On the Gromov hyperbolicity of the minimal metric

Mathematische Zeitschrift (2024) 308:24 https://doi.org/10.1007/s00209-024-03581-x Mathematische Zeitschrift On the Gromov hyperbolicity of the minimal metric Matteo Fiacchi1,2 Received: 15 November 2023 / Accepted: 15 July 2024 © The Author(s) 2024 Abstract In this paper, we study the hyperbolicity in the sense of Gromov of domains in Rd (d ≥ 3) with respect to the minimal metric introduced by Forstnerič and Kalaj (Anal PDE 17(3):981– 1003, 2024). In particular, we prove that every bounded strongly minimally convex domain is Gromov hyperbolic and its Gromov compactification is equivalent to its Euclidean closure. Moreover, we prove that the boundary of a Gromov hyperbolic convex domain does not contain non-trivial conformal harmonic disks. Finally, we study the relation between the minimal metric and the Hilbert metric in convex domains. Keywords Minimal surface · Minimal metric · Hyperbolic domain · Gromov hyperbolicity · Convex domain · Hilbert metric Mathematics Subject Classification 53C23 · 53A10 · 32Q45 · 30C80 · 31A05 1 Introduction In several complex variables, the Kobayashi metric plays a fundamental role in the study of domains in Cd and holomorphic maps between them. In recent years, the metric approach to complex analysis has led to important results, ranging from geometric function theory to complex dynamics [3, 4, 8, 9, 20, 24]. One of the main problems of this method is characterizing when the metric space (D, k D ) is Gromov hyperbolic. Gromov hyperbolicity is a coarse notion of negative curvature in the context of metric spaces and is very useful for studying metric spaces where the distance function may not arise from a Riemannian metric. The Gromov hyperbolicity of the Kobayashi metric has been extensively studied in recent years [14, 16, 19, 21–23]. The author is supported by the European Union (ERC Advanced grant HPDR, 101053085 to Franc Forstnerič) and the research program P1-0291 from ARIS, Republic of Slovenia. B Matteo Fiacchi 1 Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia 2 Institute of Mathematics, Physics and Mechanics, Jadranska 191000, Ljubljana, Slovenia 0123456789().: V,-vol 123 24 Page 2 of 20 M. Fiacchi In real Euclidean space, Forstnerič and Kalaj [15] defined the minimal metric, the analog of the Kobayashi metric in the theory of minimal surfaces. The two metrics share many similarities, but there are notable differences. For example, since conformal minimal surfaces are much more abundant than holomorphic curves, the minimal metric is significantly more rigid than the Kobayashi metric, making some typical techniques of complex analysis inapplicable in this context. Given D ⊂ Rd (d ≥ 3), we are interested in finding necessary and sufficient conditions for the minimal distance ρ D to be Gromov hyperbolic. The first natural class of domains to consider is bounded strongly minimally convex domains (see Sect. 2.2 for the definition), which are the analogs in the theory of minimal surfaces of strongly pseudoconvex domains. Theorem 1.1 Let D ⊂ Rd (d ≥ 3) be a bounded strongly minimally convex domain. Then the metric space (D, ρ D ) is Gromov hyperbolic. Moreover, the identity map D → D extends G as a homeomorphism between the Gromov compactification D and the Euclidean closure D. This answers a question of Drinovec Drnovšek and Forstnerič [13, Problem 12.7]. Inspired by Balogh’s and Bonk’s work in strongly pseudoconvex domains [5], the proof is based on estimates of the minimal metric near the boundary and the construction of the hyperbolic filling metric (4.1). The second main result involves the necessary conditions in convex domains. As already observed in Hilbert geometry [6, 18] and complex geometry [2, 16, 21–23], the “flatness” of the boundary is an obstruction to Gromov hyperbolicity. Theorem 1.2 Let D ⊂ Rd (d ≥ 3) be a hyperbolic convex domain such that (D, ρ D ) is Gromov hyperbolic. Then every conformal harmonic disk f : D → ∂ D is constant. Note that no assumptions regarding boundedness and regularity of the boundary are necessary. The proof relies on estimates of the minimal metric within convex domains and the construction of “fat” quasi-geodesic triangles near the conformal harmonic disk. Finally, we provide results concerning the Hilbert metric h D and the minimal metric g D in convex domains. In particular, we prove (Proposition 6.1) that in any convex domain D ⊂ Rd (d ≥ 3) we have h D (x, v) ≤ 2g D (x, v), x ∈ D, v ∈ Rd and we prove that in strongly convex domains, these two metrics are bilipschitz. The paper is organized as follows: Sect. 2 covers the preliminaries used throughout. In Sect. 3, we establish boundary estimates for the minimal metric in bounded strongly minimally convex domains. Sections 4 and 5 present the proofs of Theorem 1.1 and Theorem 1.2, respectively. Finally, Sect. 6 explores the relationship between the minimal and Hilbert metrics in convex domains. 2 Preliminares Notations: • Let (e j ) j=1,...,d denote the canonical basis of Rd . • For x ∈ Rd let ||x|| denote the standard Euclidean norm of x. • For u, v ∈ Rd let u, v denote the Euclidean scalar product of Rd . 123 On the Gromov hyperbolicity of the minimal metric Page 3 of 20 24 • Let D := {x ∈ R2 : ||x|| < 1} and Hd := {(x, y) ∈ R × Rd−1 : x > 0}. • If D  Rd is a domain and x ∈ Rd let δ D (x) := inf {||x − y|| : y ∈ ∂ D} . • Let G 2 (Rd ) denote the Grassman manifold of 2-planes in Rd . • Let a, b ∈ R, let denote a ∧ b := min{a, b} and a ∨ b := max{a, b}. 2.1 Minimal metric A map f : D → Rd (d ≥ 2) is said to be conformal if for all ζ ∈ D we have || f x (ζ )|| = || f y (ζ )|| and  f x (ζ ), f y (ζ ) = 0 where ζ = (x, y) are the coordinates of D ⊂ R2 . Let D ⊂ Rd (d ≥ 2) be a domain and denote by CH(D, D) the space of conformal harmonic maps f : D → D. The minimal metric of D is the function g D (x, v) = inf{1/r : f ∈ CH(D, D), f (0) = x, f x (0) = r v}, x ∈ D, v ∈ Rd . It turns out that g D is a Finsler metric, i.e. g D is not negative, upper-semicontinuous on D × Rd and absolutely homogeneous g D (x, tv) = |t|g D (x, v), t ∈ R. We can consider the intrinsic distance  1 g D (γ (t), γ̇ (t))dt, x, y ∈ D ρ D (x, y) = inf γ 0 where the infimum is over all piecewise C 1 curve γ : [0, 1] → D with γ (0) = x and γ (1) = y. The function ρ D is called minimal pseudodistance of D, and in general it may not be a distance function (for example ρRd vanishes identically). For this reason we say that a domain D ⊂ Rd is hyperbolic if ρ D is a distance and complete hyperbolic if ρ D is a complete distance (in the sense of Cauchy). See [13] for some characterizations of hyperbolicity and complete hyperbolicity. The minimal metric can be characterized in the following way: it is the largest pseudometric such that for every conformal surface M and conformal harmonic map f : M → D we have g D ( f (z), d f z (v)) ≤ κ M (z, v), z ∈ M, v ∈ Tz M, (2.1) where κ M is the hyperb (...truncated)