Four-Manifolds, Curvature Bounds, and Convex Geometry

We calculate Perelman's invariant for compact complex surfaces and a few other smooth four-manifolds. We also prove some results concerning the dependence of Perelman's invariant on the smooth structure.

An odd Seiberg-Witten invariant imposes bounds on the signature of a closed, almost complex 4-manifold with vanishing first Chern class. This applies in particular to symplectic 4-manifolds of Kodaira dimension zero.

By using the gluing formulae of the Seiberg-Witten invariant, we show the nonexistence of Einstein metric on manifolds obtained from a 4-manifold with nontrivial Seiberg-Witten invariant by performing sufficiently many connected sums or appropriate surgeries along circles or homologically trivial 2-spheres with closed oriented 4-manifolds with negative definite intersection form.

This article proves a uniform exponential decay estimate for Seiberg-Witten equations on non-compact 4-manifolds with exact symplectic ends of bounded geometry. This is an extension of the analysis for asymptotically flat almost K\"ahler (AFAK) structures by Kronheimer and Mrowka. As an application, we construct an invariant for smooth foliations without holonomy-invariant transverse measure, which takes value in the boundary-stable version of the monopole Floer homology grou...

This paper has been withdrawn by the author, due to an error in the proof of Theorem 3.8.

The study of the $k$-th elementary symmetric function of the Weyl-Schouten curvature tensor of a Riemannian metric, the so called $\sigma_k$ curvature, has produced many fruitful results in conformal geometry in recent years, especially when the dimension of the underlying manifold is 3 or 4. In these studies, the deforming conformal factor is considered to be a solution of a fully nonlinear elliptic PDE. Important advances have been made in recent years in the understanding ...

Seiberg-Witten theory is used to obtain new obstructions to the existence of Einstein metrics on 4-manifolds with conical singularities along an embedded surface. In the present article, the cone angle is required to be of the form 2(pi)/p, where p is a positive integer, but we conjecture that similar results will also hold in greater generality.

We show that every smooth closed oriented four-manifold admits a decomposition into two co- dimension zero submanifolds with common boundary. Each of these submanifolds carries a structure of a symplectic manifold with pseudo-convex boundary. This imply, in particular, that every smooth closed simply-connected four-manifold is a Stein domain in the the complement of a certain contractible 2-complex.

In this paper, we demonstrate a relation among Seiberg-Witten invariants which arises from embedded surfaces in four-manifolds whose self-intersection number is negative. These relations, together with Taubes' basic theorems on the Seiberg-Witten invariants of symplectic manifolds, are then used to prove the symplectic Thom conjecture: a symplectic surface in a symplectic four-manifold is genus-minimizing in its homology class. Another corollary of the relations is a general ...

We compute the Yamabe invariants for a new infinite class of closed $4$-dimensional manifolds by using a "twisted" version of the Seiberg-Witten equations, the $\mathrm{Pin}^-(2)$-monopole equations. The same technique also provides a new obstruction to the existence of Einstein metrics or long-time solutions of the normalised Ricci flow with uniformly bounded scalar curvature.