Global existence and convergence of a flow to Kazdan–Warner equation with non-negative prescribed function

We consider an evolution problem associated to the Kazdan–Warner equation on a closed Riemann surface $$(\Sigma ,g)$$ ( Σ , g ) $$\begin{aligned} -\Delta _{g}u=8\pi \left( \dfrac{he^{u}}{\int _{\Sigma }he^{u}\mathop {}\mathrm {d}\mu _{g}}-\dfrac{1}{\int _{\Sigma }\mathop {}\mathrm {d}\mu _{g}}\right) \end{aligned}$$ - Δ g u = 8 π h e u ∫ Σ h e u d μ g - 1 ∫ Σ d μ g where the prescribed function $$h\ge 0$$ h ≥ 0 and $$\max _{\Sigma }h>0$$ max Σ h > 0 . We prove the global existence and convergence under additional assumptions such as $$\begin{aligned} \Delta _{g}\ln h(p_0)+8\pi -2K(p_0)>0 \end{aligned}$$ Δ g ln h ( p 0 ) + 8 π - 2 K ( p 0 ) > 0 for any maximum point $$p_0$$ p 0 of the sum of $$2\ln h$$ 2 ln h and the regular part of the Green function, where K is the Gaussian curvature of $$\Sigma $$ Σ . In particular, this gives a new proof of the existence result by Yang and Zhu (Pro Am Math Soc 145:3953–3959, 2017) which generalizes existence result of Ding et al. (Asian J Math 1:230–248, 1997) to the non-negative prescribed function case.

Convergence of the Yang–Mills–Higgs flow on Gauged Holomorphic maps and applications

The symplectic vortex equations admit a variational description as global minimum of the Yang–Mills–Higgs functional. We study its negative gradient flow on holomorphic pairs [Formula: see text] where [Formula: see text] is a connection on a principal [Formula: see text]-bundle [Formula: see text] over a closed Riemann surface [Formula: see text] and [Formula: see text] is an equivariant map into a Kähler Hamiltonian [Formula: see text]-manifold. The connection [Formula: see text] induces a holomorphic structure on the Kähler fibration [Formula: see text] and we require that [Formula: see text] descends to a holomorphic section of this fibration. We prove a Łojasiewicz type gradient inequality and show uniform convergence of the negative gradient flow in the [Formula: see text]-topology when [Formula: see text] is equivariantly convex at infinity with proper moment map, [Formula: see text] is holomorphically aspherical and its Kähler metric is analytic. As applications we establish several results inspired by finite dimensional GIT: First, we prove a certain uniqueness property for the critical points of the Yang–Mills–Higgs functional which is the analogue of the Ness uniqueness theorem. Second, we extend Mundet’s Kobayashi–Hitchin correspondence to the polystable and semistable case. The arguments for the polystable case lead to a new proof in the stable case. Third, in proving the semistable correspondence, we establish the moment–weight inequality for the vortex equation and prove the analogue of the Kempf existence and uniqueness theorem.

Spectral invariants of distance functions

Calculating the spectral invariant of Floer homology of the distance function, we can find new superheavy subsets in symplectic manifolds. We show if convex open subsets in Euclidian space with the standard symplectic form are disjointly embedded in a spherically negative monotone closed symplectic manifold, their complement is superheavy. In particular, the [Formula: see text] bouquet in a closed Riemann surface with genus [Formula: see text] is superheavy. We also prove some analogous properties of a monotone closed symplectic manifold. These can be used to extend Seyfaddni’s result about lower bounds of Poisson bracket invariant.

A Note on Planarity Stratification of Hurwitz Spaces

One can easily show that any meromorphic function on a complex closed Riemann surface can be represented as a composition of a birational map of this surface to ℂℙ2and a projection of the image curve froman appropriate pointp∊ ℂℙ2to the pencil of lines throughp. We introduce a natural stratiûcation of Hurwitz spaces according to the minimal degree of a plane curve such that a given meromorphic function can be represented in the above way and calculate the dimensions of these strata. We observe that they are closely related to a family of Severi varieties studied earlier by J. Harris, Z. Ran, and I. Tyomkin.

Discrete Dirac Operators, Critical Embeddings and Ihara-Selberg Functions

The aim of the paper is to formulate a discrete analogue of the claim made by Alvarez-Gaume et al., realizing the partition function of the free fermion on a closed Riemann surface of genus $g$ as a linear combination of $2^{2g}$ Pfaffians of Dirac operators. Let $G=(V,E)$ be a finite graph embedded in a closed Riemann surface $X$ of genus $g$, $x_e$ the collection of independent variables associated with each edge $e$ of $G$ (collected in one vector variable $x$) and $\S$ the set of all $2^{2g}$ spin-structures on $X$. We introduce $2^{2g}$ rotations $rot_s$ and $(2|E|\times 2|E|)$ matrices $\Delta(s)(x)$, $s\in \Sigma$, of the transitions between the oriented edges of $G$ determined by rotations $rot_s$. We show that the generating function of the even sets of edges of $G$, i.e., the Ising partition function, is a linear combination of the square roots of $2^{2g}$ Ihara-Selberg functions $I(\Delta(s)(x))$ also called Feynman functions. By a result of Foata and Zeilberger $I(\Delta(s)(x))=\det(I-\Delta'(s)(x))$, where $\Delta'(s)(x)$ is obtained from $\Delta(s)(x)$ by replacing some entries by $0$. Thus each Feynman function is computable in a polynomial time. We suggest that in the case of critical embedding of a bipartite graph $G$, the Feynman functions provide suitable discrete analogues for the Pfaffians of Dirac operators.