Let π: M ⁿ → P ⁿ be an n-dimensional small cover over P ⁿ and λ: ℱ(P ⁿ) → ℤ₂ ⁿ be its characteristic function. The author uses the symbol c(λ) to denote the cardinal number of the image Im(λ). If c(λ) = n + 1 or n + 2, then a necessary and sufficient condition on the existence of spin structure on M ⁿ is given. As a byproduct, under some special conditions, the author uses the second Stiefel-Whitney class to detect when P ⁿ is n-colorable or (n + 1)-colorable.

In this paper, the author proves a generalized Donaldson-Uhlenbeck-Yau theorem for twisted holomorphic chain on a non-compact Kähler manifold. As an application, the author obtains a Bogomolov type Chern numbers inequality for semistable twisted holomorphic chain.

In this paper, the authors introduce a definition of the Schwarzian derivative of any locally univalent harmonic mapping defined on a simply connected domain in the complex plane. Using the new definition, the authors prove that any harmonic mapping f which maps the unit disk onto a convex domain has Schwarzian norm ∥S _f∥ ≤ 6. Furthermore, any locally univalent harmonic mapping f which maps the unit disk onto an arbitrary regular n-gon has Schwarzian norm $\left\| {{S_f}} \right\| \leq {8 \over 3}$.

For an upper triangular matrix ring, an explicit ladder of height 2 of triangle functors between homotopy categories is constructed. Under certain conditions, the author obtains a localization sequence of homotopy categories of acyclic complexes of injective modules.

In this paper, the author extends Peter Li and Tian Gang’s results on the heat kernel from projective varieties to analytic varieties. The author gets an upper bound of the heat kernel on analytic varieties and proves several properties. Moreover, the results are extended to vector bundles. The author also gets an upper bound of the heat operators of some Schröndinger type operators on vector bundles. As a corollary, an upper bound of the trace of the heat operators is obtained.

In this paper, the author characterizes the subgroups of a finite metacyclic group K by building a one to one correspondence between certain 3-tuples (k,l,β) ∈ ℕ³ and all the subgroups of K. The results are applied to compute some subgroups of K as well as to study the structure and the number of p-subgroups of K, where p is a fixed prime number. In addition, the author gets a factorization of K, and then studies the metacyclic p-groups, gives a different classification, and describes the characteristic subgroups of a given metacyclic p-group when p ≥ 3. A “reciprocity” relation on enumeration of subgroups of a metacyclic group is also given.

This paper deals with constrained trace, matrix and constrained matrix Harnack inequalities for the nonlinear heat equation ω _t = Δω + aω ln ω on closed manifolds. A new interpolated Harnack inequality for ω _t = Δω − ω ln ω+εRω on closed surfaces under ε-Ricci flow is also derived. Finally, the author proves a new differential Harnack inequality for ω _t = Δω − ω ln ω under Ricci flow without any curvature condition. Among these Harnack inequalities, the correction terms are all time-exponential functions, which are superior to time-polynomial functions.

Let M ⁿ(n ≥ 4) be an oriented compact submanifold with parallel mean curvature in an (n + p)-dimensional complete simply connected Riemannian manifold N ^n+p. Then there exists a constant δ(n, p) 2 (0, 1) such that if the sectional curvature of N satisfies ${\overline K _N} \in \;\,\left[ {\delta \left( {n,p} \right),\;1} \right]$, and if M has a lower bound for Ricci curvature and an upper bound for scalar curvature, then N is isometric to S ^n+p. Moreover, M is either a totally umbilic sphere ${S^n}({1 \over {\sqrt {1 + {H^2}} }})$, a Clifford hypersurface ${S^m}({1 \over {\sqrt {2(1 + {H^2})} }})\; \times \;{S^m}({1 \over {\sqrt {2(1 + {H^2})} }})$ in the totally umbilic sphere ${S^{n+1}}({1 \over {\sqrt {1 + {H^2}} }})$ with n = 2m, or ${\mathbb{C}{\rm{P}}^2}\left( {{4 \over 3}\left( {1 + {H^2}} \right)} \right)$ in ${S^7}({1 \over {\sqrt {1 + {H^2}} }})$. This is a generalization of Ejiri’s rigidity theorem.

In this paper the authors investigate the boundedness and almost periodicity of solutions of semilinear parabolic equations with boundary degeneracy. The equations may be weakly degenerate or strongly degenerate on the lateral boundary. The authors prove the existence, uniqueness and global exponential stability of bounded entire solutions, and also establish the existence theorem of almost periodic solutions if the data are almost periodic.