Abstract
The main result of this paper is the following maximum principle at infinity:Theorem.Let M 1 and M 2 be two disjoint properly embedded complete minimal surfaces with nonempty boundaries, that are stable in a complete flat 3-manifold. Then dist(M 1,M 2)=min(dist(∂M 1,M 2), dist(∂M 2,M 1)).In case one boundary is empty, e.g. M 1,then dist(M 1,M 2)=dist(∂M 2,M 1).If both boundaries are empty, then M 1 and M 2 are flat.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahlfors, L.;Sario, L.:Riemann Surfaces. Princeton University Press, Princeton 1960.
Bohme, R.;Tomi, F.: Zur Struktur der Lösungsmenge des Plateauproblems.Math. Z. 133 (1973), 1–29.
Caratheodory, C.:Conformal representations. Cambridge University Press, Cambridge 1963.
Courant, R.;Hilbert, D.:Methoden der Mathematischen Physik II. Springer Verlag, Berlin 1963.
Cheng, S.Y.;Yau, S.T.: Differential equations on Riemannian manifolds and their applications.Commun. Pure Appl. Math. 28 (1975), 333–354.
Finn, R.: On a class of conformal metrics, with applications to differential geometry in the large.Comment. Math. Helv. 40 (1965), 1–30.
Fischer-Colbrie, D.: On complete minimal surfaces with finite Morse index in three manifolds.Invent. Math. 82 (1985), 121–132.
Fischer-Colbrie, D.;Schoen, R.: The structure of complete minimal surfaces in 3-manifolds of non-negative scalar curvature.Commun. Pure Appl. Math. 33 (1980), 199–211.
Gilbarg, D.;Trudinger, N.S.:Elliptic partial differential equations of second order. Springer Verlag, Berlin 1983.
Huber, A.: On subharmonic functions and differential geometry in the large.Comment. Math. Helv. 32 (1957), 13–72.
Langevin, R.;Rosenberg, H.: A maximum principle at infinity for minimal surfaces and applications.Duke Math. J. 57 (1988), 819–829.
Meeks III, W.H.;Rosenberg, H.: The maximum principle at infinity for minimal surfaces in flat 3-manifolds.Comment. Math. Helv. 65 (1990), 255–270.
Nitzsche, J.:Lectures on minimal surfaces. Cambridge University Press, Cambridge 1989.
Osserman, R.:A survey of minimal surfaces. Dover Publications Inc., New York 1986.
Protter, M.H.;Weinberger, H.F.:Maximum Principles in Differential Equations. Springer Verlag, Berlin 1984.
Redheffer, R.: On the inequality Δu≥f(u, |▽u|).J. Math. Anal. Appl. 1 (1960), 277–299.
Ripoll, J.; Tomi, F.:Maximum Principles for Minimal Surfaces in ℝ3 Having Non-compact Boundary and a Uniqueness Theorem for the Helicoid. Preprint, Universität Heidelberg, Math. Inst., 1993.
Schoen, R.: Stable minimal surfaces in three manifolds. Seminar on minimal submanifolds.Ann. Math. Stud. 103 (1983), 111–125.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Soret, M. Maximum principle at infinity for complete minimal surfaces in flat 3-manifolds. Ann Glob Anal Geom 13, 101–116 (1995). https://doi.org/10.1007/BF01120326
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01120326