arXiv research

Study linear perturbations of Spin(7) metrics, finding only rank one nilpotent matrices.

We introduce and study a notion of invariant intrinsic torsion geometry which appears, for instance, in connection with the Bryant-Salamon metric on the spinor bundle over S^3. This space is foliated by six-dimensional hypersurfaces, each of which carries a particular type of SO(3)-structure; the intrinsic torsion is i…

We show that two of the Bryant-Salamon G_2-manifolds have a simple topology ; homeomorphic to the complement of some submanifolds of the 7-dimensional sphere. In this connection, we show there exists a complete Ricci-flat (non-flat) metric on the complement of an m-dimensional sphere in an n-dimensional sphere for some…

New $G_2$-instantons found on 3-sphere's spinor bundle.

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

Study on deformations of Spin(7)-structures on manifolds.

$G_2$-Monopoles are solutions to gauge theoretical equations on noncompact $7$-manifolds of $G_2$ holonomy. We shall study this equation on the $3$ Bryant-Salamon manifolds. We construct examples of $G_2$-monopoles on two of these manifolds, namely the total space of the bundle of anti-self-dual two forms over the $\ma…

Study cohomogeneity one solitons for $G_2$-structures on various manifolds.

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

We give a general description of the construction of weighted spherically symmetric metrics on vector bundle manifolds, i.e. the total space of a vector bundle $E\rightarrow M$, over a Riemannian manifold $M$, when $E$ is endowed with a metric connection. The tangent bundle of $E$ admits a canonical decomposition and t…

We study the uniqueness of minimal submanifolds and the stability of the mean curvature flow in several well-known model spaces of manifolds of special holonomy. These include the Stenzel metric on the cotangent bundle of spheres, the Calabi metric on the cotangent bundle of complex projective spaces, and the Bryant--S…

We develop a systematic approach to G_2 holonomy manifolds with an SU(2)xSU(2) isometry using maximal eight-dimensional gauged supergravity to describe D6-branes wrapped on deformed three-spheres. A quite general metric ansatz that generalizes the celebrated Bryant-Salamon metric involves nine functions. We show that o…

Constructs deformed G2-instantons on specific G2-manifolds.

We find Weitzenböck formula for the Fueter-Dirac operator which controls the infinitesimal deformations of an associative submanifold in a $7$--manifold with a $G_2$--structure. We establish a vanishing theorem to conclude rigidity under some positivity assumptions on curvature, which are particularly mild in the nearl…

We construct propose an anzatz for Spin(7) metrics as an R-bundle over closed G2 structures. These G2 structures are R3 bundles over 4-dimensional compact quaternion Kahler spaces. The inspiration for the anzatz metric comes from the Bryant-Salamon construction of G2 holonomy metrics and from the fact that the twistor …

We find a remarkable family of $\mathrm{G}_2$ structures defined on certain principal $\mathrm{SO}(3)$-bundles $P_\pm\longrightarrow M$ associated with any given oriented Riemannian 4-manifold $M$. Such structures are always cocalibrated. The study starts with a recast of the Singer-Thorpe equations of 4-dimensional ge…

Constructs new coassociative fibrations for G2 manifolds.

If a $Spin(7)$ manifold $N^8$ admits a free $S^1$ action preserving the fundamental $4$-form then the quotient space $M^7$ is naturally endowed with a $G_2$-structure. We derive equations relating the intrinsic torsion of the $Spin(7)$-structure to that of the $G_2$-structure together with the additional data of a Higg…

Study $G_2$ manifolds with symmetry, finding new submanifolds.

The paper studies deformations of calibrated subbundles in special holonomy manifolds.

We consider the deformation theory of asymptotically conical (AC) and of conically singular (CS) $G_2$-manifolds. In the AC case, we show that if the rate of convergence $ν$ to the cone at infinity is generic in a precise sense and lies in the interval $(-4, 0)$, then the moduli space is smooth and we compute its dimen…

New PDEs for $k$-harmonic maps link to calibrated fibrations.

Study of solitons in Laplacian flow on 7-manifolds.

This thesis surveys various metrics on Riemann surface spaces.

Proves existence and uniqueness of weighted metrics for smooth spaces.

New Finsler metrics constructed from $GDW$-metrics.

In this essay, we study the sufficient and necessary conditions for a Randers metrc to be of constant Ricci curvature without the restriction of strong convexity (regularity). The classification result for the case $\|β\|_α>1$ is provided, which is similar to the famous Bao-Robles-Shen's result for strongly convex Rand…

We prove the equivalences of several classical complete metrics on the Teichmüller and the moduli spaces of Riemann surfaces. We use as bridge two new Kähler metrics, the Ricci metric and the perturbed Ricci metric and prove that the perturbed Ricci metric is a complete Kähler metric with bounded negative holomorphic s…

New metric defined for bounded symmetric domains.

Introduces Finslerian convolution metrics and their properties.

Survey of recent metric geometry in Kähler metrics space.

Study on geodesics of Finsler metrics derived from Riemannian metrics.

We consider geometries on the space of Riemannian metrics conformally equivalent to the widely studied Ebin L^2 metric. Among these we characterize a distinguished metric that can be regarded as a generalization of Calabi's metric on the space of Kähler metrics to the space of Riemannian metrics, and we study its geome…

New Kähler metrics generalize Calabi's and relate to Fano manifolds.

New Finsler metrics defined by Riemannian and 1-forms are studied.

We study the geodesic equation for the Dirichlet (gradient) metric in the space of Kaehler potentials. We first solve the initial value problem for the geodesic equation of the combination metric, including the gradient metric. We then discuss a comparison theorem between it and the Calabi metric. As geometric motivati…

Sharp estimates for Finsler metrics in convex domains.

Study on special Finsler metrics with conditions for Riemannian and isotropic properties.

The study examines Lee metrics on groups and their properties.

Survey of spectral, probabilistic, and deep metric learning methods.

Study shows convergence of Lagrangian submanifolds under certain metrics.

In this paper, we study an important class of Finsler metrics--square metrics. We give two expressions of such metrics in terms of a Riemannian metric and a 1-form. We show that Einstein square metrics can be classified up to the classification of Einstein Riemannian metrics.

Defines a new Randers metric based on an existing one.

Study Kähler-Einstein metrics on singular varieties, proving metric completion properties.

Investigates O(n)-invariant metrics on SPD matrices, extending kernel metrics.

The current paper deals with some new classes of Finsler metrics with reversible geodesics. We construct weighted quasi-metrics associated with these metrics. Further, we investigate some important geometric properties of weighted quasi-metric space. Finally, we discuss the embedding of quasi-metric spaces with general…

Douglas metrics are metrics with vanishing Douglas curvature which is an important projective invariant in Finsler geometry. To find more Douglas metrics, in this paper we consider a class of Finsler metrics called general $(α,β)$-metrics, which are defined by a Riemannian metric $α=\sqrt{a_{ij}(x)y^iy^j}$ and a $1$-fo…

Proposes a method to select fair performance metrics through metric elicitation.