arXiv research

Study classifies certain 7-manifolds with specific cohomology and proves positive Ricci curvature.

The abstract discusses constructing Sasaki-Einstein metrics on specific 7-manifolds.

Study calculates deformations of instantons on a specific $Spin(7)$-manifold.

We generalize a result of the author about the classification of 1-connected 7-manifolds and demonstrate its use by two concrete applications, one to 2-connected 7-manifolds (a new proof -- and slightly different formulation -- of an up to now unpublished Theorem by Crowley and Nordstroem and one to simply connected 7-…

Study cohomotopy sets of simply connected 7-manifolds using suspension decompositions.

Study $\mathrm{Spin}(7)$-manifolds with a 4-torus action using a symmetric matrix ansatz.

The paper identifies conditions for free circle actions on specific 7-manifolds.

Study explores Laplacian coflow versions on Calabi-Yau 7-manifolds.

The paper studies curvature identities and solitons on Spin(7)-manifolds.

Study instantons on asymptotically conical Spin(7)-manifolds, identifying deformation spaces.

Study shows only two topological configurations for Spin(7)-manifold fibrations, ruling out smooth Cayley fibrations.

Study classifies 7-manifolds with specific homology and finds nonconnected moduli spaces of positive Ricci curvature metrics.

New Spin(7) manifolds created from Calabi-Yau bundles.

Study on instantons in $G_2$ and $Spin(7)$ manifolds.

New examples of non-formal Sasaki-Einstein 7-manifolds and their submanifolds found.

Study Cayley fibrations on Bryant-Salamon $Spin(7)$ manifolds.

Desingularizes conically singular Cayley submanifolds.

The study finds infinitely many 7-manifolds with non-negative curvature but not homotopy equivalent to bundles.

We find a necessary and sufficient condition for a compact 7-manifold to admit a $\tilde G_2$-structure. As a result we find a sufficient condition for an open 7-manifold to admit a closed 3-form of $\tilde G_2$-type.

New $Spin(7)$-instantons constructed on Joyce's manifold.

We present a classification theorem for closed smooth spin 2-connected 7-manifolds M. This builds on the almost-smooth classification from the first author's thesis. The main additional ingredient is an extension of the Eells-Kuiper invariant for any closed spin 7-manifold, regardless of whether the spin characteristic…

We characterize compact locally conformal parallel $G_2$ (respectively, $Spin(7)$) manifolds as fiber bundles over $S^1$ with compact nearly Kähler (respectively, compact nearly parallel $G_2$) fiber. A more specific characterization is provided when the local parallel structures are flat.

Researchers create a compact 7-manifold with unique $G_2$ structure and first Betti number 1.

Constructs Spin(7) manifolds from self-dual Einstein 4-orbifolds.

Twistor theory applied to special holonomy manifolds.

Study free circle actions on specific 7-manifolds with positive Ricci curvature.

New Sasaki-Einstein 7-manifolds found, including rational homology 7-spheres and connected sums.

Deformations of singular Cayley submanifolds studied.

The paper finds CSC Sasaki metrics on specific 7-manifolds.

Joyce constructed examples of compact eight-manifolds with holonomy Spin(7), starting with a Calabi-Yau four-orbifold with isolated singular points of a special kind. That construction can be seen as the gluing of ALE Spin(7)-manifolds to each singular point of the Calabi-Yau four-orbifold divided by an anti-holomorphi…

In this article, a six-parameter family of highly connected 7-manifolds which admit an SO(3)-invariant metric of non-negative sectional curvature is constructed and the Eells-Kuiper invariant of each is computed. In particular, it follows that all exotic spheres in dimension 7 admit an SO(3)-invariant metric of non-neg…

Study on flows of G2-structures on contact Calabi-Yau 7-manifolds.

We show that on every Spin(7) manifold there always exists a unique linear connection with totally skew-symmetric torsion preserving a nontrivial spinor and the Spin(7) structure. We express its torsion and the Riemannian scalar curvature in terms of the fundamental 4-form. We present an explicit formula for the Rieman…

Any Spin(7)-manifold admits a metric connection \nabla^c with totally skew-symmetric torsion T^c preserving the underlying structure. We classify those with \nabla^c-parallel T^c\neq0 and non-Abelian isotropy algebra iso(T^c)<spin(7). These are isometric to either Riemannian products or homogeneous naturally reductive …

We calculate the integer cohomology ring and stable tangent bundle of a family of compact, 3-Sasakian 7-manifolds constructed by Boyer, Galicki, Mann, and Rees. Previously only the rational cohomology ring was known. The most important part of the cohomology ring is a torsion group that we describe explicitly and whose…

We give a new, connected-sum-like construction of Riemannian metrics with special holonomy G_2 on compact 7-manifolds. The construction is based on a gluing theorem for appropriate elliptic partial differential equations. As a prerequisite, we also obtain asymptotically cylindrical Riemannian manifolds with holonomy SU…

We prove that the moduli space of holonomy G_2-metrics on a closed 7-manifold is in general disconnected by presenting a number of explicit examples. We detect different connected components of the G_2-moduli space by defining an integer-valued analytic refinement of the nu-invariant, a Z/48-valued defect invariant of …

In this note we construct a first example of a closed 3-form of $\tilde G_2$-type on $S^3\times S^4$. We prove that $S^3\times S^4$ does not admit a homogeneous 3-form of $\tilde G_2$-type. Thus our example is a first example of a closed 3-form of $\tilde G_2$-type on a compact 7-manifold which is not stably homogeneou…

Study of instantons on Sasakian 7-manifolds using gauge fields.

Survey of G2-holonomy manifold constructions.

We find new examples of compact Spin(7)-manifolds using a construction of Joyce. The essential ingredient in Joyce's construction is a Calabi-Yau 4-orbifold with particular singularities admitting an antiholomorphic involution, which fixes the singularities. We search the class of well-formed quasismooth hypersurfaces …

M-theory on compact eight-manifolds with $\mathrm{Spin}(7)$-holonomy is a framework for geometric engineering of 3d $\mathcal{N}=1$ gauge theories coupled to gravity. We propose a new construction of such $\mathrm{Spin}(7)$-manifolds, based on a generalized connected sum, where the building blocks are a Calabi-Yau four…

In an earlier paper we showed that the space of deformations of a smooth, compact, orientable Harvey-Lawson submanifold HL in a G2 manifold M can be identified with the direct sum of the space of smooth functions and closed 2-forms on HL. In that paper, we also introduced a new class of Lagrangian-type 4-dimensional su…

We explicitly describe the solution of the G$_2$-Laplacian flow starting from an extremally Ricci-pinched closed G$_2$-structure on a compact 7-manifold and we investigate its properties. In particular, we show that the solution exists for all real times and that it remains extremally Ricci-pinched. This result holds m…

The main purpose of this paper is to give a mathematical definition of ``mirror symmetry'' for Calabi-Yau and G_2 manifolds. More specifically, we explain how to assign a G_2 manifold (M,φ,Λ), with the calibration 3-form φand an oriented 2-plane field Λ, a pair of parametrized tangent bundle valued 2 and 3-forms of M. …

This paper explores 4-planes in Spin(7) manifolds with symplectic structures.

Geometric flows study nearly parallel G2-structures on 3-Sasakian 7-manifolds.

New G_2-manifolds created from quotients of twisted connected sums.