arXiv research

Researchers study heat flows of $G_{2}$-structures, proving short-time existence and uniqueness.

We give a classification for connected complete locally irreducible Riemannian manifolds with nonpositive curvature operator, which admit a nonzero closed or co-closed conformal Killing $L^{2}-$form. Moreover, we prove vanishing theorems for closed and co-closed conformal Killing $L^{2}-$forms on some complete Riemanni…

New insights on ACYT 6-manifolds reveal instanton conditions for curvature.

This study addresses transitions in conically singular associative submanifolds and their desingularizations.

We study the space of nearly Kähler structures on compact 6-dimensional manifolds. In particular, we prove that the space of infinitesimal deformations of a strictly nearly Kähler structure (with scalar curvature scal) modulo the group of diffeomorphisms, is isomorphic to the space of primitive co-closed (1,1)-eigenfor…

Theorem. Let M be a compact, connected, oriented smooth Riemannian n-manifold with non-empty boundary. Then the cohomology of the complex (Harm*(M),d) of harmonic forms on M is given by the direct sum H^p(Harm*(M),d) = H^p(M;R) + H^(p-1)(M;R) for p=0,1,...,n. When M is a closed manifold, a form is harmonic if and only …

We modify the Laplacian coflow of co-closed G2-structures - $\frac{d}{dt}ψ=Δψ$ where $ψ$ is the closed dual 4-form of a $G_{2}$-structure $\varphi$. The modified flow is now parabolic in the direction of closed forms upto diffeomorphisms. We then prove short time existence and uniqueness of solutions to the modified f…

Let $M$ be a closed oriented manifold of dimension $n$ and $ω$ a closed 1-form on it. We discuss the question whether there exists a Riemannian metric for which $ω$ is co-closed. For closed 1-forms with nondegenerate zeros the question was answered completely by Calabi in 1969. The goal of this paper is to give an answ…

This is a short note on generalized $G_2$-structures obtained as a consequence of a $T$-dual construction given in a previous work of the authors together with Leonardo Soriani. Given classical $G_2$-structure on certain seven dimensional manifolds, either closed or co-closed, we obtain integrable generalized $G_2$-str…

Study on dimensions of G2-structures on nilmanifolds, proving non-abelian automorphism groups.

In this paper, we study the theory of linearized gravity and prove the linear stability of Schwarzschild black holes as solutions of the vacuum Einstein equations. In particular, we prove that solutions to the linearized vacuum Einstein equations centered at a Schwarzschild metric, with suitably regular initial data, r…

The moduli space NK of infinitesimal deformations of a nearly Kähler structure on a compact 6-dimensional manifold is described by a certain eigenspace of the Laplace operator acting on co-closed primitive (1,1) forms. Using the Hermitian Laplace operator and some representation theory, we compute the space NK on all 6…

We introduce the notion of a hamiltonian 2-form on a Kaehler manifold and obtain a complete local classification. This notion appears to play a pivotal role in several aspects of Kaehler geometry. In particular, on any Kaehler manifold with co-closed Bochner tensor, the (suitably normalized) Ricci form is hamiltonian, …

It is well-known that every 6-dimensional strictly nearly Kähler manifold $(M,g,J)$ is Einstein with positive scalar curvature $scal>0$. Moreover, one can show that the space $E$ of co-closed primitive (1,1)-forms on $M$ is stable under the Laplace operator $Δ$. Let $E(a)$ denote the $a$-eigenspace of the restriction o…

We study the infinitesimal deformations of a proper nearly parallel G_2-structure and prove that they are characterized by a certain first order differential equation. In particular we show that the space of infinitesimal deformations modulo the group of diffeomorphisms is isomorphic to a subspace of co-closed $Λ^3_{27…

Nearly $G_2$-structures are unstable under a modified $G_2$-Laplacian co-flow.

The paper establishes conditions for harmonic forms on noncompact manifolds.

Given a six-dimensional symplectic manifold $(M, B)$, a nondegenerate, co-closed four-form $C$ introduces a dual symplectic structure $\widetilde{B} = *C$ independent of $B$ via the Hodge duality $*$. We show that the doubling of symplectic structures due to the Hodge duality results in two independent classes of nonc…

String backgrounds yield simplified Hull-Strominger system solutions.

The $7$-dimensional link $K$ of a weighted homogeneous hypersurface on the round $9$-sphere in $\mathbb{C}^5$ has a nontrivial null Sasakian structure which is contact Calabi-Yau, in many cases. It admits a canonical co-closed $\rm G_2$-structure $\varphi$ induced by the Calabi-Yau $3$-orbifold basic geometry. We disti…

The study constructs associative 3-folds in squashed 3-Sasakian manifolds.

Integrable flows on the Grassmannians Gr(N-1,N+1) are defined by the requirement of closedness of the differential N-1 forms $Ω_{N-1}$ of rank N-1 naturally associated with Gr(N-1,N+1). Gauge-invariant parts of these flows, given by the systems of the N-1 quasi-linear differential equations, describe coisotropic deform…

The primary aim of this thesis is to investigate metrics which are induced by a differential form and arise as a critical point of Hitchin's variational principle. Firstly, we investigate metrics associated with the structure group PSU(3) acting in its adjoint representation. We derive various obstructions to the exist…

Study solutions and singularities of G2-structures flows on specific manifolds.

New metrics link Kähler and pp-wave spacetimes.

Study linearized Schwarzschild spacetimes, proving decay of master quantities.

This paper constructs Poisson transforms and analyzes their properties on complex hyperbolic spaces.

Consider an oriented four-dimensional Lorentzian manifold $(\widetilde{M}^{3, 1}, \widetilde{g})$ and an oriented seven-dimensional Riemannian manifold $(M^{7}, g)$. We describe a class of decomposable eleven-dimensional supergravity backgrounds on the product manifold $({\mathcal{M}}^{10, 1}=\widetilde{M}^{3,1} \times…

Novel contact metric structures lead to supergravity solutions.