We associate a flow $ϕ$ to a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $ϕ$ always admits a dominated splitting and identify special cases in which $ϕ$ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of $(M,g)$.
The purpose of these notes is to show that the methods introduced by Bauer and Furuta in order to refine the Seiberg-Witten invariants of smooth 4-dimensional manifolds can also be used to obtain stable homotopy classes from Riemann surfaces, using the vortex equations on the latter.
In this article we calculate the dimension of the Hilbert space of Kahler quantization of the moduli space of vortices on a Riemann surface. This dimension is given by the holomorphic Euler characteristic of the quantum line bundle.
In this paper we prequantize the moduli space of non-abelian vortices. We explicitly calculate the symplectic form arising from the $L^2$ metric and we construct a prequantum line bundle whose curvature is proportional to this symplectic form. The prequantum line bundle turns out to be Quillen's determinant line bundle with a modified Quillen metric. Next, as in the case of abelian vortices, we construct Quillen line bundles over the moduli space whose curvatures form a family of symplectic forms which are parametrised by $Ψ_0$, a section of a certain bundle.
In this paper, we study and partially classify those Riemannian man-ifolds carrying a non-identically vanishing function f whose Hessian is minus f times the Ricci-tensor of the manifold.
We consider the vortex equations for a U(n) gauge field coupled to a Higgs field with values on the n times n square matrices. It is known that when these equations are defined on a compact Riemann surface, their moduli space of solutions is closely related to a moduli space of tau-stable holomorphic n-pairs on that surface. Using this fact and a local factorization result for the Higgs matrix, we show that the vortex solutions are entirely characterized by (1) the location in the surface of the zeros of the determinant of the Higgs matrix and (2) by the choice of a vortex internal structure a…
We study a variational Ginzburg-Landau type model depending on a small parameter $\varepsilon>0$ for (tangent) vector fields on a $2$-dimensional Riemannian manifold $S$. As $\varepsilon\to 0$, these vector fields tend to have unit length so they generate singular points, called vortices, of a (non-zero) index if the genus $\mathfrak{g}$ of $S$ is different than $1$. Our first main result concerns the characterization of canonical harmonic unit vector fields with prescribed singular points and indices. The novelty of this classification involves flux integrals constrained to a particular vorti…
In this paper we provide some local and global splitting results on complete Riemannian manifolds with nonnegative Ricci curvature. We achieve the splitting through the analysis of some pointwise inequalities of Modica type which hold true for every bounded solution to a semilinear Poisson equation. More precisely, we prove that the existence of a nonconstant bounded solution $u$ for which one of the previous inequalities becomes an equality at some point leads to the splitting results as well as to a classification of such a solution $u$.
We consider Riemannian $n$-manifolds $M$ with nontrivial $κ$-nullity "distribution" of the curvature tensor $R$, namely, the variable rank distribution of tangent subspaces to $M$ where $R$ coincides with the curvature tensor of a space of constant curvature $κ$ ($κ\in\mathbb R$) is nontrivial. We obtain classification theorems under diferent additional assumptions, in terms of low nullity/conullity, controlled scalar curvature or existence of quotients of finite volume. We prove new results, but also revisit previous ones.
In this paper we construct new solutions of the Kahler-Yang-Mills equations, by applying dimensional reduction methods to the product of the complex projective line with a compact Riemann surface. The resulting equations, that we call gravitating vortex equations, describe Abelian vortices on the Riemann surface with back reaction of the metric. As a particular case of these gravitating vortices on the Riemann sphere we find solutions of the Einstein-Bogomol'nyi equations, which physically correspond to Nielsen-Olesen cosmic strings in the Bogomol'nyi phase. We use this to provide a Geometric …
We introduce the notion of twisted gravitating vortex on a compact Riemann surface. If the genus of the Riemann surface is greater than 1 and the twisting forms have suitable signs, we prove an existence and uniqueness result for suitable range of the coupling constant generalizing the result of arXiv:1510.03810v2 in the non twisted setting. It is proved via solving a continuity path deforming the coupling constant from 0 for which the system decouples as twisted Kähler-Einstein metric and twisted vortices. Moreover, specializing to a family of twisting forms smoothing delta distribution terms…
Two metrics $g $ and $\bar g$ are geodesically equivalent, if they share the same (unparameterized) geodesics. We introduce two constructions that allow one to reduce many natural problems related to geodesically equivalent metrics, such as the classification of local normal forms and the Lie problem (the description of projective vector fields), to the case when the (1,1)-tensor $G^i_j:= g^{ik} \bar g_{kj}$ has one real eigenvalue, or two complex conjugate eigenvalues, and give first applications. As a part of the proof of the main result, we generalize Topalov-Sinjukov (hierarchy) Theorem fo…
We establish the splitting lemmas (or generalized Morse lemmas) for the energy functionals of Finsler metrics on the natural Hilbert manifolds of $H^1$-curves around a critical point or a critical $\R^1$ orbit of a Finsler isometry invariant closed geodesic. They are the desired generalization on Finsler manifolds of the corresponding Gromoll-Meyer's splitting lemmas on Riemannian manifolds (\cite{GM1, GM2}). As an application we extend to Finsler manifolds a result by Grove and Tanaka \cite{GroTa78, Tan82} about the existence of infinitely many, geometrically distinct, isometry invariant clos…
In this note we show that for the group G = U(N) the space of Hecke modifications of a rank N vector bundle over a Riemann surface C coincides with the moduli space of solutions of certain non-abelian vortex equations over C . Through the recent work of Kapustin and Witten this then leads to an isomorphism between the moduli space of vortices and the moduli space of singular monopoles on the product of C with a closed interval I .
Suppose that $M$ is a $2$-dimensional oriented Riemannian manifold, and let $γ$ be a simple closed curve on $M$. Let $m γ$ denote the curve formed by tracing $γ$ $m$ times. We prove that if $m γ$ is contractible through curves of length less than $L$, then $γ$ is contractible through curves of length less than $L$. In the last section we state several open questions about controlling length and the number of self-intersections in homotopies of curves on Riemannian surfaces.
In this paper we obtain a splitting theorem for the symmetric diffusion operator $Δ_ϕ=Δ-\left<\nablaϕ,\nabla \right>$ and a non-constant $C^3$ function $f$ in a complete Riemannian manifold $M$, under the assumptions that the Ricci curvature associated with $Δ_ϕ$ satisfies ${\rm Ric}_ϕ(\nabla f,\nabla f)\ge 0$, that $|\nabla f|$ attains a maximum at $M$ and that $Δ_ϕ$ is non-decreasing along the orbits of $\nabla f$. The proof uses the general fact that a complete manifold $M$ with a non-constant smooth function $f$ with parallel gradient vector field must be a Riemannian product $M=N\times \m…
We prove a new generalization of the Cheeger-Gromoll splitting theorem where we obtain a warped product splitting under the existence of a line. The curvature condition in our splitting is a curvature dimension inequality of the form $CD(0,1)$. Even though we have to allow warping in our splitting, we are able to recover topological applications. In particular, for a smooth compact Riemannian manifold admitting a density which is $CD(0,1)$, we show that the fundamental group of $M$ is the fundamental group of a compact manifold with nonnegative sectional curvature. If the space is also locally…
We show that a Riemannian $3$-manifold with non-negative scalar curvature is flat if it contains an area-minimizing cylinder. This scalar-curvature analogue of the classical splitting theorem of J.~Cheeger and D.~Gromollhas been conjectured by D.~Fischer-Colbrie and R.~Schoen and by M.~Cai and G.~Galloway.
It is shown that both the sinh--Gordon equation and the elliptic Tzitzeica equation can be interpreted as the Taubes equation for Abelian vortices on a CMC surface embedded in $\R^{2, 1}$, or on a surface conformally related to a hyperbolic affine sphere in $\R^3$. In both cases the Higgs field and the U(1) vortex connection are constructed directly from the Riemannian data of the surface corresponding to the sinh--Gordon or the Tzitzeica equation. Radially symmetric solutions lead to vortices with a topological charge equal to one, and the connection formulae for the resulting third Painlevé …
Let L --> X be a complex line bundle over a compact connected Riemann surface. We consider the abelian vortex equations on L when the metric on the surface has finitely many point degeneracies or conical singularities and the line bundle has parabolic structure. These conditions appear naturally in the study of vortex configurations with constraints, or configurations invariant under the action of a finite group. We first show that the moduli space of singular vortex solutions is the same as in the regular case. Then we compute the total volume and total scalar curvature of the moduli space si…
We give examples illustrating the fact that the different space/time splittings of the tangent bundle of a semi-Riemannian spin manifold give rise to non-equivalent norms on the space of compactly supported sections of the spinor bundle, and as a result, to different completions. We give a necessary and sufficient condition for two space/time splittings to define equivalent norms in terms of a generalized Doppler shift between maximal negative definite subspaces. We explore some consequences for the Noncommutative Geometry program.
We derive general expressions for the Kaehler form of the L^2-metric in terms of standard 2-forms on vortex moduli spaces. In the case of abelian vortices in gauged linear sigma-models, this allows us to compute explicitly the Kaehler class of the L^2-metric. As an application we compute the total volume of the moduli space of abelian semi-local vortices. In the strong coupling limit, this then leads to conjectural formulae for the volume of the space of holomorphic maps from a compact Riemann surface to projective space. Finally we show that the localization results of Samols in the abelian H…
We demonstrate an obstruction to finding certain splittings of four-manifolds along sufficiently twisted circle bundles over Riemann surfaces, arising from Seiberg-Witten theory. These obstructions are used to show a non-splitting result for algebraic surfaces of general type.
On a smooth line bundle $L$ over a compact Kähler Riemann surface $Σ$, we study the family of vortex equations with a parameter $s$. For each $s \in [1,\infty]$, we invoke techniques in \cite{Br} by turning the $s$-vortex equation into an $s$-dependent elliptic partial differential equation, studied in \cite{kw}, providing an explicit moduli space description of the space of gauge classes of solutions. We are particularly interested in the bijective correspondence between the open subset of vortices without common zeros and the space of holomorphic maps. For each $s$, the correspondence is uni…
Let X be a compact connected Riemann surface of genus at least two. Fix positive integers r and d. Let Q denote the Quot scheme that parametrizes the torsion quotients of {\mathcal O}^{\oplus r}_X of degree d. This Q is also the moduli space of vortices for the standard action of U(r) on {\mathbb C}^r. The group \text{PGL}(r, {\mathbb C}) acts on Q via the action of $\text{GL}(r, {\mathbb C})$ on ${\mathcal O}^{\oplus r}_X$. We prove that this subgroup $\text{PGL}(r, {\mathbb C})$ is the connected component, containing the identity element, of the holomorphic automorphism group Aut(\mathcal Q)…
We show that the number of genus $g$ embedded minimal surfaces in $\mathbb{S}^3$ tends to infinity as $g\rightarrow\infty$. The surfaces we construct resemble doublings of the Clifford torus with curvature blowing up along torus knots as $g\rightarrow\infty$, and arise from a two-parameter min-max scheme in lens spaces. More generally, by stabilizing and flipping Heegaard foliations we produce index at most $2$ minimal surfaces with controlled topological type in arbitrary Riemannian three-manifolds.
The strong maximum principle is proved to hold for weak (in the sense of support functions) sub- and super-solutions to a class of quasi-linear elliptic equations that includes the mean curvature equation for $C^0$ spacelike hypersurfaces in a Lorentzian manifold. As one application a Lorentzian warped product splitting theorem is given.
