arXiv subjects
Publications and source records attributed to Sai-Kee Yeung.
The purpose of the article is to study a foliation associated to a lattice-equivariant harmonic map of small rank from a complex ball to another. The result is related to rigidity of some complex ball quotients.
The purpose of the paper is to study some problems raised by Hayman and Gundersen about the existence of non-trivial entire and meromorphic solutions for the Fermat type functional equation $f^n+g^n+h^n=1$. Hayman showed that no non-trivial meromorphic solutions and entire solutions exist when $n \ge 9$ and $n \ge 7$ respectively. By considering the entire holomorphic curves on the Fermat surface defined by $X^n+Y^n+Z^n=W^n$ on the complex projective space $\mathbb{P}^3$ and applying the method of jet differentials, we show that no non-trivial meromorphic solutions and entire solutions exist when $n \ge 8$ and $n \ge 6$ respectively. In particular, this completes the investigation of non-trivial entire solutions for all $n$ and respectively, meromorphic solutions for all cases except for $n=7$. Finally, for the generalized Fermat type functional equation $f^n+g^m+h^l=1$, we will also prove the non-existence of non-trivial meromorphic solutions when $1/n+1/m+1/l \le 3/8$, giving the strongest result obtained so far.
In this article, we study various concrete algebraic and differential geometric properties of the Cartwright-Steger surface. In particular, we determine the genus of a generic fiber of the Albanese fibration, and deduce that the singular fibers are not totally geodesic, answering an open problem about fibrations of a complex ball quotient over a Riemann surface.
We show that there are no arithmetic fake compact hermitian symmetric spaces of type other than An for n>4.
The main goal of this article is to relate asymptotic geometric properties on a tower of coverings of a non-compact Kähler manifold of finite volume with reasonable geometric assumptions to its universal covering. Applicable examples include moduli spaces of hyperbolic punctured Riemann surfaces and Hermitian locally symmetric spaces of finite volume.
The addendum updates the results presented in the paper `Fake Projective Plane, Invent Math 168, 321-370 (2007)' and makes some additions and corrections. The fake projective planes are classified into twenty six classes. Together with a recent work of Donald Cartwright and Tim Steger, there is now a complete list of fake projective planes. There are precisely one hundred fake projective planes as complex surfaces classified up to biholomorphism.
We introduce the notion of domains with uniform squeezing property, study various analytic and geometric properties of such domains and show that they cover many interesting examples, including Teichmuller spaces and Hermitian symmetric spaces of non-compact type. The properties supported by such manifolds include pseudoconvexity, hyperconvexity, Kaehler-hyperbolicity, vanishing of cohomology groups and quasi-isometry of various invariant metrics. It also leads to nice geometric properties for manifolds covered by bounded domains and a simple criterion to provide positive examples to a problem of Serre about Stein properties of holomorphic fiber bundles.
We show that if n>5, PU(n-1,1) does not contain a cocompact arithmetic subgroup with the same Euler-Poincare characteristic (in the sense of C.T.C. Wall) as the complex projective space of dimension n-1, and show that if n=5, there are at least four such subgroups, which are in fact torsion-free. This, in particular, leads to examples of a fake projective space of dimension 4. Analogous results for arithmetic fake grassmannians Gr(m,n) with n>3 odd are also obtained.
This is an addendum to our earlier paper on the defect of an ample divisor of an abelian variety. It modifies an argument of the original paper to handle one difficulty there. At the same time the modification improves the result in the original paper by replacing the counting function by one truncated at a multiplicity given by an explicit function of the dimension of the abelian variety and the Chern number of the divisor.
The main purpose of this paper is to prove the following theorem on the defect relations for ample divisors of abelian varieties. Main Theorem. Let $A$ be an abelian variety of complex dimension $n$ and $D$ be an ample divisor in $A$. Let $f:{\bf C}\rightarrow A$ be a holomorphic map. Then the defect for the map $f$ and the divisor $D$ is zero. Corollary to Main Theorem. The complement of an ample divisor $D$ in an abelian variety $A$ is hyperbolic in the sense that there is no nonconstant holomorphic map from $\bf C$ to $A-D$.