## 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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