Related papers: Complete noncompact G2-manifolds with ALC asymptot…
This article develops the deformation theory of asymptotically cylindrical (ACyl) associative submanifolds in ACyl $G_2$-manifolds, laying the foundation for the gluing of ACyl associative submanifolds in twisted connected sum…
In an earlier paper, we proved that given an asymptotically cylindrical G_2-manifold M with a Calabi-Yau boundary X, the moduli space of coassociative deformations of an asymptotically cylindrical coassociative 4-fold C in M with a fixed…
We exhibit examples of closed Riemannian 7-manifolds with holonomy G_2 such that the underlying manifolds are diffeomorphic but whose associated G_2-structures are not homotopic. This is achieved by defining two invariants of certain…
We obtain first-order equations for G_2 holonomy of a wide class of metrics with S^3\times S^3 principal orbits and SU(2)\times SU(2) isometry, using a method recently introduced by Hitchin. The new construction extends previous results,…
Complete Riemannian metrics with holonomy group $G_2$ are constructed on the manifolds obtained by deformations of cones over $S^3 \times S^3$.
Let $B$ be a K\"ahler-Einstein Fano manifold, and $L \to B$ be a suitable root of the canonical bundle. We give a construction of complete Calabi-Yau metrics and gradient shrinking, steady, and expanding K\"ahler-Ricci solitons on the total…
Inspired by considerations in $M$-theory, we prove the equivalence between the moduli spaces of (suitably complexified) torsion free $G_{2}$-structures on 7-manifolds which are families of hyperK\"ahler ALE 4-manifolds fibered over compact…
We prove a uniqueness result for asymptotically conical (AC) gradient shrinking solitons for the Laplacian flow of closed G_2-structures: If two gradient shrinking solitons to Laplacian flow are asymptotic to the same closed G_2-cone, then…
This is the author's Ph.D. thesis, submitted to the University of Leipzig. It deals with the $L^2$ Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold. The main body of the…
It is found the most general local form of the 11-dimensional supergravity backgrounds which, by reduction along one isometry, give rise to IIA supergravity solutions with a RR field and a non trivial dilaton, and for which the condition…
We show that simply connected toric hyperK\"ahler metrics of finite topological type with maximal volume growth are generically quasi-asymptotically conical, which allows us to compute explicitly their reduced $L^2$-cohomology groups. In…
We study the space of L^2 harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type. These metrics generalize the geometric structures at infinity of several different well-known classes of metrics, including…
We investigate the collapsing geometry of hyperk\"ahler 4-manifolds. As applications we prove two well-known conjectures in the field. (1) Any collapsed limit of unit-diameter hyperk\"ahler metrics on the K3 manifold is isometric to one of…
In this paper, we investigate the geometry of asymptotically flat manifolds with controlled holonomy. We show that any end of such manifold admits a torus fibration over an ALE end. In addition, we prove a Hitchin-Thorpe inequality for…
We obtain two in a sense dual to each other results: First, that the capacity dimension of every compact, locally self-similar metric space coincides with the topological dimension, and second, that the asymptotic dimension of a metric…
We develop the deformation theory of instantons on asymptotically conical $G_2$-manifolds, where an asymptotic connection at infinity is fixed. A spinorial approach is adopted to relate the space of deformations to the kernel of a twisted…
We derive general expressions for the Kaehler form of the L^2-metric in terms of standard 2-forms on vortex moduli spaces. In the case of abelian vortices in gauged linear sigma-models, this allows us to compute explicitly the Kaehler class…
This is a survey paper. We explain the known constructions for two geometrically different classes of examples of compact Riemannian 7-manifolds with holonomy G2. One method uses resolutions of singularities of appropriately chosen…
This is our second paper in a series to study gravitational instantons, i.e. complete hyperk\"aler 4-manifolds with faster than quadratic curvature decay. We prove two main theorems: 1.The asymptotic rate of gravitational instantons to the…
We construct large families of new collapsing hyperk\"ahler metrics on the K3 surface. The limit space is the quotient of a flat 3-torus by an involution. Away from finitely many exceptional points the collapse occurs with bounded…