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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

Citations:

Zbl 0479.49026
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References:

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