Heinz-type mean curvature estimates in Lorentz-Minkowski space
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Heinz-type mean curvature estimates in Lorentz-Minkowski space Atsufumi Honda1 · Yu Kawakami2
· Miyuki Koiso3 · Syunsuke Tori4
Received: 15 April 2020 / Accepted: 23 September 2020 © The Author(s) 2020
Abstract We provide a unified description of Heinz-type mean curvature estimates under an assumption on the gradient bound for space-like graphs and time-like graphs in the Lorentz-Minkowski space. As a corollary, we give a unified vanishing theorem of mean curvature for these entire graphs of constant mean curvature. Keywords Heinz-type mean curvature estimate · Bernstein-type theorem · Space-like graph · Time-like graph · Constant mean curvature Mathematics Subject Classification Primary 53A10; Secondary 53B30 · 53C24 · 53C42
Dedicated to Professor Atsushi Kasue on the occasion of his 65th birthday The authors were partially supported by JSPS KAKENHI Grant Number JP18H04487, JP19K03463, JP19K14526, JP20H04642 and JP20H01801.
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Yu Kawakami [email protected] Atsufumi Honda [email protected] Miyuki Koiso [email protected] Syunsuke Tori [email protected]
1
Department of Applied Mathematics, Faculty of Engineering, Yokohama National University, 79-5, Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan
2
Institute of Science and Engineering, Faculty of Mathematics and Physics, Kanazawa University, Kanazawa 920-1192, Japan
3
Institute of Mathematics for Industry, Kyushu University, 744, Motooka Nishi-ku, Fukuoka 819-0395, Japan
4
Graduate School of Natural Science and Technology, Kanazawa University, Kanazawa 920-1192, Japan
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A. Honda et al.
1 Introduction The celebrated theorem of Bernstein [1] (as for a simple proof, see [2]), stating that the only entire minimal graph in the Euclidean 3-space R3 is a plane, has played a seminal role in the evolution of surface theory. For instance, by using this theorem, we can show the uniqueness theorem for an entire graph of constant mean curvature in R3 . Indeed Heinz [3] proved that if ϕ(u 1 , u 2 ) is a C 2 -differentiable function defined on the open disk B 2 (R) with center at the origin and radius R(> 0) in the (u 1 , u 2 )-plane, and the mean curvature H of the graph ϕ := {(u 1 , u 2 , ϕ(u 1 , u 2 )) ∈ R3 ; (u 1 , u 2 ) ∈ B 2 (R)} of ϕ satisfies |H | ≥ α > 0, where α is constant, then R ≤ 1/α. As a corollary, we have a vanishing theorem of mean curvature for entire graphs of constant mean curvature, that is, if ϕ is an entire graph of constant mean curvature, then H ≡ 0. Combining these results, we obtain that the only entire graph of constant mean curvature in R3 is a plane (see [4, Section 2.1]). The Heinz result was extended to the case of graphic hypersurfaces in the Euclidean (n + 1)-space Rn+1 by Chern [5] and Flanders [6]. We will study this subject for graphic hypersurfaces in the Lorentz-Minkowski space. Here we recall some basic definitions and fundamental facts. We denote by R1n+1 = (Rn+1 , , L ) the Lorentz-Minkowski (n + 1)-space with the Lorentzian metric (x 1 , . . . , x n , x n+1 ), (y 1 , .
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