Holomorphic Diffeomorphisms Research Articles

Volume functionals on pseudoconvex hypersurfaces

The focus of this paper is on a volume form defined on a pseudoconvex hypersurface [Formula: see text] in a complex Calabi–Yau manifold (that is, a complex [Formula: see text]-manifold with a nowhere-vanishing holomorphic [Formula: see text]-form). We begin by defining this volume form and observing that it can be viewed as a generalization of the affine-invariant volume form on a convex hypersurface in [Formula: see text]. We compute the first variation, which leads to a similar generalization of the affine mean curvature. In Sec. 2, we investigate the constrained variational problem, for pseudoconvex hypersurfaces [Formula: see text] bounding compact domains [Formula: see text]. That is, we study critical points of the volume functional [Formula: see text] where the ordinary volume [Formula: see text] is fixed. The critical points are analogous to constant mean curvature submanifolds. We find that Sasaki–Einstein hypersurfaces satisfy the condition, and in particular the standard sphere [Formula: see text] C[Formula: see text] does. The main work in the paper comes in Sec. 3 where we compute the second variation about the sphere. We find that it is negative in “most” directions but non-negative in directions corresponding to deformations of [Formula: see text] by holomorphic diffeomorphisms. We are led to conjecture a “minimax” characterization of the sphere. We also discuss connections with the affine geometry case and with Kähler–Einstein geometry. Our original motivation for investigating these matters came from the case [Formula: see text] and the embedding problem studied in our previous paper [S. Donaldson and F. Lehmann, Closed 3-forms in five dimensions and embedding problems, preprint (2022), arXiv:2210.16208]. There are some special features in this case. The volume functional can be defined without reference to the embedding in [Formula: see text] using only a closed “pseudoconvex” real [Formula: see text]-form on [Formula: see text]. In Sec. 4, we review this and develop some of the theory from the point of the symplectic structure on exact [Formula: see text]-forms on [Formula: see text] and the moment map for the action of the diffeomorphisms of [Formula: see text].

Yang-Mills model for centrally extended 2D gravity

A Yang-Mills theory linear in the scalar curvature for two-dimensional gravity with symmetry generated by the semidirect product formed with the Lie derivative of the algebra of diffeomorphisms with the two-dimensional Abelian algebra is formulated. As compared with dilaton models, the role of the dilaton is played by the dual field strength of a $U(1)$ gauge field. All vacuum solutions are found. They are either black holes or have constant scalar curvature. Those with constant scalar curvature have constant dual field strength. In particular, solutions with vanishing cosmological constant but nonzero scalar curvature exist. In the conformal-Lorenz gauge, the model has a conformal field theory interpretation whose residual symmetry combines holomorphic diffeomorphisms with a subclass of $U(1)$ gauge transformations while preserving two-dimensional de Sitter and anti-de Sitter boundary conditions. This is the same symmetry as in Jackiw-Teitelboim-Maxwell gravity considered by Hartman and Strominger. It is argued that this is the only nontrivial Yang-Mills model linear in the scalar curvature that exists for real Lie algebras of dimension four.

A Factorization Theorem for Harmonic Maps

Let f be a harmonic map from a Riemann surface to a Riemannian n-manifold. We prove that if there is a holomorphic diffeomorphism h between open subsets of the surface such that $$f\circ h = f$$ , then f factors through a holomorphic map onto another Riemann surface. If such h is anti-holomorphic, we obtain an analogous statement. For minimal maps, this result is well known and is a consequence of the theory of branched immersions of surfaces due to Gulliver–Osserman–Royden. Our proof relies on various geometric properties of the Hopf differential.

A decoupling property of some Poisson structures on Matn×d(C)×Matd×n(C) supporting GL(n,C)×GL(d,C) Poisson–Lie symmetry

We study a holomorphic Poisson structure defined on the linear space S(n,d)≔Matn×d(C)×Matd×n(C) that is covariant under the natural left actions of the standard GL(n,C) and GL(d,C) Poisson–Lie groups. The Poisson brackets of the matrix elements contain quadratic and constant terms, and the Poisson tensor is non-degenerate on a dense subset. Taking the d = 1 special case gives a Poisson structure on S(n, 1), and we construct a local Poisson map from the Cartesian product of d independent copies of S(n, 1) into S(n, d), which is a holomorphic diffeomorphism in a neighborhood of 0. The Poisson structure on S(n, d) is the complexification of a real Poisson structure on Matn×d(C) constructed by the authors and Marshall, where a similar decoupling into d independent copies was observed. We also relate our construction to a Poisson structure on S(n, d) defined by Arutyunov and Olivucci in the treatment of the complex trigonometric spin Ruijsenaars–Schneider system by Hamiltonian reduction.

Loewner evolution of hedgehogs and 2-conformal measures of circle maps

AbstractLet f be a germ of a holomorphic diffeomorphism with an irrationally indifferent fixed point at the origin in ${\mathbb C}$ (i.e. $f(0) = 0, f'(0) = e^{2\pi i \alpha }, \alpha \in {\mathbb R} - {\mathbb Q}$ ). Pérez-Marco [Fixed points and circle maps. Acta Math.179(2) (1997), 243–294] showed the existence of a unique continuous monotone one-parameter family of non-trivial invariant full continua containing the fixed point called Siegel compacta, and gave a correspondence between germs and families $(g_t)$ of circle maps obtained by conformally mapping the complement of these compacts to the complement of the unit disk. The family of circle maps $(g_t)$ is the orbit of a locally defined semigroup $(\Phi _t)$ on the space of analytic circle maps, which we show has a well-defined infinitesimal generator X. The explicit form of X is obtained by using the Loewner equation associated to the family of hulls $(K_t)$ . We show that the Loewner measures $(\mu _t)$ driving the equation are 2-conformal measures on the circle for the circle maps $(g_t)$ .

Hedgehogs in higher dimensions and their applications

In this paper we study the dynamics of germs of holomorphic diffeomorphisms of $(\mathbb{C}^{n},0)$ with a fixed point at the origin with exactly one neutral eigenvalue. We prove that the map on any local center manifold of $0$ is quasiconformally conjugate to a holomorphic map and use this to transport results from one complex dimension to higher dimensions.

Hedgehogs for neutral dissipative germs of holomorphic diffeomorphisms of (C^2,0)

Hedgehogs for neutral dissipative germs of holomorphic diffeomorphisms of (C^2,0)

Hedgehogs for neutral dissipative germs of holomorphic diffeomorphisms of (C^2,0)

Hedgehogs for neutral dissipative germs of holomorphic diffeomorphisms of (C^2,0)

Diffeomorphisms on the fuzzy sphere

Abstract Diffeomorphisms can be seen as automorphisms of the algebra of functions. In matrix regularization, functions on a smooth compact manifold are mapped to finite-size matrices. We consider how diffeomorphisms act on the configuration space of the matrices through matrix regularization. For the case of the fuzzy $$S^2$, we construct the matrix regularization in terms of the Berezin–Toeplitz quantization. By using this quantization map, we define diffeomorphisms on the space of matrices. We explicitly construct the matrix version of holomorphic diffeomorphisms on $$S^2$. We also propose three methods of constructing approximate invariants on the fuzzy $$S^2$. These invariants are exactly invariant under area-preserving diffeomorphisms and only approximately invariant (i.e. invariant in the large-$$N$ limit) under general diffeomorphisms.

Hedgehogs in higher dimensions and their applications

In this paper we study the dynamics of germs of holomorphic diffeomorphisms of $(\mathbb{C}^{n},0)$ with a fixed point at the origin with exactly one neutral eigenvalue. We prove that the map on any local center manifold of $0$ is quasiconformally conjugate to a holomorphic map and use this to transport results from one complex dimension to higher dimensions.

Quantum Hydrodynamics, Rotating Superfluid and Gravitational Anomaly

We present a consistent scheme of quantization of chiral flows (flows with extensive vorticity) in ideal hydrodynamics in two dimensions. Chiral flows occur in rotating superfluid, rotating turbulence, and also in electronic systems in magnetic field in the regime of a fractional Hall effect. The quantization is based on a geometric relation of chiral flows to two-dimensional quantum gravity and is implemented by the gravitational anomaly. The effect of the gravitational anomaly changes the major property of classical hydrodynamics, the Helmholtz law: vortices are no longer frozen into the flow. Effects of quantization could be cast in the form of quantum stress. We show that the quantum stress is a generator of Virasoro algebra, the centrally extended algebra of holomorphic diffeomorphisms.

Stable Manifolds of Two-dimensional Biholomorphisms Asymptotic to Formal Curves

Abstract Let $F\in\, \textrm{Diff}\, ({\mathbb{C}}^2, 0)$ be a germ of a holomorphic diffeomorphism and let $\Gamma $ be an invariant formal curve of $F$. Assume that the restricted diffeomorphism $F|_\Gamma $ is either hyperbolic attracting or rationally neutral non-periodic (these are the conditions that the diffeomorphism $F|_\Gamma $ should satisfy, if $\Gamma $ were convergent, in order to have orbits converging to the origin). Then we prove that $F$ has finitely many stable manifolds, either open domains or parabolic curves, consisting of and containing all converging orbits asymptotic to $\Gamma $. Our results generalize to the case where $\Gamma $ is a formal periodic curve of $F$.

QUANTIZATION OF HYDRODYNAMICS: ROTATING SUPERFLUID, AND GRAVITATIONAL ANOMALY

We present a consistent scheme of quantization of chiral flows (flows with extensive vorticity) in ideal hydrodynamics in two dimensions. Chiral flows occur in rotating superfluid, rotating turbulence and also in electronic systems in the magnetic field in the regime of a fractional Hall effect. The quantization is based on a geometric relation of chiral flows to two-dimensional quantum gravity and is implemented by the gravitational anomaly. The effect of the gravitational anomaly violates the major property of classical hydrodynamics, the Helmholtz law: vortices are no longer frozen into the flow. Effects of quantization could be cast in the form of quantum stress. We show that the quantum stress generates Virasoro algebra, the centrally extended algebra of holomorphic diffeomorphisms.

Semi-parabolic Bifurcations in Complex Dimension Two

Parabolic bifurcations in one complex dimension demonstrate a wide variety of interesting dynamical phenomena. In this paper we consider the bifurcations of a holomorphic diffeomorphism in two complex dimensions with a semi-parabolic, semiattracting fixed point.

Transverse Lie jets and holomorphic geometric objects on transverse bundles

Two holomorphic fields of geometric objects on a transverse Weil bundle are called equivalent if there exists a holomorphic diffeomorphism of this bundle onto itself which induces the identity transformation of the base manifold and maps one of these fields into the other. In terms of transverse Lie jets, we establish necessary and sufficient conditions for a holomorphic field of geometric objects on a transverse Weil bundle to be equivalent to the prolongation of a field of foliated geometric objects given on the base manifold. As an example, we consider a holomorphic linear connection on a transverse bundle.

Positive area and inaccessible fixed points for hedgehogs

Let $f$ be a germ of holomorphic diffeomorphism with an irrationally indifferent fixed point at the origin in $\mathbb{C}$ (i.e. $f(0)=0,f^{\prime }(0)=e^{2{\it\pi}i{\it\alpha}},{\it\alpha}\in \mathbb{R}-\mathbb{Q}$). Perez-Marco showed the existence of a unique family of non-trivial invariant full continua containing the fixed point called Siegel compacta. When $f$ is non-linearizable (i.e. not holomorphically conjugate to the rigid rotation $R_{{\it\alpha}}(z)=e^{2{\it\pi}i{\it\alpha}}z$) the invariant compacts obtained are called hedgehogs. Perez-Marco developed techniques for the construction of examples of non-linearizable germs [Fixed points and circle maps. Acta Math. 179(2) (1997), 243–294]; these were used by the author to construct hedgehogs of Hausdorff dimension one, and adapted by Cheritat [Relatively compact Siegel disks with non-locally connected boundaries, Preprint, 2009] to construct Siegel disks with pseudo-circle boundaries. We use these techniques to construct hedgehogs of positive area and hedgehogs with inaccessible fixed points.

Family of intersecting totally real manifolds of $(C^n,0)$ and germs of holomorphic diffeomorphisms

We prove the existence (and give a characterization) of a germ of complex analytic set left invariant by an abelian group of germs of holomorphic dieomorphisms at a common xed point.We also give that ensure that such a group can be linearized holomorphically near the xed point. It rests on a \small divisors condition of the family of linear parts. The second part of this article is devoted to the study families of totally real intersecting n-submanifolds of (C n ; 0). We give some conditions which allow to straighten holomorphically the family. If this is not possible to do it formally, we construct a germ of complex analytic set at the origin which interesection with the family can be holomorphically straightened. The second part is an application of the rst.

Quasi-conformal deformations of nonlinearizable germs

Let f ( z ) = e 2 π i α z + O ( z 2 ) , α ∈ R f(z) = e^{2\pi i \alpha }z + O(z^2), \alpha \in \mathbb {R} , be a germ of a holomorphic diffeomorphism in C \mathbb {C} . For α \alpha rational and f f of infinite order, the space of conformal conjugacy classes of germs topologically conjugate to f f is parametrized by the Ecalle-Voronin invariants (and in particular is infinite-dimensional). When α \alpha is irrational and f f is nonlinearizable it is not known whether f f admits quasi-conformal deformations. We show that if f f has a sequence of repelling periodic orbits converging to the fixed point, then f f embeds into an infinite-dimensional family of quasi-conformally conjugate germs, no two of which are conformally conjugate.

Generic pseudogroups on and the topology of leaves

AbstractWe show that generically a pseudogroup generated by holomorphic diffeomorphisms defined about $0\in \mathbb{C} $ is free in the sense of pseudogroups even if the class of conjugacy of the generators is fixed. This result has a number of consequences on the topology of leaves for a (singular) holomorphic foliation defined on a neighborhood of an invariant curve. In particular, in the classical and simplest case arising from local nilpotent foliations possessing a unique separatrix which is given by a cusp of the form $\{ {y}^{2} - {x}^{2n+ 1} = 0\} $, our results allow us to settle the problem of showing that a generic foliation possesses only countably many non-simply connected leaves.

Dynamics of one-resonant biholomorphisms

Our first main result is a construction of a simple formal normal form for holomorphic diffeomorphisms in \mathbb C^n whose differentials have one-dimensional family of resonances in the first m eigenvalues, m\leq n (but more resonances are allowed for other eigenvalues). Next, we provide invariants and give conditions for the existence of basins of attraction. Finally, we give applications and examples demonstrating the sharpness of our conditions.