arXiv research

The paper studies Riemannian four-manifolds and their twistor spaces using a moving frame approach.

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…

Weyl energy decreases for connected sums of certain four-manifolds.

A local classification of locally conformal flat Riemannian Einstein-like four-manifolds as well as a local classification of all locally conformal flat Riemannian four-manifolds for which all Jacobi operators have parallel eigenspaces along every geodesic is given. Non-trivial explicit examples are presented. The prob…

Study classifies certain Einstein 4-manifolds with twistorial properties.

We define a holographic dual to the Donaldson-Witten topological twist of $\mathcal{N}=2$ gauge theories on a Riemannian four-manifold. This is described by a class of asymptotically locally hyperbolic solutions to $\mathcal{N}=4$ gauged supergravity in five dimensions, with the four-manifold as conformal boundary. Und…

The paper proves properties of complex surfaces and their curvature.

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

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.

Study on metrics maximizing eigenvalues of Paneitz operator on 4-manifolds.

We classify totally geodesic and parallel hypersurfaces of four-dimensional non-reductive homogeneous pseudo-Riemannian manifolds.

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

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

We provide a local classification of self-dual Einstein Riemannian four manifolds admitting a positively oriented Hermitian structure and characterize those which carry a hyperhermitian, non-hyperkählerian structure compatible with the negative orientation. We finally show that self-dual Einstein 4-manifolds obtained a…

Non-trivial examples of Riemannian almost product structures are constructed on the product bundle of the positive and negative twistor spaces of an oriented Riemannian four-manifold. The Gil-Medrano and Naveira types of these structures are determined and a geometric interpretation of the corresponding classes is give…

Review of gravitational instantons in physics.

We define a new formal Riemannian metric on a conformal classes of four-manifolds in the context of the $σ_2$-Yamabe problem. Exploiting this new variational structure we show that solutions are unique unless the manifold is conformally equivalent to the round sphere.

In this paper we determine the Gray-Hervella classes of the compatible almost complex structures on the twistor spaces of oriented Riemannian four-manifolds considered by G. Deschamps

We develop a gluing procedure designed to obtain canonical metrics on connected sums of Einstein four-manifolds. The main application is an existence result, using two well-known Einstein manifolds as building blocks: the Fubini-Study metric on $\mathbb{CP}^2$ and the product metric on $S^2 \times S^2$. Using these met…

In this paper we describe the oriented Riemannian four-manifolds $M$ for which the Atiyah-Hitchin-Singer or Eells-Salamon almost complex structure on the twistor space ${\mathcal Z}$ of $M$ determines a harmonic map from ${\mathcal Z}$ into its twistor space.

We study almost Hermitian 4-manifolds with holonomy algebra, for the canonical Hermitian connection, of dimension at most one. We show how Riemannian 4-manifolds admitting five orthonormal symplectic forms fit therein and classify them. In this set-up we also fully describe almost Kaehler 4-manifolds.

The paper proves unique properties of Riemannian twistor spaces under specific curvature conditions.

Weakly Einstein Kähler surfaces are characterized and classified.

New examples of weakly Einstein conformal products are constructed.

Short note proves Poincaré inequality for 4-manifold forms.

We investigate the low-energy behavior of the gradient flow of the $L^2$ norm of the Riemannian curvature on four-manifolds. Specifically, we show long time existence and exponential convergence to a metric of constant sectional curvature when the initial metric has positive Yamabe constant and small initial energy.

In this paper we construct Riemannian metrics and weight functions over Casson handles. We show that the corresponding Atiyah-Hitchin-Singer complexes are Fredholm for some class of Casson handles of bounded type. Using these, the Yang-Mills moduli spaces are constructed as finite dimensional smooth manifolds over Cass…

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…

New metric found for 4-manifolds with specific properties.

Classifies Heisenberg-invariant self-dual Einstein manifolds with explicit metrics.

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…

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

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…

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

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.

This is a survey of old and new results on the problem when a compatible almost complex structure on a Riemannian manifold is a harmonic section or a harmonic map from the manifold into its twistor space. In this context, a special attention is paid to the Atiyah-Hitchin-Singer and Eells-Salamon almost complex structur…

In the complex-Riemannian framework we show that a conformal manifold containing a compact, simply-connected, null-geodesic is conformally flat. In dimension 3 we use the LeBrun correspondence, that views a conformal 3-manifold as the conformal infinity of a selfdual four-manifolds. We also find a relation between the …

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 finds a positive lower bound for the extra field in Yang-Mills equations on closed 4-manifolds.

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

In this paper we develop a bubble tree structure for a degenerating class of Riemannian metrics satisfying some global conformal bounds on compact manifolds of dimension 4. Applying the bubble tree structure, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and a diameter bound f…

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…

Let $M$ be a connected, simply connected, oriented, closed, smooth four-manifold which is spin (or equivalently having even intersection form) and put $M^\times:=M\setminus\{{\rm point}\}$.In this paper we prove that if $X^\times$ is a smooth four-manifold homeomorphic but not necessarily diffeomorphic to $M^\times$ (m…

If $M$ is the underlying smooth oriented $4$-manifold of a Del Pezzo surface, we consider the set of Riemannian metrics $h$ on $M$ such that $W^+(ω, ω)> 0$, where $W^+$ is the self-dual Weyl curvature of $h$, and $ω$ is a non-trivial self-dual harmonic $2$-form on $(M,h)$. While this open region in the space of Riemann…

Extends knot surgery to exotic four-manifolds.

We extend a result of M. Katz on conformal systoles to all four-manifolds with b^+=1 which have odd intersection form. The same result holds for all four-manifolds with b^+=1 with even intersection form and which are symplectic or satisfy the so-called 5/4-conjecture.