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Guoxin Wei

Publications and source records attributed to Guoxin Wei.

49 records · Page 3Linked to original sources
arXiv2015

Complete $λ$-hypersurfaces of weighted volume-preserving mean curvature flow

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.

math.DG↗
arXiv2014

Rigidity theorems of $λ$-hypersurfaces

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.

math.DG↗
arXiv2012

A gap theorem of self-shrinkers

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 $S<(10/7)$.

math.DG↗
arXiv2012

Lower Bounds for Laplacian and Fractional Laplacian Eigenvalues

In this paper, we investigate eigenvalues of Laplacian on a bounded domain in an $n$-dimensional Euclidean space and obtain a sharper lower bound for the sum of its eigenvalues, which gives an improvement of results due to A. D. Melas [15]. On the other hand, for the case of fractional Laplacian $(-Δ)^{α/2}|_{D}$, where $α\in(0,2]$, we obtain a sharper lower bound for the sum of its eigenvalues, which gives an improvement of results due to S.Y. Yolcu and T. Yolcu [23].

math.DG↗
arXiv2011

Estimates for Eigenvalues of Poly-harmonic Operators

In this paper, we study eigenvalues of the poly-Laplacian with arbitrary order on a bounded domain in an n-dimensional Euclidean space and obtain a lower bound for eigenvalues, which generalizes the results due to Cheng-Wei [5] and gives an improvement of results due to Cheng- Qi-Wei [3].

math.DG↗
arXiv2011

Embedded constant $m^{\text{th}}$ mean curvature hypersurfaces on spheres

In this paper, we study $n$-dimensional hypersurfaces with constant $m^{\text{th}}$ mean curvature $H_m$ in a unit sphere $S^{n+1}(1)$ and prove that if the $m^{\text{th}}$ mean curvature $H_m$ takes value between $\dfrac{1}{(\tan \fracπ{k})^m}$ and $\frac{k^2-2}{n}(\frac{k^2+m-2}{n-m})^{\frac{m-2}{2}}$ for $1\leq m\leq n-1$ and any integer $k\geq 2$, then there exists at least one $n$-dimensional compact nontrivial embedded hypersurface with constant $H_m>0$ in $S^{n+1}(1)$. When $m=1$, our results reduce to the results of Perdomo \cite{[P]}; when $m=2$ and $m=4$, our results reduce to the results of Cheng-Li-Wei \cite{[WCL]}.

math.DG↗
arXiv2010

A lower bound for eigenvalues of the poly-Laplacian with arbitrary order

In this paper, we study eigenvalues of the poly-Laplacian with arbitrary order on a bounded domain in an $n$-dimensional Euclidean space and obtain a lower bound for eigenvalues, which gives an important improvement of results due to Levine and Protter. In particular, the result of Melas is included here.

math.DG↗
arXiv2009

Estimates for lower order eigenvalues of a clamped plate problem

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.

math.DG↗
arXiv2009

Embedded hypersurfaces with constant $m^{\text{th}}$ mean curvature in a unit sphere

In this paper, we study $n$-dimensional hypersurfaces with constant $m^{\text{th}}$ mean curvature in a unit sphere $S^{n+1}(1)$ and construct many compact nontrivial embedded hypersurfaces with constant $m^{\text{th}}$ mean curvature $H_m>0$ in $S^{n+1}(1)$, for $1\leq m\leq n-1$. In particular, if the $4^{\text{th}}$ mean curvature $H_4$ takes value between $\dfrac{1}{(\tan \fracπ{k})^4}$ and $\dfrac{k^4-4}{n(n-4)}$ for any integer $k\geq3$, then there exists an $n$-dimensional ($n\geq 5$) compact nontrivial embedded hypersurface with constant $H_4$ in $S^{n+1}(1)$.

math.DG↗