We prove that a Shimura curve in the Siegel modular variety is not generically contained in the open Torelli locus as long as the rank of unitary part in its canonical Higgs bundle satisfies a numerical upper bound. As an application we show that the Coleman–Oort conjecture holds for Shimura curves associated with partial corestriction upon a suitable choice of parameters, which generalizes a construction due to Mumford.

We consider the problem of finding on a given Euclidean domain $\Omega $ of dimension $n \ge 3$ a complete conformally flat metric whose Schouten curvature A satisfies some equations of the form $f(\lambda (-A)) = 1$. This generalizes a problem considered by Loewner and Nirenberg for the scalar curvature. We prove the existence and uniqueness of such metric when the boundary $\partial \Omega $ is a smooth bounded hypersurface (of codimension one). When $\partial \Omega $ contains a compact smooth submanifold $\Sigma $ of higher codimension with $\partial \Omega {\setminus }\Sigma $ being compact, we also give a ‘sharp’ condition for the divergence to infinity of the conformal factor near $\Sigma $ in terms of the codimension.

According to Hall, a subgroup H of a group G is said to be pronormal if H and $H^g$$ are conjugate in $\langle H,H^g\rangle $$ for every $g\in G$$. In this survey, we discuss the role of pronormality for some subgroups of finite groups: Hall subgroups, subgroups of odd index, submaximal $\mathfrak X$$-subgroup, etc.

In this paper, we study the curvature estimate of the Hermitian–Yang–Mills flow on holomorphic vector bundles. In one simple case, we show that the curvature of the evolved Hermitian metric is uniformly bounded away from the analytic subvariety determined by the Harder–Narasimhan–Seshadri filtration of the holomorphic vector bundle.

In this paper, the local discontinuous Galerkin method is developed to solve the two-dimensional Camassa–Holm equation in rectangular meshes. The idea of LDG methods is to suitably rewrite a higher-order partial differential equations into a first-order system, then apply the discontinuous Galerkin method to the system. A key ingredient for the success of such methods is the correct design of interface numerical fluxes. The energy stability for general solutions of the method is proved. Comparing with the Camassa–Holm equation in one-dimensional case, there are more auxiliary variables which are introduced to handle high-order derivative terms. The proof of the stability is more complicated. The resulting scheme is high-order accuracy and flexible for arbitrary h and p adaptivity. Different types of numerical simulations are provided to illustrate the accuracy and stability of the method.

We study the phase transition of Kähler Ricci-flat metrics on some open Calabi–Yau spaces with the help of the images of moment maps of natural torus actions on these spaces.