We develop the asymptotic analysis as $ε\to 0$ for the natural gradient flow of the self-dual $U(1)$-Higgs energies $$E_ε(u,\nabla)=\int_M\left(|\nabla u|^2+ε^2|F_{\nabla}|^2+\frac{(1-|u|^2)^2}{4ε^2}\right)$$ on Hermitian line bundles over closed manifolds $(M^n,g)$ of dimension $n\ge 3$, showing that solutions converge in a measure-theoretic sense to codimension-two mean curvature flows -- i.e., integral $(n-2)$-Brakke flows -- generalizing results of the last two authors from the stationary case. Given any integral $(n-2)$-cycle $Γ_0$ in $M$, these results can be used together with the convergence theory developed in previous work of the authors to produce nontrivial integral Brakke flows starting at $Γ_0$ with additional structure, similar to those produced via Ilmanen's elliptic regularization.
We develop a new method for proving regularity for small energy stationary solutions of coupled gauge field equations. Our results duplicate those of Tian--Tao [7] for the pure Yang Mills equations, but our proof is simpler, and obtains bounded curvature without the use of Coulomb gauges. It relies instead on the Weitzenblock formulae, and an improved Kato inequality.
We show uniqueness of cylindrical blowups for mean curvature flow in all dimension and all codimension. Cylindrical singularities are known to be the most important; they are the most prevalent in any codimension. Mean curvature flow in higher codimension is a nonlinear parabolic system where many of the methods used for hypersurfaces do not apply and uniqueness of cylindrical blowups remained a major open problem. Our results imply regularity of the singular set for the system.
In this paper we investigate the moduli space of parabolic Higgs bundles over a punctured Riemann surface with varying weights at the punctures. We show that the harmonic metric depends analytically on the weights and the stable Higgs bundle. This gives a Higgs bundle generalisation of a theorem of McOwen on the existence of hyperbolic cone metrics on a punctured surface within a given conformal class, and a generalisation of a theorem of Judge on the analytic parametrisation of these metrics.
We first bound the codimension of an ancient mean curvature flow by the entropy. As a consequence, all blowups lie in a Euclidean subspace whose dimension is bounded by the entropy and dimension of the evolving submanifolds. This drastically reduces the complexity of the system. Combined with \cite{CM12}, this gives the first general bounds on generic singularities of surfaces in arbitrary codimension. We also show sharp bounds for codimension in arguably some of the most important situations of ancient flows. Namely, we prove that in any dimension and codimension any ancient flow that is cyli…
We generalize the classical Beauville-Narasimhan-Ramanan correspondence to the case of parabolic Higgs bundles with regular singularities and Higgs $V$-bundles. Using this correspondence along with Bott-Morse theoretic techniques we provide an exact component count for moduli spaces of maximal parabolic $\text{Sp}\left( 2n,\mathbb{R} \right)$-Higgs bundles with fixed parabolic structure.
We use the theory of Gaiotto, Moore and Neitzke to construct a set of Darboux coordinates on the moduli space $\mathcal{M}$ of weakly parabolic $SL(2,\mathbb{C})$-Higgs bundles. For generic Higgs bundles ($\mathcal{E},RΦ)$ with $R\gg 0$ the coordinates are shown to be dominated by a leading term that is given by the coordinates for a corresponding simpler space of limiting configurations and we prove that the deviation from the limiting term is given by a remainder that is exponentially suppressed in $R$. We then use this result to solve an associated Riemann-Hilbert problem and construct a tw…
We study parabolic G-Higgs bundles over a compact Riemann surface with fixed punctures, when G is a real reductive Lie group, and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in G with arbitrary holonomy around the punctures. Three interesting features are the relation between the parabolic degree and the geometry of the Tits boundary, the treatment of the case when the logarithm of the monodromy is on the boundary of a Weyl alcove, and the correspondence of the orbits encoding the singularity via the Kostant-Sekiguchi c…
In the present paper we carry out a systematic study about the flow of a spherical curve by the mean curvature flow with density in a 3-dimensional rotationally symmetric space with density $(M^3_w,\:g_w,\:ξ)$ where the density $ξ$ decomposes as sum of a radial part $φ$ and an angular part $ψ$. We analyse how either the parabolicity or the hyperbolicity of $(M^3_w,\:g_w)$ condition the behaviour of the flow when the solution goes to infinity.
In this paper, we consider the Yang-Mills-Higgs flow for twisted Higgs pairs over Kähler manifolds. We prove that this flow converges to a reflexive twisted Higgs sheaf outside a closed subset of codimension $4$, and the limiting twisted Higgs sheaf is isomorphic to the double dual of the graded twisted Higgs sheaves associated to the Harder-Narasimhan--eshadri filtration of the initial twisted Higgs bundle.
For a semisimple real Lie group $G$, we study topological properties of moduli spaces of polystable parabolic $G$-Higgs bundles over a Riemann surface with a divisor of finitely many distinct points. For a split real form of a complex simple Lie group, we compute the dimension of apparent parabolic Teichm{ü}ller components. In the case of isometry groups of classical Hermitian symmetric spaces of tube type, we provide new topological invariants for maximal parabolic $G$-Higgs bundles arising from a correspondence to orbifold Higgs bundles. Using orbifold cohomology we count the least number of…
We formulate a correspondence between SU(2) monopole chains and ``spectral data'', consisting of curves in $\mathbb{CP}^1\times\mathbb{CP}^1$ equipped with parabolic line bundles. This is the analogue for monopole chains of Donaldson's association of monopoles with rational maps. The construction is based on the Nahm transform, which relates monopole chains to Higgs bundles on the cylinder. As an application, we classify charge $k$ monopole chains which are invariant under actions of $\mathbb{Z}_{2k}$. We present images of these symmetric monopole chains that were constructed using a numerical…
This is a review article on some applications of generalised parabolic structures to the study of torsion free sheaves and $L$-twisted Hitchin pairs on nodal curves. In particular, we survey on the relation between representations of the fundamental group of a nodal curve and the moduli spaces of generalised parabolic bundles and generalised parabolic $L$-twisted Hitchin pairs on its normalisation as well as on an analogue of the Hitchin map for generalised parabolic $L$-twisted Hitchin pairs.
Given a generic stable strongly parabolic $SL(2,\mathbb{C})$-Higgs bundle $(\mathcal{E}, φ)$, we describe the family of harmonic metrics $h_t$ for the ray of Higgs bundles $(\mathcal{E}, t φ)$ for $t\gg0$ by perturbing from an explicitly constructed family of approximate solutions $h_t^{\mathrm{app}}$. We then describe the natural hyperKähler metric on $\mathcal{M}$ by comparing it to a simpler "semi-flat" hyperKähler metric. We prove that $g_{L^2} - g_{\mathrm{sf}} = O(\mathrm{e}^{-γt})$ along a generic ray, proving a version of Gaiotto-Moore-Neitzke's conjecture. Our results extend to weakly…
We construct explicit geometric models for moduli spaces of semi-stable strongly parabolic Higgs bundles over the Riemann sphere, in the case of rank two, four marked points, arbitrary degree, and arbitrary weights. The mechanism of construction relies on elementary geometric and combinatorial techniques, based on a detailed study of orbit stability of (in general non-reductive) bundle automorphism groups on certain carefully crafted spaces. The aforementioned techniques are not exclusive to the case we examine, and this work elucidates a general approach to construct arbitrary moduli spaces o…
The non-abelian Hodge correspondence identifies complex variations of Hodge structures with certain Higgs bundles. In this work we analyze this relationship, and some of its ramifications, when the variations of Hodge structures are determined by a (complete) one-dimensional family of compact Calabi-Yau manifolds. This setup enables us to apply techniques from mirror symmetry. For example, the corresponding Higgs bundles extend to parabolic Higgs bundles to the compactification of the base of the families. We determine the parabolic degrees of the underlying parabolic bundles in terms of the e…
Semiclassical Limits of Strongly Parabolic Higgs Bundles and Hyperpolygon Spaces
Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京数学科学研究所）；Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel（基尔大学数学系）
We consider the action of a finite subgroup of the mapping class group $Mod(S)$ of an oriented compact surface $S$ of genus $g \geq 2$ on the moduli space $\mathcal{R}(S,G)$ of representations of $π_1(S)$ in a connected semisimple real Lie group $G$. Kerckhoff's solution of the Nielsen realization problem ensures the existence of an element $J$ in the Teichmüller space of $S$ for which $Γ$ can be realised as a subgroup of the group of automorphisms of $X=(S,J)$ which are holomorphic or antiholomorphic. We identify the fixed points of the action of $Γ$ on $\mathcal{R}(S,G)$ in terms of $G$-Higg…
Given a hermitian line bundle $L\to M$ on a closed Riemannian manifold $(M^n,g)$, the self-dual Yang-Mills-Higgs energies are a natural family of functionals \begin{align*} &E_ε(u,\nabla):=\int_M\Big(|\nabla u|^2+ε^2|F_\nabla|^2+\frac{(1-|u|^2)^2}{4ε^2}\Big) \end{align*} defined for couples $(u,\nabla)$ consisting of a section $u\inΓ(L)$ and a hermitian connection $\nabla$ with curvature $F_\nabla$. While the critical points of these functionals have been well-studied in dimension two by the gauge theory community, it was shown in previous work of the second- and third-named authors that criti…
Given a Hermitian line bundle $L\to M$ over a closed, oriented Riemannian manifold $M$, we study the asymptotic behavior, as $ε\to 0$, of couples $(u_ε,\nabla_ε)$ critical for the rescalings \begin{align*} &E_ε(u,\nabla)=\int_M\Big(|\nabla u|^2+ε^2|F_\nabla|^2+\frac{1}{4ε^2}(1-|u|^2)^2\Big) \end{align*} of the self-dual Yang-Mills-Higgs energy, where $u$ is a section of $L$ and $\nabla$ is a Hermitian connection on $L$ with curvature $F_{\nabla}$. Under the natural assumption $\limsup_{ε\to 0}E_ε(u_ε,\nabla_ε)<\infty$, we show that the energy measures converge subsequentially to (the weight me…
In this paper we characterize non-collapsed limits of Ricci flows. We show that such limits are smooth away from a set of codimension $\geq 4$ in the parabolic sense and that the tangent flows at every point are given by gradient shrinking solitons, possibly with a singular set of codimension $\geq 4$. We furthermore obtain a stratification result of the singular set with optimal dimensional bounds, which depend on the symmetries of the tangent flows. Our methods also imply the corresponding quantitative stratification result and the expected $L^p$-curvature bounds. As an application of our th…
We study properly immersed ancient solutions of the codimension one mean curvature flow in $n$-dimensional Euclidean space, and classify the convex hulls of the subsets of space reached by any such flow. In particular, it follows that any compact convex ancient mean curvature flow can only have a slab, a halfspace or all of space as the closure of its set of reach. The proof proceeds via a bi-halfspace theorem (also known as a wedge theorem) for ancient solutions derived from a parabolic Omori-Yau maximum principle for ancient mean curvature flows.
