26 citations · 126 across the 34 of their papers we have counts for
37 papers
A Complex Structure on : The Genus-One Case of Calabi's Problem
Min Gao, Zizhou Tang, Wenjiao Yan
We construct a complex structure on , giving an affirmative answer to the genus-one case of Calabi's problem. The resulting compact complex threefold is non-Kähler a…
A concise proof of homogeneity of an isoparametric hypersurface with
Chao Qian, Zizhou Tang
A well-known hypersurface classification theorem states that any isoparametric hypersurface in with six principal curvatures is homogeneous. This landmark result was first…
Chern's Conjecture with Constant Cubic Trace
Huixin Tan, Zizhou Tang, Yuquan Xie +1
We prove that the values set of \(S=|A|^2\) attained by closed embedded minimal hypersurfaces in \(\mathbb S^{n+1}(1)\) with constant \(S\) and constant \(f_3=\operatorname{tr}(A^3…
Non-extendability of complex structures
Zizhou Tang, Wenjiao Yan
There exists a complex structure on a connected open subset of . The present paper proves that: (1) can be extended to a global almost complex struct…
Isoparametric foliations and complex structures
Chao Qian, Zizhou Tang, Wenjiao Yan
Explicit representations of complex structures on closed manifolds are valuable, but relatively rare in the literature. Using isoparametric theory, we construct complex structures…
Orthogonal almost complex structure and its Nijenhuis tensor
Zizhou Tang, Wenjiao Yan
In this paper, we demonstrate that on an almost Hermitian manifold , a 2-form , the pulling back of the Kähler form on the twistor bundle over $M^{2n…
Isoparametric hypersurfaces and complex structures
Zizhou Tang, Wenjiao Yan
The main purpose of this note is to construct almost complex or complex structures on certain isoparametric hypersurfaces in unit spheres. As a consequence, complex structures on $…
On two questions of James
Chao Qian, Zizhou Tang, Wenjiao Yan
64 years ago, I. M. James raised two fundamental questions about octonionic Stiefel spaces. The prime objective of this paper is to figure out partial answers to both of them.
Clifford systems, harmonic maps and metrics with non-negative curvature
Chao Qian, Zizhou Tang, Wenjiao Yan
Associated with a symmetric Clifford system on , there is a canonical vector bundle over . For and , we construc…
Topology and curvature of isoparametric families in spheres
Chao Qian, Zizhou Tang, Wenjiao Yan
An isoparametric family in the unit sphere consists of parallel isoparametric hypersurfaces and their two focal submanifolds. The present paper has two parts. The first part invest…
On the Chern conjecture for isoparametric hypersurfaces
Zizhou Tang, Wenjiao Yan
For a closed hypersurface with constant mean curvature and constant non-negative scalar curvature, the present paper shows that if $\mathrm{tr}(\mathcal{A}^…
Extrinsic geometry of the Gromoll-Meyer sphere
Chao Qian, Zizhou Tang, Wenjiao Yan
Among a family of 2-parameter left invariant metrics on Sp(2), we determine which have nonnegative sectional curvatures and which are Einstein. On the quotiente $\widetilde{N}^{11}…
Clifford algebra, isoparametric foliation and related geometric constructions
Chao Qian, Zizhou Tang
Based on representation theory of Clifford algebra, Ferus, Karcher and Münzner constructed a series of isoparametric foliations. In this paper, we will survey recent studies on iso…
Isoparametric polynomials and sums of squares
Jianquan Ge, Zizhou Tang
Hilbert's 17th problem asks that whether every nonnegative polynomial can be a sum of squares of rational functions. It has been answered affirmatively by Artin. However, the quest…
A sufficient condition for a hypersurface to be isoparametric
Zizhou Tang, Dongyi Wei, Wenjiao Yan
Let be a closed Riemannian manifold on which the integral of the scalar curvature is nonnegative. Suppose is a symmetric tensor field whose dual $(1,1)…
Isoparametric theory and its applications
Zizhou Tang, Wenjiao Yan
This is a survey on the recent progress in several applications of isoparametric theory, including an affirmative answer to Yau's conjecture on the first eigenvalue of Laplacian in…
Minimizing cones associated with isoparametric foliations
Zizhou Tang, Yongsheng Zhang
Associated with isoparametric foliations of unit spheres, there are two classes of minimal surfaces minimal isoparametric hypersurfaces and focal submanifolds. By virtue of the…
Normal scalar curvature inequality on the focal submanifolds of isoparametric hypersurfaces
Jianquan Ge, Zizhou Tang, Wenjiao Yan
An isoparametric hypersurface in unit spheres has two focal submanifolds. Condition A plays a crucial role in the classification theory of isoparametric hypersurfaces in [CCJ07], […
An exotic sphere with positive curvature
Jianquan Ge, Zizhou Tang
A metric with positive sectional curvature on the Gromoll-Meyer exotic 7-sphere is constructed explicitly. The proof relies on a 2-parameter family of left invariant metrics on Sp(…
Isoparametric foliations, a problem of Eells-Lemaire and conjectures of Leung
Chao Qian, Zizhou Tang
In this paper, two sequences of minimal isoparametric hypersurfaces are constructed via representations of Clifford algebras. Based on these, we give estimates on eigenvalues of th…
Recent progress in isoparametric functions and isoparametric hypersurfaces
Chao Qian, Zizhou Tang
This paper gives a survey of recent progress in isoparametric functions and isoparametric hypersurfaces, mainly in two directions. (1) Isoparametric functions on Riemannian manifol…
Isoparametric foliation and a problem of Besse on generalizations of Einstein condition
Zizhou Tang, Wenjiao Yan
The focal sets of isoparametric hypersurfaces in spheres with g = 4 are all Willmore submanifolds, being minimal but mostly non-Einstein ([TY1], [QTY]). Inspired by A.Gray's view,…
Isoparametric functions on exotic spheres
Chao Qian, Zizhou Tang
This paper extends widely the work in \cite{GT13}. Existence and non-existence results of isoparametric functions on exotic spheres and Eells-Kuiper projective planes are establish…
-invariant and a problem of Bérard-Bergery on the existence of closed geodesics
Zizhou Tang, Weiping Zhang
We use the -invariant of Atiyah-Patodi-Singer to compute the Eells-Kuiper invariant for the Eells-Kuiper quaternionic projective plane. By combining with a known result of Bérar…