Four-Manifolds, Curvature Bounds, and Convex Geometry

In the lecture notes, the author will survey the development of conformal geometry on four dimensional manifolds. The topic she chooses is one on which she has been involved in the past twenty or more years: the study of the integral conformal invariants on 4-manifolds and geometric applications. The development was heavily influenced by many earlier pioneer works; recent progress in conformal geometry has also been made in many different directions, here we will only present...

We introduce the notion of a special monopole class on a four-manifold. This is used to prove restrictions on the smooth structures of Einstein manifolds. As an application we prove that there are Einstein four-manifolds which are simply connected, spin, and satisfy the strict Hitchin--Thorpe inequality, and which are homeomorphic to manifolds without Einstein metrics. In the Appendix, written with J. Wehrheim, we use special monopole classes to discuss the classification o...

We introduce a new class of perturbations of the Seiberg-Witten equations. Our perturbations offer flexibility in the way the Seiberg-Witten invariants are constructed and also shed a new light to LeBrun's curvature inequalities.

This work is a sequel to our previous monograph arXiv:2010.15789 (to appear in AMS Memoirs), where we initiated our program to prove that the Bogomolov-Miyaoka-Yau inequality holds for closed, symplectic four-manifolds and, more generally, for closed, smooth four-manifolds with a Seiberg-Witten basic class. This inequality was first proved for compact, complex surfaces of general type by Miyaoka (1977) and Yau (1978). Our approach uses a version of Morse theory for a natural ...

This short paper gives a constraint on Chern classes of closed strictly pseudoconvex CR manifolds (or equivalently, closed holomorphically fillable contact manifolds) of dimension at least five. We also see that our result is ''optimal'' through some examples.

We show that the SO(3) monopole cobordism formula from Feehan and Leness (2002) implies that all smooth, closed, oriented four-manifolds with $b^1=0$ and $b^+\geq 3$ and odd with Seiberg-Witten simple type satisfy the superconformal simple type condition defined by Marino, Moore, and Peradze (1999) This implies the lower bound, conjectured by Fintushel and Stern (2001) on the number of Seiberg-Witten basic classes in terms of topological data.

Given a closed four-manifold with $b_1=0$ and a prime number $p$, we prove that for any mod $p^r$ basic class, the virtual dimension of the Seiberg-Witten moduli space is bounded above by $2r(p-1)-2$ under some conditions on $r$ and $b_2^+$. As an application, we obtain adjunction inequalities for embedded surfaces with negative self-intersection number.

We construct infinitely many examples of finite volume 4-manifolds with $T^3$ ends that do not admit any cusped asymptotically hyperbolic Einstein metrics yet satisfy a strict logarithmic version of the Hitchin-Thorpe inequality due to Dai-Wei. This is done by using estimates from Seiberg-Witten theory due to LeBrun as well as a method for constructing solutions to the Seiberg-Witten equations on noncompact manifolds due to Biquard. We also use constructions coming from the $...

The classical Seiberg-Witten equation in dimesion three and four can be generalized to an unifying framework, the generalized Seiberg-Witten (GSW) equation. It includes the anti-self dulaity (ASD) equation, the $\mathrm U(n)$-monopole equation, the Seiberg-Witten equation with multiple spinors, the Vafa-Witten equation, the complex ASD equation. In this article, we prove some vanishing results for solutions (called GSW monopoles and GSW Bogomolny monopoles, respectively) of t...

Using quantum field-theoretic arguments, Witten has established a relation between the Donaldson and Seiberg-Witten invariants of smooth four-manifolds. In this survey article, we describe the program to prove this relation using a moduli space of PU(2) = SO(3) monopoles as a cobordism between the Donaldson moduli space of anti-self-dual SO(3) connections and moduli spaces of U(1) monopoles. We provide an overview of some of our transversality and Uhlenbeck compactness result...