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Weiran Ding

Publications and source records attributed to Weiran Ding.

5 recordsLinked to original sources
arXiv2026

Pinching rigidity of surfaces with parallel mean curvature vector in spheres

Inspired by the Simon conjecture for minimal surfaces in spheres, we study closed surfaces with parallel mean curvature vector and positive Gaussian curvature immersed in unit spheres. Let $h$ be the second fundamental form, let $\mathbf{H}$ be the mean curvature vector field, and set $\tilde h=h-\mathbf{H}g$ and $\tilde S=|\tilde h|^2=|h|^2-2H^2$, where $H=|\mathbf{H}|$ and $g$ is the induced Riemannian metric on the surface $M$. We establish three Simons-type integral identities for $\tilde S$, which extend the first, second and third gap identities in the minimal case. As applications, we obtain the first two sharp endpoint gaps and several rigidity and oscillation estimates in the third interval. We further characterize the endpoint cases by combining these identities with the classification theorems of Calabi and Yau.

math.DG↗
arXiv2026

On Simon's third gap conjecture for minimal surfaces in spheres

In this paper, continuing our previous work, we investigate the third gap problem in the Simon conjecture for closed minimal surfaces in the unit sphere. By developing refined third-order Simons-type integral identities and establishing new lower bounds for higher-order curvature terms, we obtain positive gap results throughout the entire interval $\left[\frac{5}{3},\frac{9}{5}\right]$ for the squared norm of the second fundamental form, including the endpoint cases. As an application, we establish a rigidity result for closed self-shrinkers.

math.DG↗
arXiv2026

Lu's conjecture for minimal surfaces

After Chern's conjecture on the discreteness of the constant scalar curvatures of compact minimal submanifolds $M^n$ in unit spheres $\mathbb{S}^{n+q}$, Z. Q. Lu proposed a conjecture regarding the second gap, based on his ingenious refinement of the known first gap theorem. This refinement unifies Simons' first gap theorem for hypersurfaces with the corresponding theorems for high-codimensional submanifolds established by Yau, Shen, Li and Li, among others. In this paper, for arbitrary codimension, we prove Lu's conjecture for minimal 2-spheres, and for any minimal surfaces under some slight inequality conditions about the normal scalar curvature.

math.DG↗