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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 Rn+1 is isometric to either Rn, Sn(vn), or Rn-m × Sm(vm), 1 ≤ m ≤ n - 1, if the squared norm S of the second fundamental form is constant and satisfies S < 10/7. [ABSTRACT FROM AUTHOR]

In this paper, we investigate universal estimates for eigenvalues of a buckling problem. For a bounded domain in a Euclidean space, we give a positive contribution for obtaining a sharp universal inequality for eigenvalues of the buckling problem. For a domain in the unit sphere, we give an important improvement on the results of Wang and Xia. [ABSTRACT FROM AUTHOR]

In this paper, we investigate universal inequalities for eigenvalues of a clamped plate problem on a bounded domain in an $ n$-dimensional Euclidean space, Payne, Pólya 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 $ n$-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 $ \tfrac{(n-1)^4}{16}$ $ \tfrac{(n-1)^4}{16}$ [ABSTRACT FROM AUTHOR]

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, Pólya and Weinberger in [14] and this problem has been mentioned again by Ashbaugh in [1]. [ABSTRACT FROM AUTHOR]

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), if S ≥ (n-1) n(r-1)+2/n-2 + n-2/n(r-1) + 2, then 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 S > (n-1) n(r-1)+2/n-2 + n-2/n(r-1) + 2 = n. [ABSTRACT FROM PUBLISHER]