Cyclic Higgs bundles, subharmonic functions, and the Dirichlet problem

Li Q., Mochizuki T.

Fernández-Real X., Ros-Oton X.

Miyatake N.

We introduce generalized Kazdan-Warner equations on Riemannian manifolds associated with a linear action of a torus on a complex vector space. We show the existence and the uniqueness of the solution of the equation on any compact Riemannian manifold. As an application, we give a new proof of a theorem of Baraglia [5] which asserts that a cyclic Higgs bundle gives a solution of the periodic Toda equation.

Di Fratta G., Fiorenza A.

We prove a local regularity result for distributional solutions of Poisson’s equation withLpL^pdata. We use a very short argument based on Weyl’s lemma and the Riesz-Fréchet representation theorem.

Mochizuki T.

We prove the existence of an Hermitian–Einstein metric on holomorphic vector bundles with an Hermitian metric satisfying the analytic stability condition, under some assumption for the underlying Kähler manifolds. We also study the curvature decay of the Hermitian–Einstein metrics. It is useful for the study of the classification of instantons and monopoles on the quotients of four-dimensional Euclidean space by some types of closed subgroups. We also explain examples of doubly periodic monopoles corresponding to some algebraic data.

Dai S., Li Q.

In this paper, we derive a maximum principle for a type of elliptic systems and apply it to analyze the Hitchin equation for cyclic Higgs bundles. We show several domination results on the pullback metric of the (possibly branched) minimal immersion f associated to cyclic Higgs bundles. Also, we obtain a lower and upper bound of the extrinsic curvature of the image of f. As an application, we give a complete picture for maximal $$Sp(4,{\mathbb {R}})$$-representations in the $$2g-3$$ Gothen components and the Hitchin components.

Guedj V., Zeriahi A.

Guest M.A., Lin C.

Using nonlinear pde techniques, we construct a new family of globally smooth tt* structures. This includes tt* structures associated to the (orbifold) quantum cohomology of a finite number of complex projective spaces and weighted projective spaces. The existence of such magical of the tt* equations, namely smooth solutions characterized by asymptotic boundary conditions, was predicted by Cecotti and Vafa. In our situation, the tt* equations belong to a class of equations which we call the tt*-Toda lattice. Solutions of the tt*-Toda lattice are harmonic maps which have dual interpretations as Frobenius structures or variations of (semi-infinite) Hodge structures.

Simpson C.T.

Simpson C.T.

Ransford T.

Potential theory is the broad area of mathematical analysis encompassing such topics as harmonic and subharmonic functions, the Dirichlet problem, harmonic measure, Green's functions, potentials and capacity. This is an introduction to the subject suitable for beginning graduate students, concentrating on the important case of two dimensions. This permits a simpler treatment than other books, yet is still sufficient for a wide range of applications to complex analysis; these include Picard's theorem, the Phragmén–Lindelöf principle, the Koebe one-quarter mapping theorem and a sharp quantitative form of Runge's theorem. In addition there is a chapter on connections with functional analysis and dynamical systems, which shows how the theory can be applied to other parts of mathematics, and gives a flavour of some recent research. Exercises are provided throughout, enabling the book to be used with advanced courses on complex analysis or potential theory.

Hitchin N.J.

Donaldson S.K.

This paper investigates boundary value problems for Hermitian Yang—Mills equations over complex manifolds. The main result is the unique solubility of the Dirichlet problem for the Hermitian Yang—Mills equation. Connections with a number of topics are found, including the link with loop groups.

Hitchin N.J.

Donaldson S.K.