17 papers
A proof of the Simon conjecture and curvature rigidities
Weiran Ding, Jianquan Ge, Fagui Li
In this paper, we prove all gaps in the Simon conjecture for connected closed minimal surfaces immersed in unit spheres. Our proof uses the line-bundle ladder of Lin, Wang, and Xu…
Volume gap for minimal submanifolds in spheres, II
Jianquan Ge, Fagui Li
Let $f:M^n\looparrowright\Sph^{n+q}(1)$, and , be a closed, connected, non-totally-geodesic minimal immersion with second fundamental form , and put and…
Examples of Curvature Inhomogeneous Submanifolds with Constant Ricci Eigenvalues
Jianquan Ge, Yuyang Zhao
We construct two families of curvature inhomogeneous Riemannian manifolds with constant Ricci eigenvalues. The first, derived from the Einstein warped products, has two distinct Ri…
Lu's conjecture for minimal surfaces in codimension two
Jianquan Ge, Fagui Li, Yunheng Zhang
Let be a closed minimal immersion, let be the squared norm of its second fundamental form, and let be the eigenvalues of Lu's fundamental…
Compactness and Willmore Energy of Helicoidal Minimal Surfaces in the 3-Sphere
Jianquan Ge, Shilin Li
Recently, I. Castro, I. Castro-Infantes, and J. Castro-Infantes introduced a two-parameter family of helicoidal minimal surfaces in , denoted by $\operatorname{Hel}_c^…
Pinching rigidity of surfaces with parallel mean curvature vector in spheres
Weiran Ding, Jianquan Ge, Fagui Li
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 sphe…
Sharp Weitzenböck and PIC2 Estimates from Sectional-Scalar Curvature Pinching
Jian Ge
Let be an -dimensional Euclidean vector space, ,where , and . We prove the sharp pointwise estimate \[ q_2(E) \ge -\frac{2(\ell -1)…
On Chern's Conjecture for Minimal Submanifolds with Flat Normal Bundle in Spheres
Jianquan Ge, Fagui Li, Yunheng Zhang
Let be a closed minimal submanifold in the unit sphere with flat normal bundle, and let denote the squared norm of its s…
Rigidity of Closed Minimal Hypersurfaces in
Jianquan Ge, Tong Liu, Keyan Luo +1
The celebrated Chern conjecture asserts that any closed minimal hypersurface in with constant scalar curvature is isoparametric. In this paper, we resolve this c…
Hypersurfaces with Constant Ricci Eigenvalues in Real Space Forms
Jianquan Ge, Yuyang Zhao
The classification of curvature homogeneous hypersurfaces in real space forms was established by Tsukada in 1988, with the remaining rank-two cases in and $\mathbb{H…
On Simon's third gap conjecture for minimal surfaces in spheres
Weiran Ding, Jianquan Ge, Fagui Li
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 thir…
Weakly stable irreducible Yang-Mills fields over
Jianquan Ge, Lixin Xiao
Addressing Yau's conjecture (Problem 117) on , we investigate the self-duality of weakly stable Yang-Mills fields under the assumption of irreducibility. For structure groups…
Pinching rigidity theorems for normal scalar curvature
Jianquan Ge, Fagui Li, Yunheng Zhang
Let be an -dimensional closed minimal submanifold immersed in the unit sphere . Denote by and the squared norm of the second fundamental…
SDP Feasibility Problems and sos Representation Ranks for OT-FKM Type Isoparametric Polynomials
Jianquan Ge, Kai Jia, Yuyang Zhao
Semidefinite programming (SDP) provides a fundamental framework for studying properties of sum-of-squares (sos) representations of nonnegative polynomials. In this paper we study t…
Lu's conjecture for minimal surfaces
Weiran Ding, Jianquan Ge, Fagui Li +1
After Chern's conjecture on the discreteness of the constant scalar curvatures of compact minimal submanifolds in unit spheres , Z. Q. Lu proposed a conject…
Volume gap between the minimal submanifold and the unit sphere
Weiran Ding, Jianquan Ge, Fagui Li
In this paper, following the method of Cheng-Li-Yau, we first modify the coefficients in the constant to improve the volume gap. Further, we also enlarge our gap by applying…
Pinching rigidity of minimal surfaces in spheres
Weiran Ding, Jianquan Ge, Fagui Li
In this paper we give a pinching theorem of the Simon conjecture in the case s=3 and also give a new proof of the cases s=1 and s=2 by some Simons-type integral inequalities.