Kyle Broder
We introduce two new curvatures which refine and elucidate the real bisectional curvature considered by Yang-Zheng \cite{YangZheng}. A unified perspective of the various forms of the Schwarz lemma is given, leading to novel Chern-Lu, Aubin-Yau, and Chen-Cheng-Look Schwarz lemmas in both the Kähler and Hermitian category.
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A general Schwarz lemma for Hermitian manifolds
本文改进了埃尔米特流形间的Schwarz引理，引入更弱的新曲率约束，证明其在联络改变下的不变性，并由此为Gauduchon联络建立Schwarz引理，显著改进了Wu--Yau定理。
On the Schwarz Lemma for Bergman metrics of bounded domains
本文提出基于概率论Cauchy-Schwarz不等式的新Schwarz引理，用于有界域Bergman度量的研究。
We prove two results related to the Schwarz lemma in complex geometry. First, we show that if the inequality in the Schwarz lemmata of Yau, Royden and Tosatti becomes equality at one point, then the equality holds on the whole manifold. In particular, the holomorphic map is totally geodesic and has constant rank. In the second part, we study the holomorphic sectional curvature on an almost Hermitian manifold and establish a Schwarz lemma in terms of holomorphic sectional curvatures in almost Hermitian setting.
The Schwarz--Pick lemma is a fundamental result in complex analysis. It is well-known that Yau generalized it to the higher dimensional manifolds by applying his maximum principle for complete Riemannian manifolds. Jeffres obtained Schwarz lemma for volume forms of conical Kähler metrics, based on a barrier function and the maximum principle argument. In this note, we generalize Jeffres' result to general cone angles including the case when the pullback of the metric would blows up along the divisors.
Expository notes on the Schwarz lemma born out of some lectures given on the subject.
In this paper, we consider some generalized holomorphic maps between pseudo-Hermitian manifolds and Hermitian manifolds. By Bochner formulas and comparison theorems, we establish related Schwarz type results. As corollaries, Liouville theorem and little Picard theorem for basic CR functions are deduced. Finally, we study CR Carathéodory pseudodistance on CR manifolds.
The Schwarz lemmas are well-known characterizations for holomorphic maps and we exhibit two examples of their applications. For a sequence family of biholomorphisms $f_j$, it is useful to determine the location of $f_j(q)$ for a fixed point $q$ in source manifolds (see Proposition \ref{2.5}). With it, we extend the Fornaess-Stout's theorem of \citep{FS77} in monotone unions of balls to ellipsoids in Section \ref{sec2}. In Section \ref{sec3}, we discuss the curvature bounds of complete Kähler metric on $\rtimes$ domains defined in \citep{Liu004} with an idea of \citep{Ya76}.
RC-positivity, Schwarz's lemma and comparison theorems
本文针对具正曲率界的抽象埃尔米特全纯向量丛间全纯丛映射建立施瓦茨引理，将其应用于曲率张量满足RC-正性的复流形间全纯映射，并利用该引理得到新的直径和体积比较定理。
In this paper, we consider transversally harmonic maps between Riemannian manifolds with Riemannian foliations. In terms of the Bochner techniques and sub-Laplacian comparison theorem, we are able to establish a generalization of the Schwarz lemma for transversally harmonic maps of bounded generalized transversal dilatation. In addition, we also obtain a Schwarz type lemma for transversally holomorphic maps between Kähler foliations.
In this paper, we consider some generalized holomorphic maps between pseudo-Hermitian manifolds. These maps include the \emph{CR} maps and the transversally holomorphic maps. In terms of some sub-Laplacian or Hessian type Bochner formulas, and comparison theorems in the pseudo-Hermitian version, we are able to establish several Schwarz type results for both the \emph{CR} maps and the transversally holomorphic maps between pseudo-Hermitian manifolds. Finally, we also discuss the \emph{CR} hyperbolicity problem for pseudo-Hermitian manifolds.
In this paper, we obtain a Schwarz lemma for holomorphic mappings from the unit polydisc $P_m$ into the unit polydisc $P_n$, here $P_m$ and $P_n$ are endowed with $\mbox{Aut}(P_m)$-invariant Kähelr-Berwald metric $F_{t,k}$ and $\mbox{Aut}(P_n)$-invariant Kähler-Berwald metric $\tilde{F}_{\tilde{t},\tilde{k}}$ respectively. Our result generalizes the Schwarz lemma for holomorphic mappings from $P_m$ into $P_n$ whenever $P_m$ and $P_n$ are endowed with the Bergman metrics respectively. We also obtain a distortion theorem on the unit polydisc $P_m$, where $P_m$ is endowd with an $\mbox{Aut}(P_m)$…
Schwarz Lemma for VT harmonic map
本文推广了V调和映射的Schwarz引理至VT调和映射，在特定特征值及Ricci曲率下界条件下，证明了距离和体积递减性质。
In this note, we prove a Schwarz-Pick type lemma for minimal maps between negatively curved Riemannian surfaces. More precisely, we prove that if $f:M \to N$ is a minimal map with bounded Jacobian between two complete negatively curved Riemann surfaces M and N whose sectional curvatures $σ_M$ and $σ_N$ satisfy $infσ_M \ge supσ_N$, then f is area decreasing.
Boundary Value Problem and Discrete Schwarz-Pick Lemma for Generalized Hyperbolic Circle Packings
本文研究双曲几何中广义圆堆积的离散Schwarz-Pick引理，通过证明边值问题存在性与刚性，引入组合Calabi流方法，并提出极大值原理，最终证明了该引理。
We prove a Schwarz-type lemma for noncompact manifolds with possibly noncompact boundary. The result is a consequence of a suitable form of the weak maximum principle of independent interest. The paper is enriched with applications to conformal deformations of noncompact manifolds with boundary, among them a generalization of a classical result by Escobar.
Schwarz lemma on bounded symmetric domains endowed with holomorphic invariant Kähler--Berwald metrics
本文证明全局对称复Finsler空间必为Kähler-Berwald空间，并给出有界对称域上带不变Kähler-Berwald度量的Schwarz引理，其Lu常数在秩≥2时最优。
The Stochastic Schwarz lemma on Kähler Manifolds by Couplings and Its Applications
本文利用Markov耦合与Kendall-Cranston耦合建立Carathéodory距离的随机公式和比较结果，从而在完备非紧Kähler流形上证明含Ricci曲率分解的Schwarz引理，并推广到四元数Kähler流形，得到正调和函数的改进梯度估计。
We prove a version of Yau's Schwarz Lemma for general almost-complex manifolds equipped with Hermitian metrics. This requires an extension to this setting of the Laplacian comparison theorem. As an application we show that the product of two almost-complex manifolds does not admit any complete Hermitian metric with bisectional curvature bounded between two negative constants that satisfies some additional mild assumptions.
In this paper, we first establish several theorems about the estimation of distance function on real and strongly convex complex Finsler manifolds and then obtain a Schwarz lemma from a strongly convex weakly Kähler-Finsler manifold into a strongly pseudoconvex complex Finsler manifold. As applications, we prove that a holomorphic mapping from a strongly convex weakly Kähler-Finsler manifold into a strongly pseudoconvex complex Finsler manifold is necessary constant under an extra condition. In particular, we prove that a holomorphic mapping from a complex Minkowski space into a strongly pseud…
Schwarz-Pick lemma for harmonic maps which are conformal at a point
本文研究了调和映射在共形点处的微分范数估计，推广了经典的 Schwarz-Pick 引理，并证明了共形调和浸入在特定度量下是距离递减的，从而奠定了基于极小曲面的区域双曲性理论基础。
Suppose that $M$ is a Kähler manifold with a pole such that its holomorphic sectional curvature is bounded from below by a constant and its radial sectional curvature is also bounded from below. Suppose that $N$ is a strongly pseudoconvex complex Finsler manifold such that its holomorphic sectional curvature is bounded from above by a negative constant. In this paper, we establish a Schwarz lemma for holomorphic mappings $f$ form $M$ into $N$. As applications, we obtain a Liouville type rigidity result for holomorphic mappings $f$ from $M$ into $N$, as well as a rigidity theorem for bimeromorp…
We derive some consequences of the Liouville theorem for plurisubharmonic functions of L.-F. Tam and the author. The first result provides a nonlinear version of the complex splitting theorem (which splits off a factor of $\mathbb{C}$ isometrically from the simply-connected Kähler manifold with nonnegative bisectional curvature and a linear growth holomorphic function) of L.-F. Tam and the author. The second set of results concerns the so-called $k$-hyperbolicity and its connection with the negativity of the $k$-scalar curvature (when $k=1$ they are the negativity of holomorphic sectional curv…
Schwarz lemmas and rigidity of \(\bar\partial_b\)-harmonic maps
本文证明从完备伪厄米流形到负曲率Kähler流形的\\(\bar\partial_b\\)-调和映射的Schwarz引理，并利用Bochner恒等式和Siu型散度恒等式，在Sasakian源和复双曲目标上分类达到水平界的映射，导出叶状结构、多重调和性和秩刚性。
In this paper we establish two boundary versions of the Schwarz lemma. The first is for general holomorphic self maps of bounded convex domains with $C^2$ boundary. This appears to be the first boundary Schwarz lemma for general holomorphic self maps that requires no strong pseudoconvexity or finite type assumptions. The second is for biholomorphisms of domains who have an invariant Kähler metric with bounded sectional curvature. This second result applies to holomorphic homogeneous regular domains and appears to be the first boundary Schwarz lemma that makes no assumptions on the regularity o…
This is an account of some aspects of the geometry of Kähler affine metrics based on considering them as smooth metric measure spaces and applying the comparison geometry of Bakry-Emery Ricci tensors. Such techniques yield a version for Kähler affine metrics of Yau's Schwarz lemma for volume forms. By a theorem of Cheng and Yau there is a canonical Kähler affine Einstein metric on a proper convex domain, and the Schwarz lemma gives a direct proof of its uniqueness up to homothety. The potential for this metric is a function canonically associated to the cone, characterized by the property that…
We prove a number of new results on the large-scale geometry of the $L^p$-metrics on the group of area-preserving diffeomorphisms of each orientable surface. Our proofs use in a key way the Fulton-MacPherson type compactification of the configuration space of $n$ points on the surface due to Axelrod-Singer and Kontsevich. This allows us to apply the Schwarz-Milnor lemma to configuration spaces, a natural approach which we carry out successfully for the first time. As sample results, we prove that all right-angled Artin groups admit quasi-isometric embeddings into the group of area-preserving d…
Hyperbolicity and Schwarz Lemmas in Calibrated Geometry
本文在校准几何中定义了双曲性概念并推导了Schwarz引理，扩展了复几何中的相关结果。
Through the Schwarz lemma, we provide a new point of view on three well-known results of the geometry of hyperbolic surfaces. The first result deal with the length of closed geodesics on hyperbolic surfaces with boundary (Thurston, Parlier, Papadopoulos-Théret). The two others give sharp lower bounds on two metric invariants: the length of the shortest non simple closed geodesic, and the radius of the biggest embedded hyperbolic disk (Yamada). We also discuss a question of Papadopoulos and Théret about the length of arcs on surfaces with boundary. In a sequel, we use a generalization of the Sc…
We study the injectivity radius of complete Riemannian surfaces (S,g) with curvature |K(g)| bounded by 1. We show that if S is orientable with nonabelian fundamental group, then there is a point p in S with injectivity radius at least arcsinh(2/\sqrt{3}). This lower bound is sharp independently of the topology of S. This result was conjectured by Bavard who has already proved the genus zero cases. We establish a similar inequality for surfaces with boundary. The proofs rely on a version due to Yau of the Schwarz lemma, and on the work of Bavard. This article is the sequel of a previous one whe…
A refined Schwarz lemma for $V$-harmonic maps
该研究建立了$V$-调和映射的精细化Schwarz引理，推导了更紧的映射上界公式与等号成立条件，改进了此前调和映射与$f$-调和映射的相关界，并给出了主定理的应用。
Complete complex Finsler metrics and uniform equivalence of the Kobayashi metric
本文通过研究Bergman曲率有界性和Kähler流形曲率对称性，结合Schwarz引理，证明了有界强拟凸域或有界凸域上存在全纯截面曲率被负常数上界控制的完备强拟凸复Finsler度量，并进一步建立了有界强凸域上Kobayashi度量与Carathéodory度量的一致等价性。
Motivated by the recent work of Wu and Yau on the ampleness of canonical line bundle for compact Kähler manifolds with negative holomorphic sectional curvature, we introduce a new curvature notion called $\textbf{real bisectional curvature}$ for Hermitian manifolds. When the metric is Kähler, this is just the holomorphic sectional curvature $H$, and when the metric is non-Kähler, it is slightly stronger than $H$. We classify compact Hermitian manifolds with constant non-zero real bisectional curvature, and also slightly extend Wu-Yau's theorem to the Hermitian case. The underlying reason for t…
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