose of this paper is to study complete space-like hypersurfaces with constant mean curvature in a locally symmetric Lorentz space satisfying some curvature conditions. We give an optimal estimate of the squared norm of the second fundamental form of such hypersurfaces. Furthermore, the totally umbilical hypersurfaces are characterized.

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 study n-dimensional complete submanifolds with constant scalar curvature in the Euclidean space En+p and n-dimensional compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1). We prove that the totally umbilical sphere Sn(r), totally geodesic Euclidean space En and generalized cylinder Sn−1(c) × E1 are the only n-dimensional (n > 2) complete submanifolds Mn with constant scalar curvature n(n − 1)r in the Euclidean space En+p, which satisfy the following condition: where S denotes the squared norm of the second fundamental form of Mn. For compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1), we also obtain a corresponding result (see theorem 1.3).

The paper considers n-dimensional hypersurfaces with constant scalar curvature of a unit sphere Sn−1(1). The hypersurface Sk(c1)×Sn−k(c2) in a unit sphere Sn+1(1) is characterized, and it is shown that there exist many compact hypersurfaces with constant scalar curvature in a unit sphere Sn+1(1) which are not congruent to each other in it. In particular, it is proved that if M is an n-dimensional (n > 3) complete locally conformally flat hypersurface with constant scalar curvature n(n−1)r in a unit sphere Sn+1(1), then r > 1−2/n, and(1) when r ≠ (n−2)/(n−1), ifthen M is isometric to S1(√1−c2)×Sn−1(c), where S is the squared norm of the second fundamental form of M;(2) there are no complete hypersurfaces in Sn+1(1) with constant scalar curvature n(n−1)r and with two distinct principal curvatures, one of which is simple, such that r = (n−2)/(n−1) and

In this paper, we prove that n-dimensional complete and connected submanifolds with parallel mean curvature vector H in the (n+p)-dimensional Euclidean space En+p are the totally geodesic Euclidean space En, the totally umbilical sphere Sn (c) or the generalized cylinder Sn− 1 (c) ×E1 if the second fundamental form h satisfies 2≤n2|H|2/ (n− 1).

First, we shall prove that a compact connected oriented locally conformally flat n-dimensional Riemannian manifold with constant scalar curvature is isometric to a space form or a Riemannian product Sn−1(c) × S1 if its Ricci curvature is nonnegative. Second, we shall give a topological classification of compact connected oriented locally conformally flat n-dimensional Riemannian manifolds with nonnegative scalar curvature r if the following inequality is satisfied: [sum ]i,jR2ij [les ] r2/(n−1), where [sum ]i,jR2ij is the squared norm of the Ricci curvature tensor.

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.