Xu Cheng

Bio: Xu Cheng is an academic researcher from Federal Fluminense University. The author has an hindex of 2, co-authored 2 publications receiving 25 citations.

Stability properties and gap theorem for complete f-minimal hypersurfaces

Abstract: In this paper, we study complete oriented f -minimal hypersurfaces properly immersed in a cylinder shrinking soliton $$(\mathbb{S}^n \times \mathbb{R},\bar g,f)$$ .We prove that such hypersurface with L f -index one must be either $$\mathbb{S}^n \times \{ 0\}$$ or $$\mathbb{S}^{n - 1} \times \mathbb{R}$$ , where $${S}^{n - 1}$$ denotes the sphere in $$\mathbb{S}^n$$ of the same radius. Also we prove a pinching theorem for them.

Stability properties and gap theorem for complete $f$-minimal hypersurfaces

Abstract: In this paper, we study complete oriented $f$-minimal hypersurfaces properly immersed in a cylinder shrinking soliton $(\mathbb{S}^n\times \mathbb{R}, \bar{g}, f)$. We prove that such hypersurface with $L_f$-index one must be either $\mathbb{S}^n\times\{0\}$ or $\mathbb{S}^{n-1}\times\mathbb{R}$, where $\mathbb{S}^{n-1}$ denotes the sphere in $\mathbb{S}^{n}$ of the same radius. Also we prove a pinching theorem for them.

Simons-Type Equation for f-Minimal Hypersurfaces and Applications

Abstract: We derive the Simons-type equation for \(f\)-minimal hypersurfaces in weighted Riemannian manifolds and apply it to obtain a pinching theorem for closed \(f\)-minimal hypersurfaces immersed in the product manifold \(\mathbb {S}^n(\sqrt{2(n-1)})\times \mathbb {R}\) with \(f=\frac{t^2}{4}\). Also, we classify closed \(f\)-minimal hypersurfaces with \(L_f\)-index one immersed in \(\mathbb {S}^n(\sqrt{2(n-1)})\times \mathbb {R}\) with the same \(f\) as above.

On complete embedded translating solitons of the mean curvature flow that are of finite genus

Abstract: We desingularise the union of $3$ Grim paraboloids along Costa-Hoffman-Meeks surfaces in order to obtain what we believe to be the first examples in $\Bbb{R}^3$ of complete embedded translating solitons of the mean curvature flow of arbitrary non-trivial genus. This solves a problem posed by Mart\'in, Savas-Halilaj and Smoczyk.

Geometric Properties of Self-Shrinkers in Cylinder Shrinking Ricci Solitons

Abstract: In this paper, we prove some spectral properties of the drifted Laplacian of self-shrinkers properly immersed in gradient shrinking Ricci solitons. Then we use these results to prove some geometric properties of self-shrinkers. For example, we describe a collection of domains in the ambient space that cannot contain self-shrinkers.

Stability and geometric properties of constant weighted mean curvature hypersurfaces in gradient Ricci solitons

Abstract: In this paper, we study stability properties of hypersurfaces with constant weighted mean curvature (CWMC) in gradient Ricci solitons. The CWMC hypersurfaces generalize the f-minimal hypersurfaces and appear naturally in the isoperimetric problems in smooth metric measure spaces. We obtain a result about the relationship between the properness and extrinsic volume growth under the assumption of a limitation for the weighted mean curvature of the immersion. Moreover, we estimate Morse index for CWMC hypersurfaces in terms of the dimension of the space of parallel vector fields restricted to hypersurface.

Spectral properties and rigidity for self-expanding solutions of the mean curvature flows

Abstract: In this paper, we study self-expanders for mean curvature flows. First we show the discreteness of the spectrum of the drifted Laplacian on them. Next we give a universal lower bound of the bottom of the spectrum of the drifted Laplacian and prove that this lower bound is achieved if and only if the self-expander is the Euclidian subspace through the origin. Further, for self-expanders of codimension $1$, we prove an inequality between the bottom of the spectrum of the drifted Laplacian and the bottom of the spectrum of weighted stability operator and that the hyperplane through the origin is the unique self-expander where the equality holds. Also we prove the uniqueness of hyperplane through the origin for mean convex self-expanders under some condition on the square of the norm of the second fundamental form.