Given a complex analytic function f on a Whitney stratified complex analytic variety of complex dimension n, whose real part Re (f) is Morse, we prove the existence of a stratified gradient-like vector field for Re (f) such that the unstable set of a critical point p on a stratum S of complex dimension s has real dimension m (p)+ ns as was conjectured by Goresky and MacPherson.

Based on the proof of Labastida-Mari {} o-Ooguri-Vafa conjecture\cite {lmov}, we derive an infinite product formula for Chern-Simons partition functions, the generating function of quantum $\fsl_N $ invariants. Some symmetry properties of the infinite product will also be discussed.

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 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 paper we investigate the one-dimensional hyperbolic mean curvature flow for closed plane curves. More precisely, we consider a family of closed curves S 1[0, T) 2 which satisfies the following evolution equation 2 F t 2 (u, t)= k (u, t) N(u, t)- (u, t),(u, t) S 1[0, T) with the initial data F (u, 0)= F 0 (u) a n d F t (u, 0)= f (u) N 0, where k is the mean curvature and is the unit inner normal vector of the plane curve F (u, t), f (u) and are the initial velocity and the unit inner normal vector of the initial convex closed curve Fo, respectively, and is given by ( 2 F s t, F t) T, in which stands for the unit tangent vector. The above problem is an initial value problem for a system of partial differential equations for F, it can be completely reduced to an initial value problem for a single partial differential equation for its support function. The latter equation is a hyperbolic Monge-Ampere

We prove that a general miraculous cancellation formula, the divisibility of certain characteristic numbers, and some other topological results related to the generalized Rochlin invariant, the -invariant and the holonomies of certain determinant line bundles, are consequences of the modular invariance of elliptic operators on loop space.

Integral simplicial volume is a homotopy invariant of oriented closed connected manifolds, defined as the minimal weighted number of singular simplices needed to represent the fundamental class with integral coefficients. We show that odd-dimensional spheres are the only manifolds with integral simplicial volume equal to 1. Consequently, we obtain an elementary proof that, in general, the integral simplicial volume of (triangulated) manifolds is not computable.

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We will show that for any noncompact arithmetic hyperbolic m-manifold with m>3, and any compact arithmetic hyperbolic m-manifold with m > 4 that is not a 7-dimensional one defined by octonions, its fundamental group is not locally extended residually finite (LERF). The main ingredient in the proof is a study on abelian amalgamations of hyperbolic 3-manifold groups. We will also show that a compact orientable irreducible 3-manifold with empty or tori boundary supports a geometric structure if and only if its fundamental group is LERF.

We investigate spectral properties of the Laplacian in <i>L</i>² (<i>Q</i>), where <i>Q</i> is a tubular region in ³ of a fixed cross section, and the boundary conditions combined a Dirichlet and a Neumann part. We analyze two complementary situations, when the tube is bent but not twisted, and secondly, it is twisted but not bent. In the first case we derive sufficient conditions for the presence and absence of the discrete spectrum showing, roughly speaking, that they depend on the direction in which the tube is bent. In the second case we show that a constant twist raises the threshold of the essential spectrum and a local slowndown of it gives rise to isolated eigenvalues. Furthermore, we prove that the spectral threshold moves up also under a sufficiently gentle periodic twist.