arXiv subjects

Yunheng Zhang

Publications and source records attributed to Yunheng Zhang.

4 recordsLinked to original sources

arXiv2026

Lu's conjecture for minimal surfaces in codimension two

Let $M^2\to\mathbb{S}^4$ be a closed minimal immersion, let $S$ be the squared norm of its second fundamental form, and let $λ_1\geqλ_2\geq0$ be the eigenvalues of Lu's fundamental matrix. We classify all such immersions for which $S+λ_2$ is constant. We prove that the constant can only be $0$ or $2$. In the first case the image is a totally geodesic $2$-sphere; in the second case it is either a Clifford torus in a totally geodesic $\mathbb{S}^3$ or the Veronese surface in $\mathbb{S}^4$. In particular, there is no closed minimal surface in $\mathbb{S}^4$ with constant $S+λ_2>2$. Consequently, Lu's second-gap conjecture holds for minimal surfaces in codimension two. Together with the hypersurface result of Peng--Terng and the counterexamples of Li--Zhao in every codimension $m\geq3$, this completes the codimension picture for minimal surfaces.

math.DG↗

arXiv2026

On Chern's Conjecture for Minimal Submanifolds with Flat Normal Bundle in Spheres

Let $M^n$ $(n\geqslant3)$ be a closed minimal submanifold in the unit sphere $\mathbb S^{n+m}$ $(m\geqslant2)$ with flat normal bundle, and let $S$ denote the squared norm of its second fundamental form. We prove an explicit second-gap rigidity theorem for $S$. More precisely, if $S$ is constant and \[ 0\leqslant S\leqslant n+δ, \] where $δ$ is an explicit constant satisfying $δ\geqslant \frac{n}{87}$, then either $S\equiv0$ and $M$ is a totally geodesic sphere, or $S\equiv n$ and $M$ is a Clifford torus contained in a totally geodesic $\mathbb S^{n+1}\subset\mathbb S^{n+m}$. %We observe that the flat-normal-bundle assumption is necessary here. The flat-normal-bundle condition is essential in the general higher-codimensional setting: without it, the corresponding rigidity statement already fails in dimension two. This theorem provides positive evidence for Chern's conjecture in higher codimension.

math.DG↗

arXiv2026

Pinching rigidity theorems for normal scalar curvature

Let $M^n$ be an $n$-dimensional closed minimal submanifold immersed in the unit sphere $\mathbb{S}^{n+m}$. Denote by $S$ and $ρ^{\perp}$ the squared norm of the second fundamental form and the normal scalar curvature of $M^n$, respectively. Let $\{A^α\}_{α=n+1}^{n+m}$ be the shape operators of $M^n$ with respect to a local orthonormal normal frame. Denote by $λ_{1}$ the largest eigenvalue of the positive semi-definite symmetric matrix $\mathcal{A}=(\langle A^α,A^β\rangle)_{m\times m}$. We show that if $λ_{1}\leqslant n$ and $ρ^{\perp}\leqslant \left[{\sqrt{2}n(n-1)}\right]^{-1} \mathop{\inf}\limits_{p\in M}(n-λ_{1})(p)$, then $ρ^{\perp}\equiv 0$, which means the normal bundle of $M^n$ is flat, and further we give the classification of $M^n$.

math.DG↗

arXiv2025

Chern Conjecture on Minimal Willmore Hypersurfaces with Constant Scalar Curvature

In this paper, we prove that for an $n$-dimensional closed minimal Willmore hypersurface $M^n$ with constant scalar curvature in the unit sphere $\mathbb{S}^{n+1}$, the squared norm $S$ of the second fundamental form of $M^n$ satisfies $S\geqslant n+\frac{4n+9-\sqrt{4 n^{2}+60 n+81}}{2}$ if $S>n$. This proves, in the approximate sense, the Chern conjecture about the second gap ($S\geqslant 2n$ if $S>n$), which will be fully verified under a further inequality condition about the 4-th mean curvature.

math.DG↗