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We survey the known existence and non-existence results for $G_2$-instantons on non-compact cohomogeneity-1 $G_2$-manifolds and their consequences, including an explicit example of a family of $G_2$-instantons where bubbling, removable…

A GL(2, R) structure on an (n+1)-dimensional manifold is a smooth pointwise identification of tangent vectors with polynomials in two variables homogeneous of degree n. This, for even n=2k, defines a conformal structure of signature (k,…

$G_2$-monopoles are solutions to gauge theoretical equations on $G_2$-manifolds. If the $G_2$-manifolds under consideration are compact, then any irreducible $G_2$-monopole must have singularities. It is then important to understand which…

We study positive scalar curvature on the regular part of Riemannian manifolds with singular, uniformly Euclidean ($L^\infty$) metrics that consolidate Gromov's scalar curvature polyhedral comparison theory and edge metrics that appear in…

We describe some properties of noncompact Euclidean cone manifolds with cone angles less than c less than 2pi and singular locus a submanifold. More precisely, we describe its structure outside a compact set. As a corollary we classify…

We study the isoperimetric structure of Riemannian manifolds that are asymptotic to cones with non-negative Ricci curvature. Specifically, we generalize to this setting the seminal results of G. Huisken and S.-T. Yau on the existence of a…

We consider the M-theory lifts of configurations of type IIA D6-branes intersecting at angles. In supersymmetry preserving cases, the lifts correspond to special holonomy geometries, like conifolds and $G_2$ holonomy singularities.…

We provide the complete classification of seven-dimensional manifolds endowed with a closed non-parallel G$_2$-structure and admitting a transitive reductive group G of automorphisms. In particular, we show that the center of G is…

We study codimension two spacelike submanifolds contained into a general class of null hypersurfaces in generalized Robertson-Walker spacetimes, refer to as nullcones. In particular we analyze light cones and lightlike cylinders in…

We show that M-theory compactified on a compact Joyce 8-manifold of $Spin(7)$-holonomy, which yields an effective theory in $D = 3$ with $\N$ = 1 supersymmetry, admits at some special points in it moduli space a description in terms of type…

We study the existence of conformal metrics on non-compact Riemannian manifolds with non-compact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature…

Let $M_1$ and $M_2$ be two K\"ahler manifolds. We call $M_1$ and $M_2$ {\em relatives} if they share a non-trivial K\"ahler submanifold $S$, namely, if there exist two holomorphic and isometric immersions (K\"ahler immersions) $h_1: S\to…

We consider non perturbative effects in M-theory compactifications on a seven-manifold of G_2 holonomy arising from membranes wrapped on supersymmetric three-cycles. When membranes are wrapped on associative submanifolds they induce a…

Supersymmetric domain-wall spacetimes that lift to Ricci-flat solutions of M-theory admit generalized Heisenberg (2-step nilpotent) isometry groups. These metrics may be obtained from known cohomogeneity one metrics of special holonomy by…

Cohomogeneity one actions on irreducible Riemannian symmetric spaces of compact type are classified into three cases: Hermann actions, actions induced by the linear isotropy representation of a Riemannian symmetric space of rank 2, and…

It is a prominent conjecture (relating Riemannian geometry and algebraic topology) that all simply-connected compact manifolds of special holonomy should be formal spaces, i.e., their rational homotopy type should be derivable from their…

This article studies the geometry of moduli spaces of G2-manifolds, associative cycles, coassociative cycles and deformed Donaldson-Thomas bundles. We introduce natural symmetric cubic tensors and differential forms on these moduli spaces.…

We consider topology-changing transitions between 7-manifolds of holonomy G_2 constructed as a quotient of CY x S^1 by an antiholomorphic involution. We classify involutions for Complete Intersection CY threefolds, focussing primarily on…

In the first part of this note we study compact Riemannian manifolds (M,g) whose Riemannian product with R is conformally Einstein. We then consider compact 6--dimensional almost Hermitian manifolds of type W_1+W_4 in the Gray--Hervella…

This paper is concerned with the construction of special metrics on non-compact 4-manifolds which arise as resolutions of complex orbifold singularities. Our study is close in spirit to the construction of the hyperkaehler gravitational…