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Qing-Ming Cheng

Publications and source records attributed to Qing-Ming Cheng.

At least 19 recordsLinked to original sources
arXiv2026

Estimates on scalar curvature of self-shrinkers

In this paper, we study $n$-dimensional complete self-shrinkers in $\mathbb R^{n+1}$ with constant squared norm $S$ of the second fundamental form. We partially resolve the conjecture on $n$-dimensional complete self-shrinkers in $\mathbb R^{n+1}$ with constant squared norm $S$ of the second fundamental form. Furthermore, if the scalar curvature of an $n$-dimensional self-shrinker is constant, then we prove that the scalar curvature $R$ satisfies $R\leq n-1$. We also classify $n$-dimensional complete self-shrinkers in $\mathbb R^{n+1}$ with non-negative constant scalar curvature.

math.DG↗
arXiv2026

Examples of compact embedded mean convex $\lambda$-hypersurfaces

There is a well-known conjecture asserts that the round sphere should be the only compact embedded self-shrinker (i.e. $0$-hypersurface) which is diffeomorphic to a sphere. S. Brendle confirmed the conjecture for 2-dimensional $0$-hypersurfaces. For any dimensional $\lambda$-hypersurfaces, if $\lambda<0$, we constructed compact convex embedded $\lambda$-hypersurface which is diffeomorphic to a sphere and is not a round sphere. In this paper, for $\lambda>0$, we construct a compact mean convex embedded $\lambda$-hypersurface which is diffeomorphic to a sphere and is not a round sphere. In fact, for $\lambda>0$, there are no compact convex embedded $\lambda$-hypersurfaces which are diffeomorphic to spheres except a round sphere.

math.DG↗
arXiv2025

Complete two-sided $\delta$-stable minimal hypersurfaces in $\mathbf R^{n+1}$

In this paper, we study complete $\delta$-stable minimal hypersurfaces in $\mathbf R^{n+1}$. We prove that complete two-sided $\delta$-stable minimal hypersurfaces have Euclidean volume growth if $3\leq n\leq 5$ and $\delta>\delta_0(n)$, where $\delta_0(3)=1/3$, $\delta_0(4)=1/2$ and $\delta_0(5)=21/22$. We also give a sufficient condition such that complete two-sided $\delta$-stable minimal hypersurfaces in $\mathbf R^{n+1}$ is the hyperplane. Furthermore, we prove that a complete two-sided $\delta$-stable minimal hypersurface is the hyperplane if $3\leq n\leq 5$ and $\delta>\delta_1(n)$, where $\delta_1(3)=3/8$, $\delta_1(4)=2/3$ and $\delta_1(5)=21/22$.

math.DG↗
arXiv2025

Estimates for Eigenvalues of the Dirichlet Laplacian on Riemannian Manifolds

We revisit the eigenvalue problem of the Dirichlet Laplacian on bounded domains in complete Riemannian manifolds. By building on classical results like Li-Yau's and Yang's inequalities, we derive upper and lower bounds for eigenvalues. For the projective spaces and their minimal submanifolds, we also give explicit estimates on lower bounds for eigenvalues of the Dirichlet Laplacian.

math.DG↗
arXiv2024

Complete minimal hypersurfaces in a hyperbolic space $H^{4}(-1)$

In this paper, we study $n$-dimensional complete minimal hypersurfaces in a hyperbolic space $H^{n+1}(-1)$ of constant curvature $-1$. We prove that a $3$-dimensional complete minimal hypersurface with constant scalar curvature in $H^{4}(-1)$ satisfies $S\leq \frac{21}{29}$ by making use of the Generalized Maximum Principle, where $S$ denotes the squared norm of the second fundamental form of the hypersurface.

math.DG↗
arXiv2024

Embedded cylindrical and doughnut-shaped $λ$-hypersurfaces

In the paper, we construct, for $λ>0$, complete embedded and non-convex $λ$-hypersurfaces, which are diffeomorphic to a cylinder. Hence, one can not expect that $λ$-hypersurfaces share a common conclusion on the planar domain conjecture even if the planar domain conjecture of T. Ilmanen for self-shrinkers of mean curvature flow are solved by Brendle \cite{B} affirmatively. Furthermore, for a fixed $λ<0$ which may have small $|λ|$, we can construct two compact embedded $λ$-hypersurfaces which are diffeomorphic to $\mathbb{S}^{1}\times \mathbb{S}^{n-1}$, but they are not isometric to each other.

math.DG↗
arXiv2023

Examples of compact embedded $λ$-hypersurfaces

In the paper, we construct compact embedded $λ$-hypersurfaces which are diffeomorphic to a sphere and are not isometric to a standard sphere. Hence, one can not expect to have Alexandrov type theorem for $λ$-hypersurfaces.

math.DG↗
arXiv2023

$3$-dimensional complete vacuum static spaces

In this paper, we study complete Vacuum Static Spaces. A complete classification of 3-dimensional complete Vacuum Static Spaces with non-negative scalar curvature and constant squared norm of Ricci curvature tensor is given by making use of the generalized maximum principle.

math.DG↗
arXiv2022

Complete hypersurfaces with $w$-constant mean curvature in the unit spheres

In this paper, we study $4$-dimensional complete hypersurfaces with $w$-constant mean curvature in the unit sphere. We give a lower bound of the scalar curvature for $4$-dimensional complete hypersurfaces with $w$-constant mean curvature. As a by-product, we give a new proof of the result of Deng-Gu-Wei under the weaker topological condition.

math.DG↗
arXiv2021

Chern conjecture on minimal hypersurfaces

In this paper, we study $n$-dimensional complete minimal hypersurfaces in a unit sphere. We prove that an $n$-dimensional complete minimal hypersurface with constant scalar curvature in a unit sphere with $f_3$ constant is isometric to the totally geodesic sphere or the Clifford torus if $S\leq 1.8252 n-0.712898$, where $S$ denotes the squared norm of the second fundamental form of this hypersurface.

math.DG↗
arXiv2021

The second gap on complete self-shrinkers

In this paper, we study complete self-shrinkers in Euclidean space and prove that an $n$-dimensional complete self-shrinker in Euclidean space $\mathbb{R}^{n+1}$ is isometric to either $\mathbb{R}^{n}$, $S^{n}(\sqrt{n})$, or $S^k (\sqrt{k})\times\mathbb{R}^{n-k}$, $1\leq k\leq n-1$, if the squared norm $S$ of the second fundamental form, $f_3$ are constant and $S$ satisfies $S<1.83379$. We should remark that the condition of polynomial volume growth is not assumed.

math.DG↗
arXiv2019

Stability and area growth of $λ$-hypersurfaces

In this paper, We 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 area for complete and non-compact $λ$-hypersurfaces are studied.

math.DG↗
arXiv2019

Area of minimal hypersurfaces

A well-known conjecture of Yau states that the area of one of Clifford minimal hypersurfaces $S^k\big{(}\sqrt{\frac{k}{n}}\, \big{)}\times S^{n-k}\big{(}\sqrt{\frac{n-k}{n}}\, \big{)}$ gives the lowest value of area among all non-totally geodesic compact minimal hypersurfaces in the unit sphere $S^{n+1}(1)$. The present paper shows that Yau conjecture is true for minimal rotational hypersurfaces, more precisely, the area $|M^n|$ of compact minimal rotational hypersurface $M^n$ is either equal to $|S^n(1)|$, or equal to $|S^1(\sqrt{\frac{1}{n}})\times S^{n-1}(\sqrt{\frac{n-1}{n}})|$, or greater than $2(1-\frac{1}π)|S^1(\sqrt{\frac{1}{n}})\times S^{n-1}(\sqrt{\frac{n-1}{n}})|$. As the application, the entropies of some special self-shrinkers are estimated.

math.DG↗
arXiv2018

Complete $λ$-surfaces in $\mathbb R^3$

The purpose of this paper is to study complete $λ$-surfaces in Euclidean space $\mathbb R^3$. A complete classification for 2-dimensional complete $λ$-surfaces in Euclidean space $\mathbb R^3$ with constant squared norm of the second fundamental form is given.

math.DG↗
arXiv2018

Complete Lagrangian self-shrinkers in $\mathbf R^4$

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.

math.DG↗