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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).

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).

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 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.

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].

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 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).

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.

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.