Inspired by the Gromov-Hausdorff distance, we define a new notion called the intrinsic flat distance between oriented m dimensional Riemannian manifolds with boundary by isometrically embedding the manifolds into a common metric space, measuring the flat distance between them and taking an infimum over all isometric embeddings and all common metric spaces. This is made rigorous by applying Ambrosio-Kirchheim’s extension of FedererFleming’s notion of integral currents to arbitrary metric spaces. We prove the intrinsic flat distance between two compact oriented Riemannian manifolds is zero iff they have an orientation preserving isometry between them. Using the theory of AmbrosioKirchheim, we study converging sequences of manifolds and their limits, which are in a class of metric spaces that we call integral current spaces. We describe the properties of such spaces including the fact that they are countably Hm rectifiable spaces and present numerous examples.

In this paper we survey certain aspects of the classical Weil-Petersson metric and its generalizations. Being a natural L^2 metric on the parameter space of a family of complex manifolds or holomorphic vector bundles which admit some canonical metrics, the Weil-Petersson metric is well defined when the automorphism group of each fiber is discrete and the curvature of the Weil-Petersson metric can be computed via certain integrals over each fiber. We will discuss the case when these fibers have continuous automorphism groups. We also discuss the relation between the Weil-Petersson metric and energy of harmonic maps.

We extend Donaldson’s diagonalization theorem to intersection forms with certain local coefficients, under some constraints. This provides new examples of non-smoothable topological 4-manifolds.

Inthisarticle,weprovideasufficientandnecessaryconditiononparabolic isometries of positive translation lengths on complete visibility CAT(0) spaces. One of the consequences is that each parabolic isometry of a complete simply connected visibility manifold of nonpositive sectional curvature has zero translation length. Applications on the geometry of open negatively curved manifolds will also be discussed.

In this paper, we establish an analytic foundation for a fully non-linear equation 2 1 = f on manifolds with metrics of positive scalar curvature and apply it to give a (rough) classification of such manifolds. A crucial point is a simple observation that this equation is a degenerate elliptic equation without any condition on the sign of f and it is elliptic not only for f > 0 but also for f < 0. By defining a Yamabe constant Y2,1 with respect to this equation, we show that a manifold with metrics of positive scalar curvature admits a conformal metric of positive scalar curvature and positive 2-scalar curvature if and only if Y2,1 > 0. We give a complete solution for the corresponding Yamabe problem. Namely, let g0 be a positive scalar curvature metric, then in its conformal class there is a conformal metric with 2(g) = 1(g), for some constant . Using these analytic results, we give a rough classification of the space of manifolds with metrics of positive scalar curvature.