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Xueyuan Wan

Publications and source records attributed to Xueyuan Wan.

51 records · Page 3Linked to original sources
arXiv2019

Norm estimates and asymptotic faithfulness of the quantum $SU(n)$ representations of the mapping class groups

We give a direct proof for the asymptotic faithfulness of the quantum $SU(n)$ representations of the mapping class groups using peak sections in Kodaira embedding. We give also estimates on the norm of the parallell transport of the projective connection on the Verlinde bundle. The faithfulness has been proved earlier in [1] using Toeplitz operators of compact Kähler manifolds and in [10] using skein theory.

math.DG↗
arXiv2019

New Kähler metric on quasifuchsian space and its curvature properties

Let $QF(S)$ be the quasifuchsian space of a closed surface $S$ of genus $g\geq 2$. We construct a new mapping class group invariant Kähler metric on $QF(S)$. It is an extension of the Weil-Petersson metric onthe Teichmüller space $\mathcal T(S)\subset QF(S)$. We also calculate its curvature and prove some negativity for the curvature along the tautological directions.

math.GT↗
arXiv2019

Plurisuperharmonicity of reciprocal energy function on Teichmuller space and Weil-Petersson metrics

We consider harmonic maps$u(z): \mathcal{X}_z\to N$ in a fixed homotopy class from Riemann surfaces $\mathcal{X}_z$ of genus $g\geq 2$ varying in the Teichmü{}ller space $\mathcal T$ to a Riemannian manifold $N$ with non-positive Hermitian sectional curvature. The energy function $E(z)=E(u(z))$ can be viewed as a function on $\mathcal T$ and we study its first and the second variations. We prove that the reciprocal energy function $E(z)^{-1}$ is plurisuperharmonic on Teichmüller space. We also obtain the (strict) plurisubharmonicity of $\log E(z)$ and $E(z)$. As an application, we get the following relationship between the second variation of logarithmic energy function and the Weil-Petersson metric if the harmonic map $u(z)$ is holomorphic or anti-holomorphic and totally geodesic, i.e., $$ \sqrt{-1}\p\b{\p}\log E(z)=\frac{ω_{WP}}{2π(g-1)}. $$ We consider also the energy function $E(z)$ associated to the harmonic maps from a fixed compact Kähler manifold $M$ to Riemann surfaces ${\mathcal{X}_z\}_{z\in\mathcal{T}}$ in a fixed homotopy class. If $u(z)$ is holomorphic or anti-holomorphic, then the above equation is also proved.

math.DG↗
arXiv2018

Geometry of logarithmic forms and deformations of complex structures

We present a new method to solve certain $\bar{\partial}$-equations for logarithmic differential forms by using harmonic integral theory for currents on Kahler manifolds. The result can be considered as a $\bar{\partial}$-lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness of logarithmic forms, give geometric and simpler proofs of Deligne's degeneracy theorem for the logarithmic Hodge to de Rham spectral sequences at $E_1$-level, as well as certain injectivity theorem on compact Kahler manifolds. Furthermore, for a family of logarithmic deformations of complex structures on Kahler manifolds, we construct the extension for any logarithmic $(n,q)$-form on the central fiber and thus deduce the local stability of log Calabi-Yau structure by extending an iteration method to the logarithmic forms. Finally we prove the unobstructedness of the deformations of a log Calabi-Yau pair and a pair on a Calabi-Yau manifold by differential geometric method.

math.AG↗
arXiv2018

On local stabilities of $p$-Kähler structures

By use of a natural extension map and a power series method, we obtain a local stability theorem for p-Kähler structures with the $(p,p+1)$-th mild $\partial\bar\partial$-lemma under small differentiable deformations.

math.CV↗
arXiv2018

Plurisubharmonicity and geodesic convexity of energy function on Teichmüller space

Let $π:\mc{X}\to \mc{T}$ be Teichmüller curve over Teichmüller space $\mc{T}$, such that the fiber $\mc{X}_z=π^{-1}(z)$ is exactly the Riemann surface given by the complex structure $z\in \mc{T}$. For a fixed Riemannian manifold $M$ and a continuous map $u_0: M\to \mc{X}_{z_0}$, let $E(z)$ denote the energy function of the harmonic map $u(z):M\to \mc{X}_z$ homotopic to $u_0$, $z\in \mathcal T$. We obtain the first and the second variations of the energy function $E(z)$, and show that $\log E(z)$ is strictly plurisubharmonic on Teichmüller space, from which we give a new proof on the Steinness of Teichmüller space. We also obtain a precise formula on the second variation of $E^{1/2}$ if $\dim M=1$. In particular, we get the formula of Axelsson-Schumacher on the second variation of the geodesic length function. We give also a simple and corrected proof for the theorem of Yamada, the convexity of energy function $E(t)$ along Weil-Petersson geodesics. As an application we show that $E(t)^c$ is also strictly convex for $c>5/6$ and convex for $c=5/6$ along Weil-Petersson geodesics. We also reprove a Kerckhoff's theorem which is a positive answer to the Nielsen realization problem.

math.DG↗
arXiv2018

Geodesic-Einstein metrics and nonlinear stabilities

In this paper, we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic-Einstein metric on this line bundle, which generalize the classical stabilities and Hermitian-Einstein metrics of holomorphic vector bundles. We introduce a Donaldson type functional and show that this functional attains its absolute minimum at geodesic-Einstein metrics, and we also discuss the relations between the existence of geodesic-Einstein metrics and the nonlinear stabilities of the line bundle. As an application, we will prove that a holomorphic vector bundle admits a Finsler-Einstein metric if and only if it admits a Hermitian-Einstein metric, which answers a problem posed by S. Kobayashi.

math.DG↗
arXiv2018

Chern forms of holomorphic Finsler vector bundles and some applications

In this paper, we present two kinds of total Chern forms $c(E,G)$ and $\mathcal{C}(E,G)$ as well as a total Segre form $s(E,G)$ of a holomorphic Finsler vector bundle $π:(E,G)\to M$ expressed by the Finsler metric $G$, which answers a question of J. Faran (\cite{Faran}) to some extent. As some applications, we show that the signed Segre forms $(-1)^ks_k(E,G)$ are positive $(k,k)$-forms on $M$ when $G$ is of positive Kobayashi curvature; we prove, under an extra assumption, that a Finsler-Einstein vector bundle in the sense of Kobayashi is semi-stable; we introduce a new definition of a flat Finsler metric, which is weaker than Aikou's one (\cite{Aikou}) and prove that a holomorphic vector bundle is Finsler flat in our sense if and only if it is Hermitian flat.

math.DG↗
arXiv2018

A Donaldson type functional on a holomorphic Finsler vector bundle

In this paper, we solve a problem of Kobayashi posed in \cite{Ko4} by introducing a Donaldson type functional on the space $F^+(E)$ of strongly pseudo-convex complex Finsler metrics on $E$ -- a holomorphic vector bundle over a closed Kähler manifold $M$. This Donaldson type functional is a generalization in the complex Finsler geometry setting of the original Donaldson functional and has Finsler-Einstein metrics on $E$ as its only critical points, at which this functional attains the absolute minimum.

math.DG↗
arXiv2018

The asymptotics of the $L^2$-curvature and the second variation of analytic torsion on Teichmüller space

We consider the relative canonical line bundle $K_{\mathcal{X}/\mathcal{T}}$ and a relatively ample line bundle $(L, e^{-ϕ})$ over the total space $ \mathcal{X}\to \mathcal{T}$ of fibration over the Teichmüller space by Riemann surfaces. We consider the case when the induced metric $\sqrt{-1}\partial\bar{\partial}ϕ|_{\mathcal{X}_y}$ on $\mathcal{X}_y$ has constant scalar curvature and we obtain the curvature asymptotics of $L^2$-metric and Quillen metric of the direct image bundle $E^k=π_*(L^k+K_{\mathcal{X}/\mathcal{T}})$. As a consequence we prove that the second variation of analytic torsion $τ_k(\bar{\partial})$ satisfies \begin{align*} \partial\bar{\partial}\logτ_k(\bar{\partial})=o(k^{-l}) \end{align*} at the point $y\in\mathcal{T}$ for any $l\geq 0$ as $k\to\infty$.

math.DG↗
arXiv2018

Holomorphic sectional curvature of complex Finsler manifolds

In this paper, we get an inequality in terms of holomorphic sectional curvature of complex Finsler metrics. As applications, we prove a Schwarz Lemma from a complete Riemannian manifold to a complex Finsler manifold. We also show that a strongly pseudoconvex complex Finsler manifold with semi-positive but not identically zero holomorphic sectional curvature has negative Kodaira dimension under an extra condition.

math.DG↗
arXiv2016

Logarithmic vanishing theorems on compact Kähler manifolds I

In this paper, we first establish an $L^2$-type Dolbeault isomorphism for logarithmic differential forms by Hörmander's $L^2$-estimates. By using this isomorphism and the construction of smooth Hermitian metrics, we obtain a number of new vanishing theorems for sheaves of logarithmic differential forms on compact Kähler manifolds with simple normal crossing divisors, which generalize several classical vanishing theorems, including Norimatsu's vanishing theorem, Gibrau's vanishing theorem, Le Potier's vanishing theorem and a version of the Kawamata-Viehweg vanishing theorem.

math.AG↗
arXiv2014

Rational curves on Hermitian manifolds

By using analytic method, we prove that there exist rational curves on compact Hermitian manifolds with positive holomorphic bisectional curvature. It confirms a question of S.-T. Yau. It is well-known that Mori proved in \cite{Mori79} that every compact complex manifold $N$ with $c_1(N)>0$ contains at least one rational curve. However, as a borderline example, we show that the standard Hopf surface $S^1\times S^3$ has a Hermitian metric with non-negative holomorphic bisectional curvature (in particular, $c_1(S^1\times S^3)\geq 0$), but it contains no rational curve.

math.DG↗