Several identities similar to the Schlaefli formula are established for tetrahedra in a space of constant curvature.

We use the construction of unfolded Seiberg-Witten Floer spectra of general 3-manifolds defined in our previous paper to extend the notion of relative Bauer-Furuta invariants to general 4-manifolds with boundary. One of the main purposes of this paper is to give a detailed proof of the gluing theorem for the relative invariants.

In this paper the authors study the hyperbolic geometric flow on Riemann surfaces. This new nonlinear geometric evolution equation was recently introduced by the first two authors, motivated by Einstein equation and Hamilton's Ricci flow. We prove that, for any given initial metric on ² in certain class of metrics, one can always choose suitable initial velocity symmetric tensor such that the solution exists for all time, and the scalar curvature corresponding to the solution metric <i>g</i> _<i>ij</i> keeps uniformly bounded for all time; moreover, if the initial velocity tensor is suitably large", then the solution metric <i>g</i> _<i>ij</i> converges to the flat metric at an algebraic rate. If the initial velocity tensor does not satisfy the condition, then the solution blows up at a finite time, and the scalar curvature <i>R</i>(<i>t</i>, <i>x</i>) goes to positive infinity as (<i>t</i>, <i>x</i>) tends to the blowup points, and a flow with surgery has

Fix a number $g>1$, let $S$ be a close surface of genus $g$ and $\Teich(S)$ be the Teichm\"uller space of $S$ endowed with the Weil-Petersson metric. In this paper we show that the Riemannian sectional curvature operator of $\Teich(S)$ is non-positive definite. As an application we show that any twist harmonic map from rank-one hyperbolic spaces $H_{Q,m}=Sp(m,1)/Sp(m)\cdot Sp(1)$ or $H_{O,2}=F_{4}^{-20}/SO(9)$ into $\Teich(S)$ is a constant.

For fixed subgroups Fix(ϕ) of automorphisms ϕ on hyperbolic 3-manifold groups π1(M), we observed that rk(Fix(ϕ))<2rk(π1(M)) and the constant 2 in the inequality is sharp; we also classify all possible groups Fix(ϕ).