arXiv research

We generalize the ancient solutions of the Ricci flow on certain principal ${\rm SO}(3)$ bundles over compact quaternionic Kähler manifolds constructed by Bakas, Kong, and Ni to certain $RP^3$ fibre bundles over a product of two compact quaternionic Kähler manifolds. The ancient solutions are of Type I, $κ$-noncollapse…

Complete solutions found for Toda equations on non-compact surfaces.

We generalize the circle bundle examples of ancient solutions of the Ricci flow discovered by Bakas, Kong, and Ni to a class of principal torus bundles over an arbitrary finite product of Fano Kähler-Einstein manifolds studied by Wang and Ziller in the context of Einstein geometry. As a result, continuous families of $…

Study local perturbations of vector bundles with polynomial curvature solutions.

Ancient solutions to Ricci flow on torus bundles have additional symmetries.

We prove that a given Calabi-Yau threefold with a stable holomorphic vector bundle can be perturbed to a solution of the Strominger system provided that the second Chern class of the vector bundle is equal to the second Chern class of the tangent bundle. If the Calabi-Yau threefold has strict SU(3) holonomy then the eq…

We construct a class of stable SU(5) bundles on an elliptically fibered Calabi-Yau threefold with two sections, a variant of the ordinary Weierstrass fibration, which admits a free involution. The bundles are invariant under the involution, solve the topological constraint imposed by the heterotic anomaly equation and …

The paper introduces new functionals and equations for complex vector bundles.

Study Ricci flow on torus bundles and related manifolds.

Solves Demailly's system for direct sums of ample line bundles on Riemann surfaces.

We construct new smooth solutions to the Hull-Strominger system, showing that the Fu-Yau solution on torus bundles over K3 surfaces can be generalized to torus bundles over K3 orbifolds. In particular, we prove that, for $13 \leq k \leq 22$ and $14\leq r\leq 22$, the smooth manifolds $S^1\times \sharp_k(S^2\times S^3)$…

Study generalizes Hermitian-Einstein equation for cyclic Higgs bundles, proving existence and inequality.

Paper solves quantum differential equations for projective bundles using Borel multitransforms.

The paper studies twistor sections of Dirac bundles and proves vanishing theorems.

Unique solution found for Demailly's equation on stable bundles.

We provide an algebraic framework for quantization of Hermitian metrics that are solutions of the Hitchin equation for Higgs bundles over a projective manifold. Using Geometric Invariant Theory, we introduce a notion of balanced metrics in this context. We show that balanced metrics converge at the quantum limit toward…

Let $C\to M$ be the bundle of connections of a principal bundle on $M$. The solutions to Hamilton-Cartan equations for a gauge-invariant Lagrangian density $Λ$ on $C$ satisfying a weak condition of regularity, are shown to admit an affine fibre-bundle structure over the set of solutions to Euler-Lagrange equations for …

A twisted Higgs bundle on a Kähler manifold $X$ is a pair $(E,φ)$ consisting of a holomorphic vector bundle $E$ and a holomorphic bundle morphism $φ\colon M\otimes E \to E$ for some holomorphic vector bundle $M$. Such objects were first considered by Hitchin when $X$ is a curve and $M$ is the tangent bundle of $X$, and…

Develops higher-order Euler-Poincaré field equations for principal G-bundles.

Introduces a new Yang-Mills functional for connections and scalars over circle bundles.

Solves Dirac equation coupled to vector bundles.

Study on positivity properties of vector bundle Monge-Ampère equation.

Classifies solutions of Toda equations near singularities.

New solutions found for $G_2$ system using K3 orbifolds.

Paper improves uncertainty quantification in PINNs using error bounds and solution bundles.

An explicit construction of surfaces with flat normal bundle in the Euclidean space (unit hypersphere) in terms of solutions of certain linear system is proposed. In the case of 3-space our formulae can be viewed as the direct Lie sphere analog of the generalized Weierstrass representation of surfaces in conformal geom…

Extends Yang-Mills theory to non-integrable Lie algebroids.

In this paper, we show that the bundle method can be applied to solve semidefinite programming problems with a low rank solution without ever constructing a full matrix. To accomplish this, we use recent results from randomly sketching matrix optimization problems and from the analysis of bundle methods. Under strong d…

Paper solves vortex equations on complex surfaces, linking to Higgs bundle stability.

This paper studies the behavior of sequences of solutions to Seiberg-Witten like equations for a pair consisting of a Hermitian connection on a line bundle over a 4-dimensional manifold and a section of the self-dual spinor bundle of a complex Clifford module on the manifold. Examples include the cases where the Cliffo…

We construct new examples of solutions of the Hull-Strominger system on non-Kähler torus bundles over K3 surfaces, with the property that the connection $\nabla$ on the tangent bundle is Hermite-Yang-Mills. With this ansatz for the connection $\nabla$, we show that the existence of solutions reduces to known results ab…

We derive modified Perelman-type monotonicity formulas for solutions to the generalized Ricci flow equation with symmetry on principal bundles, which lead to rigidity and classification results for nonsingular solutions.

We consider Laplacians acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants. We argue that the obtained formal s…

We study the N=1 supersymmetric solutions of D=11 supergravity obtained as a warped product of four-dimensional anti-de-Sitter space with a seven-dimensional Riemannian manifold M. Using the octonion bundle structure on M we reformulate the Killing spinor equations in terms of sections of the octonion bundle on M. The …

New solutions found for complex structures on specific manifolds.

Investigates $J$-equation on holomorphic vector bundles over Kähler manifolds.

In the spirit of [10,2], we study the Calabi-Yau equation on $T^2$-bundles over $\mathbb{T}^2$ endowed with an invariant non-Lagrangian almost-Kähler structure showing that for $T^2$-invariant initial data it reduces to a Monge-Ampère equation having a unique solution. In this way we prove that for every total space $M…

Geometric equation defines canonical metrics on vector bundle families.

In this work, we study and solve the normalized Ricci flow equation for circle bundles over surfaces. Moreover, we study the asymptotic behavior of the solutions and their connections to some model geometries.

We introduce $Z$-critical connections for holomorphic vector bundles and prove their existence under stability conditions.

Constructing solutions to the heterotic G$_2$ system on specific types of manifolds.

Paper fills in technical details for Hitchin's self-duality equations proof.

The paper classifies Toda equations for noncompact symmetric spaces and their solutions.

Solves Dirichlet problem for generalized Hitchin's equation on cyclic Higgs bundles.

We prove a gluing theorem for solutions of Hitchin's self-duality equations with logarithmic singularities on a rank-2 vector bundle over a noded Riemann surface representing a boundary point of Teichmüller moduli space.

Construct Hermitian-Einstein metrics on stable holomorphic vector bundles using dynamical methods.

Study parallel tractors and cotractors on almost Grassmannian structures.

Proves well-posedness of gradient solitons on bundle gerbe.