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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 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.