arXiv subjects
Publications and source records attributed to Qing-Ming Cheng.
The purpose of this paper is to study complete self-shrinkers of mean curvature flow in Euclidean spaces. In the paper, we give a complete classification for 2-dimensional complete Lagrangian self-shrinkers in Euclidean space $\mathbb R^4$ with constant squared norm of the second fundamental form.
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 $n$.
This paper studies eigenvalues of the clamped plate problem on a bounded domain in an $n$-dimensional Euclidean space. We give an estimate for the gap between $\sqrt {Γ_{k+1}-Γ_{1}}$ and $\sqrt {Γ_{k}-Γ_{1}}$, for any positive integer $k$. According to the asymptotic formula of Agmon and Pleijel, we know, the gap between $\sqrt {Γ_{k+1}-Γ_{1}}$ and $\sqrt {Γ_{k}-Γ_{1}}$ is bounded by a term with a lower order $k^{\frac1n}$ in the sense of the asymptotic formula of Agmon and Peijel, where $Γ_j$ denotes the $j^{^{\text{th}}}$ eigenvalue of the clamped plate problem.
In this paper, we construct compact embedded $λ$-hypersurfaces with the topology of torus which are called $λ$-torus in Euclidean spaces $\mathbb {R}^{n+1}$.
In this paper, we introduce a definition of $λ$-hypersurfaces of weighted volume-preserving mean curvature flow in Euclidean space. We prove that $λ$-hypersurfaces are critical points of the weighted area functional for the weighted volume-preserving variations. Furthermore, we classify complete $λ$-hypersurfaces with polynomial area growth and $H-λ\geq 0$, which are generalizations of the results due to Huisken, Colding-Minicozzi. We also define a $\mathcal{F}$-functional and study $\mathcal{F}$-stability of $λ$-hypersurfaces, which extend a result of Colding-Minicozzi. Lower bound growth and upper bound growth of the area for complete and non-compact $λ$-hypersurfaces are also studied.
It is our purpose to study complete self-shrinkers in Euclidean space. First of all, we show some examples of complete self-shrinkers without polynomial volume growth. 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 $\mathbb R^3$. In \cite{DX2}, Ding and Xin have proved this result under the assumption of polynomial volume growth, which is removed in our theorem.
In this paper, our purpose is to study rigidity theorems for $λ$-hypersurfaces in Euclidean space under Gauss map. As a Bernstein type problem for $λ$-hypersurfaces, we prove that an entirely graphic $λ$-hypersurface in Euclidean space is a hyperplane.
Since $n$-dimensional $λ$-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 $λ$-hypersurfaces. We give a gap theorem of complete $λ$-hypersurfaces with polynomial area growth. By making use of the generalized maximum principle for $\mathcal L$ of $λ$-hypersurfaces, we prove a rigidity theorem of complete $λ$-hypersurfaces.
In this paper we study eigenvalues of the closed eigenvalue problem of the Witten-Laplacian on an $n$-dimensional compact Riemannian manifold. Estimates for eigenvalues are given. As applications, we give a sharp upper bound for the $k^{\text{th}}$ eigenvalue and for isoparametric minimal hypersurfaces in the unit sphere, an explicit upper bound of the $(n+3)^{\text{th}}$ eigenvalue of the Laplacian is obtained. Furthermore, we generalize the Reilly's result on the first eigenvalue of the Laplacian.
In this paper, we study eigenvalues of the closed eigenvalue problem of the differential operator $ L$, which is introduced by Colding and Minicozzi in [4], on an $n$-dimensional compact self-shrinker in ${R}^{n+p}$. Estimates for eigenvalues of the differential operator $ L$ are obtained. Our estimates for eigenvalues of the differential operator $ L$ are sharp. Furthermore, we also study the Dirichlet eigenvalue problem of the differential operator $ L$ on a bounded domain with a piecewise smooth boundary in an $n$-dimensional complete self-shrinker in $ {R}^{n+p}$. For Euclidean space $ {R}^{n}$, the differential operator $ L$ becomes the Ornstein-Uhlenbeck operator in stochastic analysis. Hence, we also give estimates for eigenvalues of the Ornstein-Uhlenbeck operator.
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 $S<(10/7)$.
For an $n$-dimensional polytope $Ω$ in $\mathbb{R}^{n}$, we study lower bounds for eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. In the asymptotic formula on the average of the first $k$ eigenvalues, Li and Yau (1983) obtained the first term with the order $k^{\frac2n}$, which is optimal. The next landmark goal is to give the second term with the order $k^{\frac1n}$ in the asymptotic formula. For this purpose, Kovař\'ık, Vugalter and Weidl (2009) have made an important breakthrough in the case of dimension 2. It is our purpose to study the $n$-dimensional case for arbitrary dimension $n$. We obtain the second term in the asymptotic sense.
For an $n$-dimensional compact submanifold $M^n$ in the Euclidean space $\mathbf R^{N}$, we study estimates for eigenvalues of the Paneitz operator on $M^n$. Our estimates for eigenvalues are sharp.
It is our purpose to study complete self-shrinkers in Euclidean space. By introducing a generalized maximum principle for $\mathcal{L}$-operator, we give estimates on supremum and infimum of the squared norm of the second fundamental form of self-shrinkers without assumption on \emph{polynomial volume growth}, which is assumed in Cao and Li. Thus, we can obtain the rigidity theorems on complete self-shrinkers without assumption on \emph{polynomial volume growth}. For complete proper self-shrinkers of dimension 2 and 3, we give a classification of them under assumption of constant squared norm of the second fundamental form.
In this paper, we study estimates for eigenvalues of the clamped plate problem. A sharp upper bound for eigenvalues is given and the lower bound for eigenvalues in [10] is improved.
In this paper, we investigate the Dirchlet eigenvalue problems of poly-Laplacian with any order and quadratic polynomial operator of the Laplacian. We give some estimates for lower bounds of the sums of their first $k$ eigenvalues which improve the previous results.
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 [J. Funct. Anal. 245(2007), 334-352].
For a bounded domain $Ω$ with a piecewise smooth boundary in an $n$-dimensional Euclidean space $\mathbf{R}^{n}$, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. First we give a general inequality for eigenvalues of the Laplacian. As an application, we study lower order eigenvalues of the Laplacian and derive the ratios of lower order eigenvalues of the Laplacian.