arXiv subjects

Ya Tao

Publications and source records attributed to Ya Tao.

4 recordsLinked to original sources

arXiv2026

Minimal Hypersurfaces with constant scalar curvature in $\mathbf{S}^6$

In this paper, we propose certain assumptions on the principal curvatures for a closed minimal hypersurface $M^5$ in $\mathbf{S}^6$ to be isoparametric, provided that the functions $S, f_3,f_4$ are constants. Our result removes the nonnegative scalar curvature assumption as in Tang and Yan \cite{TY}. Finally, as a rigidity result, if $M^5\subset \mathbf{S}^6$ has a point with exactly two distinct principal curvatures, then it must be a Clifford torus.

math.DG↗

arXiv2025

Complete hypersurfaces in $R^{n+1}$ with constant mean and scalar curvature

In this paper, we investigate the rigidity problems of complete hypersurfaces with constant mean curvature and constant scalar curvature in Euclidean spaces. Firstly, under some conditions of Gaussian-Kronecker curvature, we provide characterizations for the unsolved cases of N\'u\~nez's theorems in dimensions 4 and 5, as well as several rigidity results under some conditions of $r$-th mean curvatures. Moreover, for the case of dimension 6, we also present analogous rigidity results. Finally, for general dimensions, we offer a rigidity theorem under similar pinching conditions.

math.DG↗

arXiv2024

Rigidity Results for Compact Submanifolds with Pinched Ricci Curvature in Euclidean and Spherical Space Forms

For compact submanifolds in Euclidean and Spherical space forms with Ricci curvature bounded below by a function $\alpha(n,k,H,c)$ of mean curvature, we prove that the submanifold is either isometric to the Einstein Clifford torus, or a topological sphere for the maximal bound $\alpha(n,[\frac{n}{2}],H,c)$, or has up to $k$-th homology groups vanishing. This gives an almost complete (except for the differentiable sphere theorem) characterization of compact submanifolds with pinched Ricci curvature, generalizing celebrated rigidity results obtained by Ejiri, Xu-Tian, Xu-Gu, Xu-Leng-Gu, Vlachos, Dajczer-Vlachos.

math.DG↗

arXiv2024

Normal Scalar Curvature Inequality on a Class of Austere Submanifolds

In this paper, we establish new normal scalar curvature inequalities on a class of austere submanifolds by proving sharper DDVV-type inequalities on associated austere subspaces. We also provide some examples of austere submanifolds in this class and point out one of them achieves the equality in our normal scalar curvature inequality everywhere. As a byproduct, we obtain a Simons-type gap theorem for closed austere submanifolds in unit spheres which belong to that class.

math.DG↗