Let M³ be a three-dimensional almost coKähler manifold such that the Ricci curvature of the Reeb vector field is invariant along the Reeb vector field. In this paper, we obtain some classification results of M³ for which the Ricci tensor is η-parallel or the Riemannian curvature tensor is harmonic.

In this paper we use the Bott residue formula in equivariant cohomology to show a formula for the algebraic degree in semidefinite programming.

We show that one-dimensional circle is the only case for closed smooth metric measure spaces with nonnegative Bakry-Émery Ricci curvature whose spectrum of the weighted Laplacian has an optimal positive upper bound. This result extends the work of Hang-Wang in the manifold case (Int. Math. Res. Not. 18 (2007), Art. ID rnm064, 9pp).

Let f denote a transcendental meromorphic function with N(r, f) = S(r, f) and k be an integer. By using methods different from others, we have been able to derive several new results and pose some new conjectures that relate to the yet to be resolved conjecture concerning the quantitative estimates on the zeros of ff^(k)-b, for a non-vanishing small function b.

A compact Riemann surface X of genus g ≥ 2 is called asymmetric or pseudo-real if it admits an anticonformal automorphism but no anticonformal involution. The order d = #(δ) of an anticonformal automorphism δ of such a surface is divisible by 4. In the particular case where d = 4, δ is a pseudo-symmetry and the surface is called pseudo-symmetric.

A Riemann surface X is said to be p-hyperelliptic if it admits a conformal involution ρ for which the orbit space X/<ρ> has genus p. This notion is the particular case of so called cyclic (q,n)-gonal surface which is defined as the one admitting a conformal automorphism φ of prime order n such that X/φ has genus q. We are interested in possible values of n and q for which an asymmetric surface of given genus g ≥ 2 is (q,n)-gonal, and possible values of p for which the surface is p-hyperelliptic. Up till now, this problem was solved in the case where the surface is asymmetric and pseudo-symmetric. If an asymmetric Riemann surface X is not pseudo-symmetric then any anticonformal automorphism of X has order divisible by 2^sn for s ≥ 3 and n = 1 or n being an odd prime. In this paper we give the necessary and sufficient conditions on the existence of an asymmetric Riemann surface with the full automorphism group being G = Z_2^sn, and we study (q,n)-gonal automorphisms and p-hyperelliptic involutions in G.

We generalize Earle-Li's polydisk theorem and embedding theorem, and study isometries from the unit disk to infinite dimensional Teichmüller spaces. We also give a simple proof that for any non-Strebel point τ, there exist infinitely many real analytic geodesic disks through τ and the basepoint in infinitely dimensional Teichmüller spaces.

Let {a,b} and {c,d} be two pairs of bounding simple closed curves on an oriented surface which intersect nontrivialy. We prove that if these pairs are invariant under the action of an orientation reversing involution, then the corresponding bounding pair maps generate a free group. This supports the conjecture stated by C. Leininger and D. Margalit that any pair of elements of the Torelli group either commute or generate a free group.

In this paper, we investigate the initial value problem (IVP henceforth) associated with the generalized damped Boussinesq equation with double rotational inertia $$\left\{\begin{array}{ll} u_{tt}+\gamma\Delta^2 u_{tt}-a\Delta u_{tt}-2b\Delta u_t-\alpha\Delta^3u+\beta\Delta^2 u-\Delta u=\Delta f(u),\quad x \in\mathbf{R}^n, \; t>0, \\ u(x,0)=u_0(x),\quad u_t(x,0)=u_1(x),\quad x \in\mathbf{R}^n. \end{array}\right.$$ Based on decay estimates of solutions to the corresponding linear equation, we establish the decay estimates and the pointwise estimates by using Fourier transform. Under small condition on the initial data, we obtain the existence and asymptotic behavior of global solutions in the corresponding Sobolev spaces by time weighted norms technique and the contraction mapping principle.

We discuss complete noncompact hypersurfaces in the Euclidean space Rⁿ⁺¹ with finite total curvature. We obtain vanishing result and finiteness theorem for the space of L² harmonic 2-forms. These results are generalized versions of results for L² harmonic 1-forms.

Let u: (M, g) → (N, h) be a map between Riemannian manifolds (M, g) and (N, h). The p-bienergy of u is τ_p(u) = ∫_M|τ(u)|^p dν_g, where τ(u) is the tension field of u and p > 1. Critical points of τ_p are called p-biharmonic maps and isometric p-biharmonic maps are called p-biharmonic submanifolds. When p = 2, p-biharmonic submanifolds are biharmonic submanifolds and in recent years many nonexistence results are found for biharmonic submanifolds in nonpositively curved manifolds. In this paper we will study the nonexistence result for general p-biharmonic submanifolds.

The fold-and-cut theorem states that one can find a flat folding of paper, so that one complete straight cut on the folding creates any desired polygon. We extend this problem to curved origami for piecewise C¹ simple closed curves. Many of those curves on paper turn out to be cut by a straight plane after we fold the paper into a conical shape—the surface consists of half-lines with a common vertex. Let γ: I → R² be a piecewise C¹ simple closed curve such that there exists a parametrization γ(ψ)= (r(ψ) cos ψ, r(ψ) sin ψ) on ψ $\in$ [0,2π) for a Lipschitz continuous function r: R→ (0, ∞). We prove that there exists a conical folding of the plane so that γ can be cut by a plane on the folding if a certain condition on angular total variation holds.

In this paper, we prove the sublinear tracking property in Thurston's metric for sample paths of random walks on mapping class group.

A trajectory for a Sasakian magnetic field, which is a generalization of geodesics, on a real hypersurface in a complex hyperbolic space CHⁿ is said to be extrinsic circular if it can be regarded as a circle as a curve in CHⁿ. We study how the moduli space of extrinsic circular trajectories, which is the set of their congruence classes, on a totally η-umbilic real hypersurface is contained in the moduli space of circles in CHⁿ. From this aspect we characterize tubes around totally geodesic complex hypersurfaces CH^n-1 in CHⁿ by some properties of such trajectories.