General Schwarz Lemmata and their applications

Ni, Lei

doi:10.1142/s0129167x1940007x

Cited by 15 publications

(16 citation statements)

References 23 publications

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“…Moreover, if f is an immersion at some point, then f is an isometric immersion. We should remark that in a more recent paper [24] the estimate (A.1) has been generalized to a family of interpolating estimates that connects Theorem 1.4 with Yau-Royden's result. The rigidity for the equality case also holds for the estimates there.…”

Section: Extensionsmentioning

confidence: 99%

Liouville Theorems and a Schwarz Lemma for Holomorphic Mappings Between Kähler Manifolds

2021

Comm Pure Appl Math

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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 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 curvature and Kobayashi hyperbolicity) introduced recently in [33] by F. Zheng and the author. We lastly prove a new Schwarz‐lemma‐type estimate in terms of only the holomorphic sectional curvatures of both domain and target manifolds. © 2020 Wiley Periodicals LLC.

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Section: Extensionsmentioning

confidence: 99%

Liouville Theorems and a Schwarz Lemma for Holomorphic Mappings Between Kähler Manifolds

2021

Comm Pure Appl Math

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“…Once we have the global equality of v, by the formulas in Proposition 4 in [17] (see also inequality (A.3) in [15]) and (5.41) in [19], then ∇df = 0. So f is totally geodesic and the rank of df is constant in all cases [21].…”

Section: The Equality Case Of Schwarz Lemmatamentioning

confidence: 95%

“…The second part of the paper is to derive a new Schwarz lemma on almost Hermitian manifolds, following the work of Ni [15] . Recently, by applying a viscosity consideration from PDE theory, Ni proves a Schwarz lemma on Kähler manifolds as follows (see also related work in [4]): any holomorphic map from a complete Kähler manifold with holomorphic sectional curvature bounded from below by K 1 ≤ 0 into a Kähler manifold with holomorphic sectional curvature bounded from above by K 2 < 0 is distance decreasing up to a constant K 1 K 2 , provided that the bisectional curvature of the domain is bounded from below.…”

Section: Here Dsmentioning

confidence: 99%

“…In complex geometry, the Ahlfors-Schwarz lemma has been generalized to higher dimensional manifolds and to more general settings, by the work of Chern [5], Lu [12], Yau [24], Chen-Cheng-Look [4], Royden [17], Tosatti [19] and Ni [15] [16] etc (see [9] and references therein). In the first part of the paper, we study the equality case of the Schwarz lemmata obtained by Yau, Royden and Tosatti.…”

Section: Introductionmentioning

confidence: 99%

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Schwarz lemma: the case of equality and an extension

Chen¹,

Nie²

2021

Preprint

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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.

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“…Since then, there have been a lot of developments on the generalizations and applications of Schwarz lemma. We refer to [9] [17] [18] [19] and especially to [14] [15] for the recent advances.…”

Section: Introductionmentioning

confidence: 99%

Liouville Theorems for holomorphic maps on pseudo-Hermitian manifolds

Chen¹,

Ren²

2021

Preprint

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We prove some Liouville type results for generalized holomorphic maps in three classes: maps from pseudo-Hermitian manifolds to almost Hermitian manifolds, maps from almost Hermitian manifolds to pseudo-Hermitian manifolds and maps from pseudo-Hermitian manifolds to pseudo-Hermitian manifolds, assuming that the domains are compact. For instance, we show that any (J, J N ) holomorphic map from a compact pseudo-Hermitian manifold M with nonnegative (resp. positive) pseudo-Hermitian sectional curvature to an almost Hermitian manifold N with negative (resp. nonpositive) holomorphic sectional curvature is constant. We also construct explicit almost CR structures on a complex vector bundle over an almost CR manifold.

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General Schwarz Lemmata and their applications

Cited by 15 publications

References 23 publications

Liouville Theorems and a Schwarz Lemma for Holomorphic Mappings Between Kähler Manifolds

Liouville Theorems and a Schwarz Lemma for Holomorphic Mappings Between Kähler Manifolds

Schwarz lemma: the case of equality and an extension

Liouville Theorems for holomorphic maps on pseudo-Hermitian manifolds

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