Related papers: The vortex equation on affine manifolds

If $X\subset\operatorname{Gr}(2,6)$ is the Fano variety of lines of a smooth cubic fourfold, then we show that the restriction to $X$ of any Schur functor of the tautological quotient bundle is modular and slope polystable. Moreover it is…

This work explores the geometry of stable wild Vafa-Witten bundles over the complex projective plane $\mathbb{P}^2$. Specifically, we consider stable rank-two pairs $(E,\Phi)$, with $E\to\mathbb{P}^2$ a rank-two holomorphic vector bundle…

In this paper, we study the moduli space of Higgs pairs, which can be considered as a generalization of holomorphic pairs. Higgs pairs are an example of quiver bundles. We introduce the notion of $\tau$-stability of Higgs pairs for…

For the model of a compressible barotropic fluid on a two dimensional rotating Riemmanian manifold we discuss a special class of smooth solutions having a form of a steady non-singular vortex moving with a bearing field. The model can be…

We elaborate one- and two-dimensional (1D and 2D) models of media with self-repulsive cubic nonlinearity, whose local strength is subject to spatial modulation that admits the existence of flat-top solitons of various types, including…

A complex manifold $X$ of dimension $n$ together with an ample vector bundle $E$ on it will be called a {\sf generalized polarized variety}. The adjoint bundle of the pair $(X,E)$ is the line bundle $K_X + det(E)$. We study the positivity…

We solve the geodesic equation in the space of K\"ahler metrics under the setting of asymptotically locally Euclidean (ALE) K\"ahler manifolds and we prove global $\mathcal{C}^{1,1}$ regularity of the solution. Then, we relate the solution…

We show that a projective manifold is stable if and only if the Mabuchi energy is proper on the space of algebraic metrics. We show that stability implies finite automorphism group.

Let $E\to M$ be a holomorphic vector bundle over a compact Kaehler manifold $(M, \omega)$. We prove that if $E$ admits a $\omega$-balanced metric (in X. Wang's terminology) then it is unique. This result together with a result of L.…

Let $(M,\mathsf{g})$ be a connected and compact Riemannian manifold admitting an isometric action by a compact Lie group $G$ whose principal orbits have codimension one. We show that any $G$-invariant, smooth, and divergence-free vector…

The equation of motion of a quantized vortex filament in a trapped Bose-Einstein condensate [A. A. Svidzinsky and A. L. Fetter, Phys. Rev. A {\bf 62}, 063617 (2000)] has been generalized to the case of an arbitrary anharmonic anisotropic…

Let $X$ be a projective K3 surfaces. In two examples where there exists a fine moduli space $M$ of stable vector bundles on $X$, isomorphic to a Hilbert scheme of points, we prove that the universal family $\mathcal{E}$ on $X\times M$ can…

We prove that any holomorphic vector bundle admitting a holomorphic connection, over a compact K\"ahler Calabi-Yau manifold, also admits a flat holomorphic connection. This addresses a particular case of a question asked by Atiyah and…

We study instability of a vortex soliton $e^{i(m\theta+\omega t)}\phi_{\omega,m}(r)$ to $$iu_t+\Delta u+|u|^{p-1}u=0,\quad\text{for $x\in\R^n$, $t>0$,}$$ where $n=2$, $m\in\N$ and $(r,\theta)$ are polar coordinates in $\R^2$. Grillakis…

This article concerns a question asked by M. V. Nori on homotopy of sections of Projective modules defined on the polynomial algebra over a smooth affine domain $R$. While this question has an affirmative answer, it is known that the…

We investigate the behaviour of local perturbations of a wide class of geometric PDEs on holomorphic Hermitian vector bundles over a compact complex manifold. Our main goal is to study the existence of solutions near an initial solution…

We study topological vortex solutions in a generalized Abelian Higgs model with non-polynomial dielectric and potential functions. These quantities are chosen by requiring integrability of the self-dual limit of the theory for all values of…

The classical Cauchy rigidity theorem for convex polytopes reads that if two convex polytopes have isometric developments then they are congruent. In other words, we can decide whether two polyhedra are isometric or not by using their…

For a compact group G, we give a sufficient condition for embedding one G-equivariant vector bundle into another one and for a stable isomorphism between two such bundles to imply an isomorphism. Our criteria involve multiplicities of…

The Hitchin-Kobayashi correspondence for vector bundles, established by Donaldson, Kobayashi, Luebke, Uhlenbeck and Yau, states that an indecomposable holomorphic vector bundle over a compact Kaehler manifold is stable in the sense of…