In this paper, we observe that the brane functional studied in hep-th/9910245 can be used to generalize some of the works that Schoen and I [4] did many years ago. The key idea is that if a three dimensional manifold M has a boundary with strongly positive mean curvature, the effect of this mean curvature can influence the internal geometry of M. For example, if the scalar curvature of M is greater than certain constant related to this boundary effect, no incompressible surface of higher genus can exist.

In this article we show that for every finite area hyperbolic surface X of type (g,n) and any harmonic Beltrami differential μ on X , then the magnitude of μ at any point of small injectivity radius is uniform bounded from above by the ratio of the Weil–Petersson norm of μ over the square root of the systole of X up to a uniform positive constant multiplication. We apply the uniform bound above to show that the Weil–Petersson Ricci curvature, restricted at any hyperbolic surface of short systole in the moduli space, is uniformly bounded from below by the negative reciprocal of the systole up to a uniform positive constant multiplication. As an application, we show that the average total Weil–Petersson scalar curvature over the moduli space is uniformly comparable to -g as the genus g goes to infinity.

In this paper we establish the short-time existence and uniqueness theorem for hyperbolic geometric flow, and prove the nonlinear stability of hyperbolic geometric flow defined on the Euclidean space with dimension larger than 4. Wave equations satisfied by the curvatures are derived. The relations of the hyperbolic geometric flow with the Einstein equations and the Ricci flow are discussed.

We study the collapsing behavior of the Kähler–Ricci flow on a compact Kähler manifold X admitting a holomorphic submersion inherited from its canonical bundle. We show that the flow metric degenerates at exactly the rate of e^{-t} as predicted by the cohomology information, and so the fibres collapse at the optimal rate. Consequently, it leads to some analytic and geometric extensions to the regular case of works by J. Song and G. Tian. Its applicability to general Calabi–Yau fibrations will also be discussed in local settings.

Let φ∈C^0∩W_{1,2}(Σ,X) where Σ is a compact Riemann surface, X is a compact locally CAT(1) space, and W_{1,2}(Σ,X) is defined as in Korevaar-Schoen. We use the technique of harmonic replacement to prove that either there exists a harmonic map u: Σ → X homotopic to φ or there exists a nontrivial conformal harmonic map v: S^2 → X. To complete the argument, we prove compactness for energy minimizers and a removable singularity theorem for conformal harmonic maps.