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Register a new userA Compactness Theorem for Embedded Measured Riemann Surface Laminations
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We prove a compactness theorem for embedded measured hyperbolic Riemann surface laminations in a compact almost complex manifold $(X, J)$. To prove compactness result, we show that there is a suitable topology on the space of measured Riemann surface laminations induced by Levy-Prokhorov metric. As an application of the compactness theorem, we show that given a biholomorphism of $phi $ of a closed complex manifold $X$, some power $phi^k $ ($k>0$) fixes a measured Riemann surface lamination in $X$.
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We study the dynamics of the geodesic and horocycle flows of the unit tangent bundle $(hat M, T^1mathcal{F})$ of a compact minimal lamination $(M,mathcal F)$ by negatively curved surfaces. We give conditions under which the action of the affine group generated by the joint action of these flows is minimal, and examples where this action is not minimal. In the first case, we prove that if $mathcal F$ has a leaf which is not simply connected, the horocyle flow is topologically transitive.
A Riemann surface $X$ is said to be of emph{parabolic type} if it supports a Greens function. Equivalently, the geodesic flow on the unit tangent of $X$ is ergodic. Given a Riemann surface $X$ of arbitrary topological type and a hyperbolic pants decomposition of $X$ we obtain sufficient conditions for parabolicity of $X$ in terms of the Fenchel-Nielsen parameters of the decomposition. In particular, we initiate the study of the effect of twist parameters on parabolicity. A key ingredient in our work is the notion of textit{non standard half-collar} about a hyperbolic geodesic. We show that the modulus of such a half-collar is much larger than the modulus of a standard half-collar as the hyperbolic length of the core geodesic tends to infinity. Moreover, the modulus of the annulus obtained by gluing two non standard half-collars depends on the twist parameter, unlike in the case of standard collars. Our results are sharp in many cases. For instance, for zero-twist flute surfaces as well as half-twist flute surfaces with concave sequences of lengths our results provide a complete characterization of parabolicity in terms of the length parameters. It follows that parabolicity is equivalent to completeness in these cases. Applications to other topological types such as surfaces with infinite genus and one end (a.k.a. the infinite Loch-Ness monster), the ladder surface, Abelian covers of compact surfaces are also studied.
We extend the Boutet de Monvel Toeplitz index theorem to complex manifold with isolated singularities following the relative $K$-homology theory of Baum, Douglas, and Taylor for manifold with boundary. We apply this index theorem to study the Arveson-Douglas conjecture. Let $ball^m$ be the unit ball in $mathbb{C}^m$, and $I$ an ideal in the polynomial algebra $mathbb{C}[z_1, cdots, z_m]$. We prove that when the zero variety $Z_I$ is a complete intersection space with only isolated singularities and intersects with the unit sphere $mathbb{S}^{2m-1}$ transversely, the representations of $mathbb{C}[z_1, cdots, z_m]$ on the closure of $I$ in $L^2_a(ball^m)$ and also the corresponding quotient space $Q_I$ are essentially normal. Furthermore, we prove an index theorem for Toeplitz operators on $Q_I$ by showing that the representation of $mathbb{C}[z_1, cdots, z_m]$ on the quotient space $Q_I$ gives the fundamental class of the boundary $Z_Icap mathbb{S}^{2m-1}$. In the appendix, we prove with Kai Wang that if $fin L^2_a(ball^m)$ vanishes on $Z_Icap ball ^m$, then $f$ is contained inside the closure of the ideal $I$ in $L^2_a(ball^m)$.
We present analytical implementation of conformal field theory on a compact Riemann surface. We consider statistical fields constructed from background charge modifications of the Gaussian free field and derive Ward identities which represent the Lie derivative operators in terms of the Virasoro fields and the puncture operators associated with the background charges. As applications, we derive Eguchi-Ooguris version of Wards equations and certain types of BPZ equations on a torus.
This paper is devoted to the study of reducing subspaces for multiplication operator $M_phi$ on the Dirichlet space with symbol of finite Blaschke product. The reducing subspaces of $M_phi$ on the Dirichlet space and Bergman space are related. Our strategy is to use local inverses and Riemann surface to study the reducing subspaces of $M_phi$ on the Bergman space, and we discover a new way to study the Riemann surface for $phi^{-1}circphi$. By this means, we determine the reducing subspaces of $M_phi$ on the Dirichlet space when the order of $phi$ is $5$; $6$; $7$ and answer some questions of Douglas-Putinar-Wang cite{DPW12}.
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