## A bridge principle for minimal and constant mean curvature submanifolds of $$R^ N$$.(English)Zbl 0637.49020

The bridge principle states that if $$N_ 1$$ and $$N_ 2$$ are two stable minimal discs, then there exists a minimal disc which is close to $$N_ 1$$ and $$N_ 2$$ joined by a thin strip. While soap films act this way, it seems that it took thirty years until an acceptable version was stated, proved, and published by W. W. Meeks and S.-T. Yau [Math. Z. 179, 151-168 (1982; Zbl 0479.49026)] for oriented stable minimal surfaces in $${\mathbb{R}}^ 3.$$ In this paper the author generalizes the statement of the Meeks-Yau bridge principle to arbitrary dimensions and codimensions. The stability hypothesis is also weakened to the assumption that $$N_ 1$$ and $$N_ 2$$ have no Jacobi fields. A further result extends the bridge principle to codimension one surfaces of constant mean curvature.
The method of proof is to first establish a bridge of width $$\epsilon$$ which, while it does not form a minimal surface, is well adapted to the subsequent construction. The new connected surfaces is called the approximate solution because the $$L^ p$$ norm of the mean curvature is small. The proof of the theorem is thereby reduced to solving an elliptic boundary value problem for small enough $$\epsilon$$. This is done by using the Schauder fixed point theorem, for which the author proves a substantial number of technical estimates.
Reviewer: H.Parks

### MSC:

 49Q05 Minimal surfaces and optimization 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Zbl 0479.49026
Full Text:

### References:

 [1] [A] Allard, W.K.: On the first variation of a varifold. Ann. Math. 95, 417-491 (1972) · Zbl 0252.49028 [2] [A, S] Almgren, F.J., Solomon, B.: How to connect minimal surfaces by bridges. Am. Math. Soc. [Abstracts] April, 1980, p. 255 [3] [Au] Aubin, T.: Nonlinear Analysis on Manifolds. Monge-Ampere Equations. Berlin Heidelberg New York: Springer 1982 [4] [B] Berger, M.: Nonlinearity and Functional Analysis. New York: Academic Press 1970 [5] [C] Courant, R.: Dirichlet’s Principle, Conformal Mapping and Minimal Surfaces. New York: Interscience, 1950 · Zbl 0040.34603 [6] [DoC] doCarmo, M.: Differential Geometry of Curves and Surfaces. Eaglewood Cliffs, New Jersey: Prentice Hall Inc 1976 [7] [G, T] Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Berlin Heidelberg New York: Springer 1977 · Zbl 0361.35003 [8] [H] Hass, J.: The geometry of the slice-ribbon problem. Math. Proc. Camb. Philos. Soc.94, 101-108 (1983) · Zbl 0535.57004 [9] [Ka] Kapouleas, N.: Complete constant mean curvature surfaces. in Euclidean 3-space (Preprint) [10] [Kr] Kruskal, M.: The bridge theorem for minimal surfaces. Comm. Pure Appl. Math.7, 297-316 (1954) · Zbl 0055.39602 [11] [La] Lang, S.: Real Analysis. Redding, MA: Addison Wesley 1983 · Zbl 0502.46003 [12] [Law] Lawson, H.B.: Lectures on Minimal Submanifolds. Vol. 1. Boston: Publish or Perish Inc. 1980 · Zbl 0434.53006 [13] [M, Y] Meeks, W.H., Yau, S.T.: The existence of embedded minimal surfaces and the problem of uniqueness. Math. Z.179, 151-168 (1982) · Zbl 0479.49026 [14] [M, S] Michael, J.H., Simon, L.: Sobolev and mean value inequalities on generalized submanifolds ofR n . Comm. Pure Appl. Math.26, 361-397 (1983) · Zbl 0256.53006 [15] [M] Morrey, C.B.: Multiple Integrals in the Calculus of Variations. Berlin Heidelberg New York: Springer 1966 · Zbl 0142.38701 [16] [N] Nitsche, J.C.C.: Vorlesungen über Minimalflächen. Berlin Heidelberg New York: Springer 1975 · Zbl 0319.53003 [17] [T] Taubes, C.H.: Self-dual Yang-Mills connections on non-selfdual 4-manifolds. J. Diff. Geom.17, 139-170 (1982) · Zbl 0484.53026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.