arXiv research

Taubes proved that all compact oriented four-manifolds admit non-flat instantons. We show that there exists a non-compact oriented four-manifold having no non-flat instanton.

The study of rigidity theorems on 4-manifolds with boundary.

Study pinched self-dual Weyl curvature in compact 4-manifolds.

Classifies self-dual almost-Kähler 4-manifolds, proving uniqueness up to rescaling.

The study finds conditions for almost-Kähler 4-manifolds to be Kähler.

We show that under certain conditions, a nontrivial Riemannian submersion from positively curved four manifolds does not exist. This gives a partial answer to a conjecture due to Fred Wilhelm. We also prove a rigidity theorem for Riemannian submersions with totally geodesic fibers from compact four-dimensional Einstein…

Study classifies special 4D shapes with certain curvature.

We explain a new phenomenon on non compact complete Riemannian four manifolds, where d^+ image of one forms can not exhaust densely on L^2 self dual forms on each compact subset, if a certain L^2 self dual harmonic form exists. This leads to construct a new functional analytic framework on the Seiberg-Witten map.

The paper proves properties of complex surfaces and their curvature.

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.

Study classifies certain Einstein 4-manifolds with twistorial properties.

In this paper we study the Ricci flow on compact four-manifolds with positive isotropic curvature and with no essential incompressible space form. Our purpose is two-fold. One is to give a complete proof of Hamilton's classification theorem on four-manifolds with positive isotropic curvature and with no essential incom…

Review of gravitational instantons in physics.

Trisections help compute Heegaard Floer homology invariants.

Einstein 4-manifolds become conformally Kähler with positive scalar curvature.

In this note we prove that, under a weak condition, small deformations of a compact balanced manifold are also balanced. This condition is satisfied on the twistor space over a compact self-dual four manifold.

We prove that any compact four-manifold admits a Riemannian metric with negative isotropic curvature in the sense of Micallef and Moore.

No spin structures found in a hyperbolic 4D space.

Study pinches Weyl curvature on 4-manifolds, proving anti-self-duality.

New steady gradient Ricci solitons found on specific four-manifolds.

If a (possibly finite) compact Lie group acts effectively, locally linearly, and homologically trivially on a closed, simply-connected four-manifold with second Betti number at least three, then it must be isomorphic to a subgroup of S^1 x S^1, and the action must have nonempty fixed-point set. Our results strengthen a…

We consider a set of gauge-theoretic equations on closed oriented four-manifolds, which was introduced by Vafa and Witten. The equations involve a triple consisting of a connection and extra fields associated to a principal bundle over a closed oriented four-manifold. They are similar to Hitchin's equations over compac…

This research proves a topological inequality for symplectic four-manifolds using non-Abelian monopoles.

We study generalized Kaehler manifolds for which the corresponding complex structures commute and classify completely the compact generalized Kaehler four-manifolds for which the induced complex structures yield opposite orientations.

In this paper we prove the path connectedness of the moduli spaces of metrics with positive isotropic curvature on certain compact four-dimensional manifolds.

We generalize bridge trisections to surfaces in four-manifolds, linking them to braided links.

New operators and curvatures derived from embedded manifolds.

Study topological twists of massive SQCD with gauge group SU(2) and 3 or fewer fundamental hypermultiplets.

In this paper, we completely classify all compact 4-manifolds with positive isotropic curvature. We show that they are diffeomorphic to $\mathbb{S}^4,$ or $\mathbb{R}\mathbb{P}^4$ or quotients of $\mathbb{S}^3\times \mathbb{R}$ by a cocompact fixed point free subgroup of the isometry group of the standard metric of $\m…

Study evaluates topological contributions in massive SQCD on compact 4-manifolds.

The Kapustin-Witten equations relate to nonabelian Hodge theory on Kähler surfaces.

In this paper we classify the four dimensional gradient shrinking solitons under certain curvature conditions satisfied by all solitons arising from finite time singularities of Ricci flow on compact four manifolds with positive isotropic curvature. As a corollary we generalize a result of Perelman on three dimensional…

In the present paper, we establish a gluing construction for the Nahm pole solutions to the Kapustin-Witten equations over manifolds with boundaries and cylindrical ends. Given two Nahm pole solutions with some convergence assumptions on the cylindrical ends, we prove that there exists an obstruction class for gluing t…

Paper characterizes tamed and weakened tamed four-manifolds using a new technique.

We prove that Hitchin's generalized Kaehler structure on the moduli space of instantons over a compact, even generalized Kaehler four-manifold may be obtained by generalized Kaehler reduction, in analogy with the usual Kaehler case. The underlying reduction of Courant algebroids is a realization of Donaldson's $μ$-map …

We first provide an alternative proof of the classical Weitzneböck formula for Einstein four-manifolds using Berger curvature decomposition, motivated by which we establish a unified framework for a Weitzenböck formula for a large class of canonical metrics on four-manifolds (or a Weitzenböck formula for "Einstein metr…

This article presents a new and more elementary proof of the main Seiberg-Witten-based obstruction to the existence of Einstein metrics on smooth compact 4-manifolds. It also introduces a new smooth manifold invariant which conveniently encapsulates those aspects of Seiberg-Witten theory most relevant to the study of R…

Research on the least complex surface in certain 4D shapes.

In a recent paper, Park constructs certain exotic simply-connected four-manifolds with small Euler characteristics. Our aim here is to prove that the four-manifolds in his constructions are minimal.

We characteristize those Einstein four manifolds which are locally symmetric spaces of noncompact type. Namely they are four manifolds which admit solutions to the (non-Abelian) Seiberg Witten equations and satisty certain characterisitc number equality.

The study refines known counterexamples in 4D to satisfy certain inequalities.

We develop a notion of Einstein manifolds with skew torsion on compact, orientable Riemannian manifolds of dimension four. We prove an analogue of the Hitchin-Thorpe inequality and study the case of equality. We use the link with self-duality to study the moduli space of 1-instantons on the 4-sphere for a family of met…

In this note we prove that a four-dimensional compact oriented half-confor\-mally flat Riemannian manifold $M^4$ is topologically $\mathbb{S}^{4}$ or $\mathbb{C}\mathbb{P}^{2},$ provided that the sectional curvatures all lie in the interval $[\frac{3\sqrt{3}-5}{4},\,1].$ In addition, we use the notion of biorthogonal (…

We develop the calculus for hypersurface variations based on variation of the hypersurface defining function. This is used to show that the functional gradient of a new Willmore-like, conformal hypersurface energy agrees exactly with the obstruction to smoothly solving the singular Yamabe problem for conformally compac…

Survey on knot theory's impact on four-dimensional topology.

This article analyzes the interplay between symplectic geometry in dimension four and the invariants for smooth four-manifolds constructed using holomorphic triangles introduced in math.SG/0110169. Specifically, we establish a non-vanishing result for the invariants of symplectic four-manifolds, which leads to new proo…

This paper explores parallels between minimal surfaces and Einstein manifolds.

In this paper we develop analysis of the monopole maps over the universal covering space of a compact four manifold. We induce a property on local properness of the covering monopole map under the condition of closeness of the AHS complex. In particular we construct a higher degree of the covering monopole map when the…