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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 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 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 .

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 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 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 .