arXiv research

The paper finds new $\mathrm{Spin}(7)$ metrics with specific orbits.

Sharp inequality for complex projective space's systoles under scalar curvature bounds.

New transitions found in Spin(7) holonomy metrics related to a dynamical system.

The article constructs Spin(7) metrics with Aloff--Wallach spaces as orbits.

The paper proves a mass theorem for non-spin manifolds with low regularity curvature.

Classifies Spin(7) structures on compact 8-manifolds with abelian fundamental group.

Sharp inequalities for Kähler manifolds' systolic invariants are established.

Let $(M,g)$ be a pseudo-Riemannian manifold of signature $(p,q)$. We compute the obstruction for a vector bundle $S$ over $(M,g)$ to admit a Dirac operator whose principal symbol induces on $S$ the structure of a bundle of irreducible real Clifford modules of complex type, that is, a real spinor bundle of irreducible c…

The paper extends Gromov's K-cowaist to complete foliated manifolds and estimates leafwise scalar curvature.

Classifies ancient and expanding Ricci flows with specific groups.

Let $(M,Ω)$ be a closed $8$-dimensional manifold equipped with a generically non-integrable $\mathrm{Spin}(7)$-structure $Ω$. We prove that if $\mathrm{Hom}(H^{3}(M,\mathbb{Z}), \mathbb{Z}_{2}) = 0$ then the moduli space of irreducible $\mathrm{Spin}(7)$-instantons on $(M,Ω)$ with gauge group $\mathrm{SU}(r)$, $r\geq 2…

We describe the different classes of $\mathrm{Spin(7)}$ structures in terms of spinorial equations. We relate them to the spinorial description of $\mathrm{G}_2$ structures in some geometrical situations. Our approach enables us to analyze invariant $\mathrm{Spin(7)}$ structures on quasi abelian Lie algebras.

The dimensions of the spaces of $k$-homogeneous $\mathrm{Spin}(9)$-invariant valuations on the octonionic plane are computed using results from the theory of differential forms on contact manifolds as well as octonionic geometry and representation theory. Moreover, a valuation on Riemannian manifolds of particular inte…

Study harmonic flow of Spin(7)-structures on compact 8-manifolds.

Study local topological constraints on Berry curvature in spin-orbit coupled Bose-Einstein condensates.

Study spin chains and sigma models on flag manifolds, calculating spectra and geodesics.

New spinor fields reveal local or global geometric properties of manifolds.

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…

Abstract: Mapping 3-manifold bordisms to topological orders and domain walls.

New flow for G2-structures helps find torsion-free structures.

Study finds metrics with positive intermediate Ricci curvature on specific low-dimensional manifolds.

The paper constructs manifolds without smooth psc metrics but with $\mathrm{L}^\infty$-metrics that are psc outside singular points.

We give a differential-geometric construction of compact manifolds with holonomy $\mathrm{Spin}(7)$ which is based on Joyce's second construction of compact $\mathrm{Spin}(7)$-manifolds in \cite{Joyce00} and Kovalev's gluing construction of $G_2$-manifolds in \cite{Kovalev03}. We also give some examples of compact $\ma…

The Hermitian symmetric space $M=\mathrm{EIII}$ appears in the classification of complete simply connected Riemannian manifolds carrying a parallel even Clifford structure. This means the existence of a real oriented Euclidean vector bundle $E$ over it together with an algebra bundle morphism $\varphi:\mathrm{Cl}^0(E) …

Constructs differential models for twisted Spin^c-bordism and its dual, defining a new anomaly map.

We show that the bar version of the $\mathrm{Pin}(2)$-monopole Floer homology of a three-manifold $Y$ equipped with a self-conjugate spin$^c$ structure $\mathfrak{s}$ is determined by the triple cup product of $Y$ together with the Rokhlin invariants of the spin structures inducing $\mathfrak{s}$. This is a manifestati…

Study infinitesimal deformations of Killing spinors on nearly parallel G2-manifolds.

Constructs explicit solutions to Spin(7)-structures gradient flow.

The paper proves rigidity for mapping class group actions on metrics of positive scalar curvature.

Solves sigma model on U(3)/U(1)^3, describing geodesics and spectrum.

Study isotropic embeddings of Lie group orbits and their application to magnetic geodesic flows.

We study $\mathrm{Spin}(7)$-manifolds with an effective multi-Hamiltonian action of a four-torus. On an open dense set, we provide a Gibbons-Hawking type ansatz that describes such geometries in terms of a symmetric $4\times4$-matrix of functions. This description leads to the first known $\mathrm{Spin}(7)$-manifolds w…

Let $M$ be a closed spin manifold which supports a positive scalar curvature metric. The set of concordance classes of positive scalar curvature metrics on $M$ forms an abelian group $P(M)$ after fixing a positive scalar curvature metric. The group $P(M)$ measures the size of the space of positive scalar curvature metr…

Sharp distance estimates for compact spin manifolds using Dirac operator.

Researchers found a maximal antipodal set of three elements in a 7x7 sphere space.

A quantum field theory for Spin(7)-instantons derived from moduli spaces.

Researchers compute the ν-invariant for specific G2-structures on nilmanifolds.

Scalar curvature rigidity for products of convex hypersurfaces

Computes homotopy groups of diffeomorphism spaces for high-dimensional manifolds.

Study of multi-toric geometries with larger compact symmetry groups.

Alternative proof and description of orientations for instanton moduli spaces.

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

Topology of isometric classes and flows of geometric structures

We construct a continuous 1-parameter family of smooth complete Ricci-flat metrics of cohomogeneity one on vector bundles over $\mathbb{CP}^2$, $\mathbb{HP}^2$ and $\mathbb{OP}^2$ with respective principal orbits $G/K$ the Wallach spaces $SU(3)/T^2$, $Sp(3)/(Sp(1)Sp(1)Sp(1))$ and $F_4/\mathrm{spin}(8)$. Almost all the …

Study on representation varieties of RAAGs for different Lie groups.

It is shown that on a compact spin symmetric space with a Kähler or Quaternion-Kähler structure, the first eigenvalue of the Dirac operator is linked to a ''{lowest}'' action of the holonomy, given by the fiberwise action on spinors of the canonical forms characterized by this holonomy. The result is also verified for …

The paper classifies hypersurfaces in Spin$^c$ manifolds that satisfy a specific inequality.

Quadratic Killing tensors on Lie groups are always decomposable.