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Bifurcation of minimal surfaces in Riemannian manifolds. (English) Zbl 0835.53010

Trans. Am. Math. Soc. 347, No. 1, 51-62 (1995); correction ibid. 349, No. 11, 4689-4690 (1997).
The authors study the question if a closed minimal submanifold \(M\) of a Riemannian manifold \(N\) persists under perturbations of the metric \(g_0\) of the ambient space \(N\). By a standard procedure, the condition for the minimality of nearby submanifolds with respect to nearby metrics \(g\) is reduced to a system of equations in a space of finite dimension \(k\) (the space of normal Jacobi fields along \(M\)) containing \(g\) as a parameter. Under a nondegeneracy condition concerning the Taylor expansion of this \(k\)-dimensional system the authors show that there are metrics \(g\) arbitrarily close to \(g_0\) such that the \(g\)-minimal submanifolds sufficiently close to \(M\) are in one-to-one correspondence with the roots of a polynomial \(k \times k\)-system, all of which are non- degenerate. In the special case \(k = 1\) the connection with elementary catastrophes is discussed. The proof of lemma 2 makes no sense as it stands; the authors have, however, communicated a corrected version to the reviewer.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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