arXiv subjects
Publications and source records attributed to Xueyuan Wan.
Using spinors, we show a dihedral type rigidity for polyhedral initial data sets. This rigidity connects spacetime positive mass theorem, dihedral rigidity and capillary marginally trapped surfaces. Our method is to extend the rigidity analysis of spacetime positive mass theorem due to Beig-Chrusciel to the settings of a twisted spinor bundle.
A theorem of Llarull says that if a smooth metric $g$ on the $n$-sphere $\mathbb{S}^n$ is bounded below by the standard round metric and the scalar curvature $R_g$ of $g$ is bounded below by $n (n - 1)$, then the metric $g$ must be the standard round metric. We prove a spectral Llarull theorem by replacing the bound $R_g \geq n (n - 1)$ by a lower bound on the first eigenvalue of an elliptic operator involving the Laplacian and the scalar curvature $R_g$. We utilize two methods: spinor and spacetime harmonic function.
We show that every integer in the interval $[2pχ(Σ), -2pχ(Σ)]$ is achieved by the signature of a rank $2p$ flat symplectic bundle over a surface with boundary $Σ$. When $p=1$, one can prescribe the type (elliptic, parabolic, hyperbolic) of the holonomy along the boundary.
By exploiting the conformality of a warped product metric with a direct product metric, we develop a new connection on a twisted spinor bundle and its associated Dirac operator. We obtain a Llarull type scalar curvature rigidity for a general class of domains in a warped product. Also, we are able to address Gromov dihedral rigidity in hyperbolic space assuming matching angles.
This paper deals with the representations of the fundamental groups of compact surfaces with boundary into classical simple Lie groups of Hermitian type. We relate work on the signature of the associated local systems of Atiyah-Patodi-Singer, to Burger-Iozzi-Wienhard's Toledo invariant. To measure the difference, we extend Atiyah-Patodi-Singer's rho invariant, initially defined on $\mathrm{U}(p)$, to discontinuous class functions, first on $\mathrm{U}(p,q)$, and then on other classical groups via embeddings into $\mathrm{U}(p,q)$. In this way, we present three different invariants -- signature, Toledo and rho invariant -- in a unifying way, which is a version of the classical signature formula of Atiyah-Patodi-Singer for manifolds with boundary.
In this paper, we define two types of strongly decomposable positivity, which serve as generalizations of (dual) Nakano positivity and are stronger than the decomposable positivity introduced by S. Finski. We provide the criteria for strongly decomposable positivity of type I and type II and prove that the Schur forms of a strongly decomposable positive vector bundle of type I are weakly positive, while the Schur forms of a strongly decomposable positive vector bundle of type II are positive. These answer a question of Griffiths affirmatively for strongly decomposably positive vector bundles. Consequently, we present an algebraic proof of the positivity of Schur forms for (dual) Nakano positive vector bundles, which was initially proven by S. Finski.
In this paper, we consider a special relative Kähler fibration that satisfies a homogenous Monge-Ampère equation, which is called a Monge-Ampère fibration. There exist two canonical types of generalized Weil-Petersson metrics on the base complex manifold of the fibration. For the second generalized Weil-Petersson metric, we obtain an explicit curvature formula and prove that the holomorphic bisectional curvature is non-positive, the holomorphic sectional curvature, the Ricci curvature, and the scalar curvature are all bounded from above by a negative constant. For a holomorphic vector bundle over a compact Kähler manifold, we prove that it admits a projectively flat Hermitian structure if and only if the associated projective bundle fibration is a Monge-Ampère fibration. In general, we can prove that a relative Kähler fibration is Monge-Ampère if and only if an associated infinite rank Higgs bundle is Higgs-flat. We also discuss some typical examples of Monge-Ampère fibrations.
For a holomorphic vector bundle $E$ over a Hermitian manifold $M$ there are two important notions of curvature positivity, the Griffiths positivity and Nakano positivity. We study the consequence of these positivities and the relevant estimates. If $E$ is Griffiths negative over Kähler manifold, then there is a Kähler metric on its total space $E$, and we calculate the curvature and prove the non-positivity of the curvature along the tautological direction. The Nakano positivity can be formulated as a positivity for the Nakano curvature operator and we give estimate the Nakano curvature operator associated with a Nakano positive direct image bundle. As applications we construct a mapping class group invariant Kähler metric on the quasi-Fuchsian space QF$(S)$, which extends the Weil-Petersson metric on the Teichmüller space $\mathcal{T}(S)\subset {\rm QF}(S)$, and we obtain estimates for the Nakano curvature operator for the dual Weil-Petersson metric on the holomorphic cotangent bundle of Teichmüller space.
Let $(M,g)$ be a complete connected $n$-dimensional Riemannian spin manifold without boundary such that the scalar curvature satisfies $R_g\geq -n(n-1)$ and $\mathcal{E}\subset M$ be an asymptotically hyperbolic end, we prove that the mass functional of the end $\mathcal{E}$ is timelike future-directed or zero. Moreover, it vanishes if and only if $(M,g)$ is isometric to the hyperbolic space. We also consider the mass of an asymptotically hyperbolic manifold with compact boundary, we prove the mass is timelike future-directed if the mean curvature of the boundary is bounded from below by a function defined using distance estimates. As an application, the mass is timelike future-directed if the mean curvature of the boundary is bounded from below by $-(n-1)$ or the scalar curvature satisfies $R_g\geq (-1+κ)n(n-1)$ for any positive constant $κ$ less than one.
We generalize a band width estimate of Gromov to CMC initial data sets. We give three independent proofs: via the stability of a hypersurface with prescribed null expansion, via a perturbation of the spacetime harmonic function and via the Dirac operator.
In this paper, by using Atiyah-Patodi-Singer index theorem, we obtain a formula for the signature of a flat symplectic vector bundle over a surface with boundary, which is related to the Toledo invariant of a surface group representation in the real symplectic group and the Rho invariant on the boundary. As an application, we obtain a Milnor-Wood type inequality for the signature. In particular, we give a new proof of the Milnor-Wood inequality for the Toledo invariant in the case of closed surfaces and obtain some modified inequalities for the surface with boundary.
Let $π:\mathcal{X}\to M$ be a holomorphic fibration with compact fibers and $L$ a relatively ample line bundle over $\mathcal{X}$. We obtain the asymptotic of the curvature of $L^2$-metric and Qullien metric on the direct image bundle $π_*(L^k\otimes K_{\mathcal{X}/M})$ up to the lower order terms than $k^{n-1}$ for large $k$. As an application we prove that the analytic torsion $τ_k(\bar{\partial})$ satisfies $\partial\bar{\partial}\log(τ_k(\bar{\partial}))^2=o(k^{n-1})$, where $n$ is the dimension of fibers.
In this paper, we introduce a flow over the projective bundle $p:P(E^*)\to M$, which is a natural generalization of both Hermitian-Yang-Mills flow and Kähler-Ricci flow. We prove that the semipositivity of curvature of the hyperplane line bundle $\mathcal{O}_{P(E^*)}(1)$ is preserved along this flow under the null eigenvector assumption. As applications, we prove that the semipositivity is preserved along the this flow if the base manifold $M$ is a curve, which implies that the Griffiths semipositivity is preserved along the Hermitian-Yang-Mills flow over a curve. And we also reprove that the nonnegativity of holomorphic bisectional curvature is preserved under Kähler-Ricci flow.
In this paper, we introduce the associated geodesic-Einstein flow for a relatively ample line bundle $L$ over the total space $\mathcal{X}$ of a holomorphic fibration and obtain a few properties of that flow. In particular, we prove that the pair $(\mathcal{X}, L)$ is nonlinear semistable if the {associated} Donaldson type functional is bounded from below and the geodesic-Einstein flow has long-time {existence property}. We also define the associated $S$-classes and $C$-classes for $(\mathcal{X}, L)$ and obtain two inequalities between them when $L$ admits a geodesic-Einstein metric. Finally, in the appendix of this paper, we prove that a relatively ample line bundle is geodesic-Einstein if and only if an associated infinite rank bundle is Hermitian-Einstein.
By use of a natural map introduced recently by the first and third authors from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the small differentiable deformation of this manifold, we will give a power series proof for Kodaira-Spencer's local stability theorem of Kähler structures. We also obtain two new local stability theorems, one of balanced structures on an $n$-dimensional balanced manifold with the $(n-1,n)$-th mild $\partial\bar\partial$-lemma by power series method and the other one on $p$-Kähler structures with the deformation invariance of $(p,p)$-Bott-Chern numbers.
In the paper, we consider the harmonic maps between surfaces $Σ$ and $S$ in the homotopy class of a (branched) covering map $u_0$. We prove the uniqueness of critical points of energy function and the injectivity of Hopf differential if $u_0$ is a covering map. On the other hand, if $u_0$ is a branched covering, we show that the uniqueness of critical points fails if $u_0$ is a non-simple branched covering, and prove the injectivity of Hopf differential $Φ:\mc{T}(S)\to \op{QD}(Σ,g)$ when $g=[u_0^* h]$ for some hyperbolic metric $h$ on $S$.
In this short note, we prove that a complex Finsler vector bundle with positive Kobayashi curvature must be ample, which partially solves a problem of S. Kobayashi posed in 1975. As applications, a strongly pseudoconvex complex Finsler manifold with positive Kobayashi curvature must be biholomorphic to the complex projective space; we also show that all Schur polynomials are numerically positive for complex Finsler vector bundles with positive Kobayashi curvature.
For a fixed smooth map $u_0$ between two Riemann surfaces $Σ$ and $S$ with non-zero degree, we consider the energy function on Teichmüller space $\mc{T}$ of $Σ$ that assigns to a complex structure $t\in \mc{T}$ on $Σ$ the energy of the harmonic map $u_t:Σ_t:=(Σ,t) \to S$ homotopic to $u_0$. We prove that the energy function is convex at its critical points. If $t_0\in\mc{T}$ is a critical point such that $du_{t_0}$ is never zero, then the energy function is strictly convex at this point. As an application, in the case that $u_0$ is a covering map, we prove that there exists a unique critical point $t_0\in \mc{T}$ minimizing the energy function. Moreover, the energy density satisfies $\frac{1}{2}|du|^2(t_0)\equiv 1$ and the Hessian of the energy function is positive definite at this point.