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The spaces of index one minimal surfaces and stable constant mean curvature surfaces embedded in flat three manifolds. (English) Zbl 0867.53007
Let $$M$$ be an orientable surface immersed in a flat three-dimensional manifold $$N$$. Minimal immersions and nonzero constant mean curvature immersions are critical points for the area variation of $$M$$, but while the former are critical points for all variations with compact support, the latter are critical points only for compactly supported variations that preserve the volume enclosed by the surface $$M$$. Variational problems of this kind are called isoperimetric problems. For the analysis of the stability of these immersions, we need to consider the second variation formula for the area, namely the quadratic formula called index form. If $$M$$ is an orientable surface immersed with constant mean curvature in an orientable flat three manifold, then $$M$$ is stable if for any smooth function $$u$$ with compact support such that $$\int_M u.dA=0$$ we have $$I(u)\geq 0$$, where $$I$$ is expressed by means of the Laplacian on $$M$$ and by the squared norm of the second fundamental form and $$u$$ is the normal component of the variation vector.
After reporting some of the major results existing in the literature in the past fifteen years on the relationship between the values of the index, the genus $$g$$, the total curvature and the geometry of the immersions, the authors study the spaces of index one minimal surfaces and the space of constant mean curvature surfaces with $$g\geq 2$$ in nonfixed flat 3-manifolds. Using appropriate topological, group-theoretical, analytical and differential methods, the authors prove important compactness theorems, namely:
Theorem. From every sequence of compact orientable index one minimal surfaces embedded in flat 3-tori one can extract, up to scaling, a convergent subsequence from both the tori and the surfaces to an index one minimal surface embedded in the limit three-torus. The above surfaces converge in the $$C^k$$ topology, $$k\geq 2$$, and their topology is preserved in the limit.
For the space of constant mean curvature surfaces the authors prove the following Theorem. Let $$M_n\subset N_n$$ be a sequence of compact, orientable, stable surfaces embedded in complete, orientable, flat 3-manifolds. Assume that $$\text{genus}(M_n)\geq 2$$ and that the induced morphisms $$\pi_1(M_n)\to \pi_1(N_n)$$ between the fundamental groups are surjective. Then one can extract, up to dilatations, convergent subsequences of both the surfaces and the ambient manifolds to a compact stable surface weakly embedded in the limit manifold. Moreover, the compactness is strong in the sense that these limits preserve the topological type of the surfaces and the affine diffeomorphism class of the ambient manifolds, but it is proved that there is no compactness in a strong sense for the space of genus one stable embedded surfaces.
A corollary of these important theorems of compactness is a nonexistence result for a certain kind of stable surfaces with $$g\geq 2$$ embedded in most flat 3-manifolds, which is very interesting for isoperimetric problems as it implies restrictions on the type of solutions of these problems.

MSC:
 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 49Q20 Variational problems in a geometric measure-theoretic setting
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References:
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