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The min-max construction of minimal surfaces. (English) Zbl 1051.53052

Yau, S.-T. (ed.), Surveys in differential geometry. Lectures on geometry and topology held in honor of Calabi, Lawson, Siu, and Uhlenbeck at Harvard University, Cambridge, MA, USA, May 3–5, 2002. Somerville, MA: International Press (ISBN 1-57146-114-0/hbk). Surv. Differ. Geom. 8, 75-107 (2003).
In this survey article, the authors give complete proofs for results about constructing closed embedded minimal surfaces in a close \(3\)-dimensional manifold. Those results including for instance results of J. Pitts, F. Smith, and L. Simon and F. Smith are well-known in some sense, but proofs are not easy to find. The basic idea of constructing minimal surfaces is to use min-max arguments and sweep-out. This idea goes back to Birkhoff, who used such a method to find simple closed geodesics on spheres. The difficulties in generalizing this method to get embedded minimal surfaces in \(3\)-manifolds are regularity, embeddedness, and good genus problems.
Let \(M\) be a closed \(3\)-dimensional Riemannian manifold. A family \(\{\Sigma_t\}_{t \in [0,1]}\) of subsets of \(M\) is said to be a {generalized family of surfaces} if there are a finite subset \(T\) of \([0,1]\) and a finite set of points \(P\) in \(M\) such that (i) the Hausdorff measure or area \({\mathcal H}^2(\Sigma_t)\) is continuous on \(t\); (ii) \(\Sigma_t \to \Sigma_{t_0}\) in the Hausdorff topology whenever \(t \to t_0\); (iii) \(\Sigma_t\) is a surface for every \(t \notin T\); (iv) For \(t \in T\), \(\Sigma_t \) is a surface in \(M - P\).
Given a generalized family \(\{\Sigma_t\}\) we can generate new generalized families via families of diffeomorphisms from \(M\) onto itself. In fact, take a map \(\psi \in C^\infty([0,1]\times M, M)\) such that \(\psi(t, \cdot) \in \text{Diff}_0\), the identity component of the diffeomorphism group of \(M\) for each \(t\). Defining \(\{\Sigma'_t\}\) by \(\Sigma'_t = \psi(t, \Sigma_t)\), one can obtain a new generalized family of surfaces of \(M\). We say that a set \(\Lambda\) of generalized families is {saturated} if it is closed under this operation.
Given a family \(\{\Sigma_t\} \subset \Lambda\), we denote by \({\mathcal F}(\{\Sigma_t\})\) the area of its maximal slice and by \(m_0(\Lambda)\) the infimum of \(\mathcal F\) taken over all families of \(\Lambda\); that is,
\[ {\mathcal F}(\{\Sigma_t\}) = \max_{t\in [0,1]} {\mathcal H}^2(\Sigma_t)\qquad \text{and} \qquad m_0(\Lambda) = \inf_{\Lambda} \mathcal F = \inf_{\{\Sigma_t\}\in \Lambda} \left[ \max_{t\in [0,1]} {\mathcal H}^2(\Sigma_t)\right]. \]
If \(\lim_n {\mathcal F}(\{\Sigma_t\}^n) = m_0(\Lambda)\), then we say that the sequence of generalized families of surfaces \(\{\{\Sigma_t\}^n\}\) is a minimizing sequence. In this case, if \(\{t_n\}\) is a sequence of parameters and \({\mathcal H}^2(\Sigma^n_{t_n}) \to m_0(\Lambda)\), then we say that \(\{\Sigma^n_{t_n}\}\) is a min-max sequence.
An important point in the min-max construction is to find a saturated set \(\Lambda\) of generalized families of surfaces with \(m_0(\Lambda) > 0\). The authors prove that if \(M\) is a closed \(3\)-manifold with a Riemannian metric and \(\{\Sigma_t\}\) the level sets of a Morse function, then the smallest saturated set \(\Lambda\) containing the family \(\{\Sigma_t\}\) has \(m_0(\Lambda) > 0\).
The main result in this article is that for any saturated set of generalized families of surfaces \(\Lambda\) of a closed \(3\)-manifold with a Riemannian metric, there is a min-max sequence obtained from \(\Lambda\) converging in the sense of varifolds to a smooth embedded minimal surface with area \(m_0(\Lambda)\) (multiplicity being allowed). An easy corollary of two results mentioned right above is the existence of a smooth embedded minimal surface in any closed Riemannian \(3\)-manifold.
Another important result proved by the authors is that if \(\{\Sigma^n_{t_n}\}\) is the min-max sequence of the main theorem and \(\Sigma^\infty\) is its limit, then \[ {\mathbf{g}}(\Sigma^\infty) \leq \liminf_{n\to \infty} {\mathbf{g}}(\Sigma^n_{t_n}), \] where \({\mathbf{g}}(\Sigma^\infty)\) and \({\mathbf{g}}(\Sigma^n_{t_n})\) denote the genus of \(\Sigma^\infty\) and \(\Sigma^n_{t_n}\), respectively.
For the entire collection see [Zbl 1034.53003].

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
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