International Journal of Mathematics

This is a special volume in honor of the 110th anniversary of the late Professor Shiing-Shen Chern, who was the founding advisor of this journal.

The symmetric difference of the $q$-binomial coefficients $F_{n,k}(q)=\left[\genfrac{}{}{0ex}{}{n+k}{k}\right]-q^n\left[\genfrac{}{}{0ex}{}{n+k-2}{k-2}\right]$ was introduced by Reiner and Stanton. They proved that $F_{n,k}(q)$ is symmetric and unimodal for $k\ge 2$ and $n$ even by using the representation theory for Lie algebras. Based on Sylvester’s proof of the unimodality of the Gaussian coefficients, as conjectured by Cayley, we find an interpretation of the unimodality of $F_{n,k}(q)$ in terms of semi-invariants. In the spirit of the strict unimodality of the Gaussian coefficients due to Pak and Panova, we prove the strict unimodality of the symmetric difference $G_{n,k,r}(q)=\left[\genfrac{}{}{0ex}{}{n+k}{k}\right]-q^{nr/2}\left[\genfrac{}{}{0ex}{}{n+k-r}{k-r}\right]$, except for the two terms at both ends, where $n,r\ge 8$, $k\ge r$ and at least one of $n$ and $r$ is even.

We present a potential generalization of the Kervarie semi-characteristic (with real coefficient) to the case of non-orientable manifolds.

Starting with the complexes, quaternions and the octonions we study five families of groups and homogeneous spaces each building geometrically on the previous families. For each family a particular theorem is studied from the perspective of the structure developed. The evolution of the spaces and the corresponding results form a sort of mathematical memoir of a career spanning more than 50 years.

In April 2003, Chern began a study of almost-complex structures on the six-sphere, with the idea of exploiting the special properties of its well-known almost-complex structure invariant under the exceptional group ${\mathrm{\text{G}}}_2$. While he did not solve the (currently still open) problem of determining whether there exists an integrable almost-complex structure on $S^6$, he did prove a significant identity that resolves the question for an interesting class of almost-complex structures on $S^6$.

By studying a complex Monge–Ampère equation, we present an alternate proof to a recent result of Chu-Lee-Tam concerning the projectivity of a compact Kähler manifold $N^n$ with ${\mathrm{\text{Ric}}}_k<0$ for some integer $k$ with $1<k<n$, and the ampleness of the canonical line bundle $K_N$.

Differential geometry, especially the use of curvature, plays a central role in modern Hodge theory. The vector bundles that occur in the theory (Hodge bundles) have metrics given by the polarizations of the Hodge structures, and the sign and singularity properties of the resulting curvatures have far reaching implications in the geometry of families of algebraic varieties. A special property of the curvatures is that they are $1^{\mathrm{\text{st}}}$ order invariants expressed in terms of the norms of algebro-geometric bundle mappings. This partly expository paper will explain some of the positivity and singularity properties of the curvature invariants that arise in the Hodge theory with special emphasis on the norm property.

General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the Minkowski sum boundary of $m$ arbitrary ellipsoids in $N$-dimensional Euclidean space. Expressions for the principal curvatures of these Minkowski sums are also derived. These results are then used to obtain upper and lower volume bounds for the Minkowski sum of ellipsoids in terms of their defining matrices; the lower bounds are sharper than the Brunn–Minkowski inequality. A reverse isoperimetric inequality for convex bodies is also given.

In 1978, Rabinowitz proved the existence of a non-constant $T$-periodic solution for nonlinear Hamiltonian systems on ${\mathbf{\text{R}}}^{2n}$ with Hamiltonian function being super-quadratic at the infinity and zero for any given $T>0$. Since the minimal period of this solution may be $T/k$ for some positive integer $k$, he proposed the question whether there exists a solution with $T$ as its minimal period for such a Hamiltonian system. This is the so-called Rabinowitz minimal periodic solution conjecture. In the last more than 40 years, this conjecture has been deeply studied by many mathematicians. But under the original structural conditions of Rabinowitz, the conjecture is still open when $n\ge 2$. In this paper, I give a brief survey on the studies of this conjecture and hope to lead to more interests on it.

The torsion of every metric connection on a Riemannian manifold has three components: one totally skew-symmetric, one of vectorial type and one of twistorial type, which is also called the traceless cyclic component. In this paper we classify complete simply connected Riemannian manifolds carrying a metric connection whose torsion is parallel, has nonzero vectorial component and vanishing twistorial component.

Let $M$ be an even-dimensional spin manifold with action of a compact Lie group $G$. We review the construction of bouquets of analytic equivariant cohomology classes and of $K$-integration. Given an elliptic curve $E$, we introduce, as in Grojnowski, elliptic bouquets of germs of holomorphic equivariant cohomology classes on $M$. Following Bott–Taubes and Rosu, we show that integration of an elliptic bouquet is well defined. In particular, this imply Witten’s rigidity theorem. Our systematic use of the Chern–Weil construction of equivariant classes allows us to reduce proofs to algebraic identities.

We use loop group factorizations to construct Darboux transformations, Permutability formulas, Scaling Transformations, and explicit soliton solutions of the $1$-d Schrödinger flows on compact Hermitian symmetric spaces.

This article is a survey of recent results about the existence of Riemannian metrics of positive scalar curvature on some open $3$-manifolds. This results culminate in a rigidity theorem obtained using the theory of stable minimal surfaces.

Motivated by the notion of the algebraic hyperbolicity, we introduce the notion of Nevanlinna hyperbolicity for a pair $(X,D)$, where $X$ is a projective variety and $D$ is an effective Cartier divisor on $X$. This notion links and unifies the Nevanlinna theory, the complex hyperbolicity (Brody and Kobayashi hyperbolicity), the big Picard-type extension theorem (more generally the Borel hyperbolicity). It also implies the algebraic hyperbolicity. The key is to use the Nevanlinna theory on parabolic Riemann surfaces recently developed by P$\mathrm{\text{ă}}$un and Sibony [Value distribution theory for parabolic Riemann surfaces, preprint (2014), arXiv:1403.6596].

In this paper, we classify compact manifolds of dimension three equipped with a path structure and a fixed contact form (which we refer to as a strict path structure) under the hypothesis that their automorphism group is non-compact. We use a Cartan connection associated to the structure and show that its curvature is constant.

The aim of this paper is to explain how to parametrize the equations of the facets of Kirwan polytopes using the notion of Ressayre’s pairs.

We discuss the behavior of the bottom of the spectrum of scalar Schrödinger operators under Riemannian coverings of orbifolds. We apply our results to geometrically finite and to conformally compact orbifolds.