Chern-Osserman inequality for minimal surfaces in 𝐇ⁿ

Chen, Qing; Cheng, Yi

doi:10.1090/s0002-9939-99-05334-4

Cited by 7 publications

(22 citation statements)

References 6 publications

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“…(1) Sup t>0 to 0 as the distance r(q) goes to infinity (see Theorem 2.1 in [22]), so we have that A S (q) < h b (r(q)) outside a compact set K ⊂ S and we recover the complete statement of the main theorem in [7].…”

Section: Introduction and Main Resultssupporting

confidence: 66%

“…In contrast to what happens with cmi surfaces in R n , the total Gaussian curvature of surfaces S 2 immersed in the hyperbolic space H n (b) is always infinite, by the Gauss equation. However, it is possible to consider surfaces S 2 ⊆ H n (b) with finite total extrinsic curvature S B S 2 dσ < ∞, and this is what Chen Qing and Chen Yi did in [6] and [7].…”

Section: Introduction and Main Resultsmentioning

confidence: 94%

“…In the papers [6] and [7], a Chern-Osserman type inequality was studied for a completely, properly and minimally immersed surface (cmi for short) in the Hyperbolic space, extending the classical result originally established by S.S. Chern and R. Osserman in [4] for cmi surfaces in the Euclidean space to this strictly negatively curved setting.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Our proof of Theorem 1.1 basically follows the lines of argument used in the proofs given in [6] and [7]. A basic fact used in these proofs is the monotonicity property satisfied by the volume growth of the extrinsic balls in minimal surfaces that are properly immersed in the real space forms K n (b) with b ≤ 0, namely, that the function Vol(Et) Vol(B b,2 t ) is a non-decreasing function of r. We have the same monotonicity property when we consider the extrinsic balls on a surface S that is properly immersed in a Cartan-Hadamard manifold N with negative and variable sectional curvature bounded from above by b < 0.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Following the break with the framework given by the constant curvature of the ambient space H n (b) in the works [6] and [7], we have had to overcome several analytical and topological difficulties.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 4 more Smart Citations

The Chern–Osserman inequality for minimal surfaces in a Cartan–Hadamard manifold with strictly negative sectional curvatures

Esteve

Palmer

2014

Ark. Mat.

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We state and prove a Chern-Osserman-type Inequality in terms of the volume growth for minimal surfaces S which have finite total extrinsic curvature and are properly immersed in a Cartan-Hadamard manifold N with sectional curvatures bounded from above by a negative quantity K N ≤ b < 0 and such that they are not too curved (on average) with respect to the Hyperbolic space with constant sectional curvature given by the upper bound b. We have also proven the same Chern-Osserman-type Inequality for minimal surfaces with finite total extrinsic curvature and properly immersed in an asymptotically hyperbolic Cartan-Hadamard manifold N with sectional curvatures bounded from above by a negative quantity K N ≤ b < 0.2000 Mathematics Subject Classification. Primary 53A15, 53C20; Secondary 53C42.is the geodesic t-ball centered at the pole o in the ambient space N , and B b,2 t denotes the geodesic t-ball in H 2 (b). Remark 1.2. The main theorem in [7] is a corollary of Theorem 1.1. In fact, note that condition (1.4) is superfluous when the ambient manifold is H n (b). On the other hand, when the ambient manifold is H n (b), then condition (1.3) implies that A S (q) goes

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Section: Introduction and Main Resultssupporting

confidence: 66%

Section: Introduction and Main Resultsmentioning

confidence: 94%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 3 more Smart Citations

The Chern–Osserman inequality for minimal surfaces in a Cartan–Hadamard manifold with strictly negative sectional curvatures

Esteve

Palmer

2014

Ark. Mat.

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On the Fundamental Tone of Minimal Submanifolds with Controlled Extrinsic Curvature

Gimeno

2013

Potential Anal

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The aim of this paper is to obtain the fundamental tone for minimal submanifolds of the Euclidean or hyperbolic space under certain restrictions on the extrinsic curvature. We show some sufficient conditions on the norm of the second fundamental form that allow us to obtain the same upper and lower bound for the fundamental tone of minimal submanifolds in a Cartan-Hadamard ambient manifold. As an intrinsic result, we obtain a sufficient condition on the volume growth of a Cartan-Hadamard manifold to achieve the lowest bound for the fundamental tone given by McKean.

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Extrinsic isoperimetry and compactification of minimal surfaces in Euclidean and hyperbolic spaces

Gimeno

Palmer

2012

Isr. J. Math.

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We study the topology of (properly) immersed complete minimal surfaces P 2 in Hyperbolic and Euclidean spaces which have finite total extrinsic curvature, using some isoperimetric inequalities satisfied by the extrinsic balls in these surfaces, (see [12]). We present an alternative and partially unified proof of the Chern-Osserman inequality satisfied by these minimal surfaces, (in R n and in H n (b)), based in the isoperimetric analysis above alluded. Finally, we show a Chern-Osserman type equality attained by complete minimal surfaces in the Hyperbolic space with finite total extrinsic curvature.2000 Mathematics Subject Classification. Primary 53C20 ; Secondary 53C42, 49Q05.

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Chern-Osserman inequality for minimal surfaces in 𝐇ⁿ

Abstract: Abstract. We obtain Chern-Osserman's inequality of a complete properly immersed minimal surface in hyperbolic n-space, provided the L 2 -norm of the second fundamental form of the surface is finite.

Cited by 7 publications

References 6 publications

The Chern–Osserman inequality for minimal surfaces in a Cartan–Hadamard manifold with strictly negative sectional curvatures

The Chern–Osserman inequality for minimal surfaces in a Cartan–Hadamard manifold with strictly negative sectional curvatures

On the Fundamental Tone of Minimal Submanifolds with Controlled Extrinsic Curvature

Extrinsic isoperimetry and compactification of minimal surfaces in Euclidean and hyperbolic spaces

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