Totally geodesic discs in strongly convex domains

Gaussier, Hervé; Seshadri, Harish

doi:10.1007/s00209-012-1063-3

Cited by 12 publications

(10 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(8.11) Since z n ∈ Ω there exists a unique complex geodesic ϕ n of the domain Ω so that ϕ n (0) = z n and ϕ n (1) = e 1 . As a consequence of [17,Lemma 3.5], the restriction α n : [0, 1) → C q of ϕ n extends C 1 -smoothly to the closed interval [0, 1] and α ′ (1) ∈ T e 1 ∂Ω, hence as the real number t increases to 1, the point ϕ n (t) converges to e 1 non-tangentially.…”

Section: Existence Of Backward Orbitsmentioning

confidence: 95%

“…Complex geodesics are isometries between (D, k D ) and (Ω, k Ω ), and therefore map real geodesic of D to real geodesic in Ω. On the other hand every real geodesic of Ω is contained in some complex geodesic [17,Lemma 3.3], and its pullback is a real geodesic in D. Hence every real geodesic in Ω is C ∞ , and for all x = y ∈ Ω the geodesic segment joining x to y is unique up to isometries of the interval I. Moreover for every p ∈ Ω and ζ ∈ ∂Ω there exists a unique geodesic ray connecting the two points.…”

Section: Real Geodesicsmentioning

confidence: 99%

See 1 more Smart Citation

Canonical models on strongly convex domains via the squeezing function

Altavilla,

Arosio,

Guerini

2020

Preprint

View full text Add to dashboard Cite

We prove that if a holomorphic self-map f : Ω → Ω of a bounded strongly convex domain Ω ⊂ C q with smooth boundary is hyperbolic then it admits a natural semi-conjugacy with a hyperbolic automorphism of a possibly lower dimensional ball B k . We also obtain the dual result for a holomorphic self-map f : Ω → Ω with a boundary repelling fixed point. Both results are obtained by rescaling the dynamics of f via the squeezing function.

show abstract

Section: Existence Of Backward Orbitsmentioning

confidence: 95%

Section: Real Geodesicsmentioning

confidence: 99%

Canonical models on strongly convex domains via the squeezing function

Altavilla,

Arosio,

Guerini

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Recent work on the rigidity of local Bergman isometries may be found in [40]. Isometries of the Kobayashi metric between a strongly pseudoconvex domain and the ball are also shown to be rigid in [33] while a more recent result in [24] proves the rigidity of an isometry between a pair of strongly convex domains even in the non equidimensional case; the choice of either the Kobayashi or the Carathéodory metric is irrelevant here since the two coincide. However, this seems to be unknown for isometries of the Kobayashi or the Carathéodory metric between a pair of strongly pseudoconvex domains.…”

Section: Setmentioning

confidence: 98%

Bounds for invariant distances on pseudoconvex Levi corank one domains and applications

Balakumar¹,

Mahajan²,

Verma³

2015

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

View full text Add to dashboard Cite

Let D ⊂ C n be a smoothly bounded pseudoconvex Levi corank one domain with defining function r, i.e., the Levi form ∂∂r of the boundary ∂D has at least (n − 2) positive eigenvalues everywhere on ∂D. The main goal of this article is to obtain bounds for the Carathéodory, Kobayashi and the Bergman distance between a given pair of points p, q ∈ D in terms of parameters that reflect the Levi geometry of ∂D and the distance of these points to the boundary. Applications include an understanding of Fridman's invariant for the Kobayashi metric on Levi corank one domains, a description of the balls in the Kobayashi metric on such domains that are centered at points close to the boundary in terms of Euclidean data and the boundary behaviour of Kobayashi isometries from such domains.

show abstract

“…However, the proof given there is incorrect. In fact, by carefully reading that proof, one can see that what the authors of [GS13] claimed is essentially the following (with notations fixed there): Let Ω be a bounded strongly convex domain with C 3 -smooth boundary and let p ∈ ∂Ω. Let φ be a complex geodesic of Ω with φ(1) = p and ψ be a holomorphic mapping from ∆ into Ω such that ψ(1) = p and ψ ′ (1) = φ ′ (1).…”

Section: Uniqueness Of Complex Geodesics With Prescribed Boundary Datamentioning

confidence: 99%

Complex geodesics and complex Monge-Ampère equations with boundary singularity

Huang¹,

Wang²

2020

Preprint

View full text Add to dashboard Cite

We study complex geodesics and complex Monge-Ampère equations on bounded strongly linearly convex domains in C n . More specifically, we prove the uniqueness of complex geodesic discs in such domains with prescribed boundary value and direction, when the boundaries of these domains have minimal regularity. The existence of such complex geodesic discs was proved by the first author in the 1990s, but the uniqueness was left open. Using this uniqueness result and a uniform C 1, 1/2 -estimate of complex geodesics and their dual mappings, as well as some other previously known results, we solve a homogeneous complex Monge-Ampère equation with prescribed boundary singularity, which was first considered by Bracci-Patrizio-Trapani on bounded strongly convex domains in C n with smooth boundary in their two important papers.

show abstract

Totally geodesic discs in strongly convex domains

Abstract: International audienc

Cited by 12 publications

References 15 publications

Canonical models on strongly convex domains via the squeezing function

Canonical models on strongly convex domains via the squeezing function

Bounds for invariant distances on pseudoconvex Levi corank one domains and applications

Complex geodesics and complex Monge-Ampère equations with boundary singularity

Contact Info

Product

Resources

About