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

In this paper, we study the Nakano-positivity and dual-Nakano-positivity of certain adjoint vector bundles associated to ample vector bundles. As applications, we get new vanishing theorems about ample vector bundles. For example, we prove that if E is an ample vector bundle over a compact Khler manifold E , $ S^ kE\ts\det E $ is both Nakano-positive and dual-Nakano-positive for any E . Moreover, $ H^{n, q}(X, S^ kE\ts\det E)= H^{q, n}(X, S^ kE\ts\det E)= 0$ for any E . In particular, if E is a Griffiths-positive vector bundle, the naturally induced Hermitian vector bundle $(S^ kE\ts\det E, S^ kh\ts\det h) $ is both Nakano-positive and dual-Nakano-positive for any E .

In this paper we introduce the hyperbolic mean curvature flow and prove that the corresponding system of partial differential equations is strictly hyperbolic, and based on this, we show that this flow admits a unique short-time smooth solution and possesses the nonlinear stability defined on the Euclidean space with dimension larger than 4. We derive nonlinear wave equations satisfied by some geometric quantities related to the hyperbolic mean curvature flow. Moreover, we also discuss the relation between the equations for hyperbolic mean curvature flow and the equations for extremal surfaces in the Minkowski spacetime.

In this note we derive some consequences from the Mario-Vafa formula and the cut-and-join equation, these include unified simple proofs of the \lambda_g conjecture and some other Hodge integral identities. We also describe a proof of the ELSV formula relating Hurwitz numbers and Hodge integrals by using the cut-and-join equation, following our proof of the Mario-Vafa formula.

In this paper, we study effective recursion formulae for computing intersection numbers of mixed \psi and \psi classes on moduli spaces of curves. By using the celebrated WittenKontsevich theorem, we generalize MulaseSafnuk form of Mirzakhani's recursion and prove a recursion formula of higher WeilPetersson volumes. We also present recursion formulae to compute intersection pairings in the tautological rings of moduli spaces of curves.