Jeffrey Streets

Generalized Kähler geometry and the pluriclosed flow

Abstract In Streets and Tian (2010) [1] the authors introduced a parabolic flow for pluriclosed metrics, referred to as pluriclosed flow. We also demonstrated in Streets and Tian (2010) (preprint) [2] that this flow, after certain gauge transformations, gives a class of solutions to the renormalization group flow of the nonlinear sigma model with B-field. Using these transformations, we show that our pluriclosed flow preserves generalized Kahler structures in a natural way. Equivalently, when coupled with a nontrivial evolution equation for the two complex structures, the B-field renormalization group flow also preserves generalized Kahler structure. We emphasize that it is crucial to evolve the complex structures in the right way to establish this fact.

Pluriclosed flow, Born-Infeld geometry, and rigidity results for generalized Kähler manifolds

ABSTRACT We prove long time existence and convergence results for the pluriclosed flow, which imply geometric and topological classification theorems for generalized Kähler structures. Our approach centers on the reduction of pluriclosed flow to a degenerate parabolic equation for a (1, 0)-form, introduced in [30]. We observe a number of differential inequalities satisfied by this system which lead to a priori L∞ estimates for the metric along the flow. Moreover we observe an unexpected connection to “Born-Infeld geometry” which leads to a sharp differential inequality which can be used to derive an Evans-Krylov type estimate for the degenerate parabolic system of equations. To show convergence of the flow we generalize Yaus oscillation estimate to the setting of generalized Kähler geometry.

Evans–Krylov Estimates for a nonconvex Monge–Ampère equation

We establish Evans–Krylov estimates for certain nonconvex fully nonlinear elliptic and parabolic equations by exploiting partial Legendre transformations. The equations under consideration arise in part from the study of the “pluriclosed flow” introduced by Streets and Tian (Int Math Res Not 16:3101–3133, 2010).

A Conformally Invariant Gap Theorem in Yang–Mills Theory

We show a sharp conformally invariant gap theorem for Yang–Mills connections in dimension 4 by exploiting an associated Yamabe-type problem.

A formal Riemannian structure on conformal classes and uniqueness for the

We define a new formal Riemannian metric on a conformal classes of four-manifolds in the context of the

\sigma_{2}

\sigma_2

-Yamabe problem

-Yamabe problem. Exploiting this new variational structure we show that solutions are unique unless the manifold is conformally equivalent to the round sphere.

Collapsing in the L 2 Curvature Flow

We show some results for the L2 curvature flow linked by the theme of addressing collapsing phenomena. First we show long time existence and convergence of the flow for SO(3)-invariant initial data on S3, as well as a long time existence and convergence statement for three-manifolds with initial L2 norm of curvature chosen small with respect only to the diameter and volume, which are both necessary dependencies for a result of this kind. In the critical dimension n = 4 we show a related low-energy convergence statement with an additional hypothesis. Finally we exhibit some finite time singularities in dimension n ≥ 5, and show examples of finite time singularities in dimension n ≥ 6 which are collapsed on the scale of curvature.