##
**Rigidity of min-max minimal spheres in three-manifolds.**
*(English)*
Zbl 1260.53079

The main results concern the use of mini-max methods to prove some rigidity results on metrics on 3-spheres with scalar curvature \(R\geq 6\), namely the existence of a minimal surface of index at most one and area \(\leq 4\pi\).

Given a metric \(g\) on the 3-sphere, the authors introduce the concept of width of \((S^3,g)\), given by the mini-max invariant \[ W(S^3,g)=\inf_{\{\Sigma_t\} \in \bar{\Lambda}} \sup_{t\in [-1,1]} |\Sigma_t|, \] where \(\Sigma_t=F_t(\bar{\Sigma}_t)\), for some one-parameter family of diffeomorphisms \(F_t\) of \(S^3\), isotopic to the identity, and \(\bar{\Sigma}_t=x_4^{-1}(t)\) is the level set for the last component function of the standard immersion into \(\mathbb{R}^4\). This concept is generalized to a saturated collection \(\Lambda\) of generalized families of surfaces (sweep-out) on a 3-dimensional Riemannian manifold \(M\), possibly with boundary, satisfying certain natural conditions, that allows to apply similar results of C. de Lellis and F. Pellandini [J. Reine Angew. Math. 644, 47–99 (2010; Zbl 1201.53009)], defining the width \(W(M,\Lambda)\) associated with \(\Lambda\).

If \(M\) has a connected boundary \(\partial M\) of positive mean curvature, and assuming the existence of a saturated set \(\Lambda\) with width \(>|\partial M|\), by the following similar pull-tight procedure developed by T.H. Colding and C. De Lellis [in: S.-T. Yau (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, 2002. Somerville, MA: International Press. Surv. Differ. Geom. 8, 75–107 (2003; Zbl 1051.53052)], the authors prove that there is a mini-max sequence of slices \(\{\Sigma^n_{t_n}\}\in \Lambda\) such that the squares of the two-dimensional Hausdorff measures satisfy \(\mathcal{H}^2(\Sigma_{t_n}^n)\to W(M, \Lambda)\), and that it converges to a stationary varifold, and regularity holds. A suitable vector field is defined in the radial direction defined by the distance function from \(\partial M\). This generates a one-parameter family of diffeomorphisms that deform the initial family to obtain a new family that, for \(t>t_0\), lies in the interior of \(M\), reducing the problem to these cases, obtaining a sequence that converges to a minimal surface \(\Sigma\) contained in the interior of \(M\) and of area \( W(M, \Lambda)\).

Assuming \(M\) closed, the authors prove that a sweepout of oriented surfaces of a saturated collection \(\Lambda\) with a distinguished element at \(t=0\), with area \(W(M,\Lambda,g)\) and of strictly larger area \(|\Sigma_0|>|\Sigma_t|=:f(t)\), and satisfying \(f''(0)<0\), then \(\Sigma_0\) is an embedded minimal surface of index one. The proof consists in assuming that the index is greater than one leads to a contradiction using the Jacobi operator and suitable Jacobi fields. The authors say that \((M,g)\) satisfies the \((*)_h\)-condition, where \(h\) is a nonnegative integer, if \(M\) does not contain embedded nonorientable surfaces and no minimal embedded surface of genus less or equal to \(h\) is stable. Taking the Heegaard genus \(h\) of \(M\), i.e., the lowest possible genus of a surface \(\Sigma\) that splits \(M\backslash \Sigma\) into two connected components that are both diffeomorphic to a solid ball with handles attached, they prove that there exist an orientable embedded minimal surface \(\Sigma_0\) of index one and genus \(h\) with area equal to \(W(M, \Lambda^h)\), where \(\Lambda^h\) is the union af all saturated sets defined by certain families associated to this Heegaard genus \(h\), and contained in a sweepout with \(f(t)\) satisfying the above properties. This leads to the conclusion that if \(M\) is closed and every embedded surface is oriented, then there exists an embedded minimal surface of index \(\leq 1\) and genus \(\leq \tilde{h}\), where \(\tilde{h}\) is the Heegaard genus of \(M\). In case \(M\) admits an nonorientable embedded surface they take \(\tilde{h}\) as the lowest possible genus among all nonorientable embedded surfaces and obtain the same result using a similar argument as in [H. Bray et al., Commun. Pure Appl. Math. 63, No. 9, 1237–1247 (2010; Zbl 1200.53053)]. As a corollary, they conclude that, if \(M= S^3\), there exists a minimal sphere of index at most one.

In order to obtain further results on \(S^3\) with a metric \(g\) of scalar curvature \(R\geq 6\), the authors consider on \(M\) a family of metrics \(g(t)\) defined by the Ricci flow. Then, they show that the width of \(M\) of a saturated family \(\Lambda\) w.r.t. \(g(t)\) defines a Lipschitz continuous function on \(t\), and some estimation is given. They prove that, if the scalar curvature of \(M\) w.r.t \(g\) is greater or equal to \(6\) and the Ricci is positive, then there exists an embedded minimal surface \(\Sigma\) of index less or equal to one with area \(|\Sigma|\leq 4\pi\). Using a maximum principle and short-time existence for the Ricci flow, they show that the infimum of \(|\Sigma|\), with \(\Sigma\) running over all minimal surfaces with index \(\leq 1\), equals \(4\pi\) if and only if \(M=S^3\) and \(g\) has constant sectional curvature. If \(g\) is any metric on \(S^3\) of positive Ricci curvature and \(R\geq 6\), then there exists an embedded minimal sphere \(\Sigma\) of index one and area \(W(S^3,g)\) less or equal to \(4\pi\), and equality holds if and only if \(g\) has constant scalar curvature. Dropping the assumption of positiveness of the Ricci tensor and using an argument of stability for minimal spheres, they prove that, if \(g\) is not of constant sectional curvature 1, such embedding exists with area strictly smaller than \(4\pi\) and of index zero or one.

Similar results are obtained for the more general ambient space \(M=S^3/\Gamma\) of positive Ricci curvature and scalar curvature \(R\geq 6\).

Given a metric \(g\) on the 3-sphere, the authors introduce the concept of width of \((S^3,g)\), given by the mini-max invariant \[ W(S^3,g)=\inf_{\{\Sigma_t\} \in \bar{\Lambda}} \sup_{t\in [-1,1]} |\Sigma_t|, \] where \(\Sigma_t=F_t(\bar{\Sigma}_t)\), for some one-parameter family of diffeomorphisms \(F_t\) of \(S^3\), isotopic to the identity, and \(\bar{\Sigma}_t=x_4^{-1}(t)\) is the level set for the last component function of the standard immersion into \(\mathbb{R}^4\). This concept is generalized to a saturated collection \(\Lambda\) of generalized families of surfaces (sweep-out) on a 3-dimensional Riemannian manifold \(M\), possibly with boundary, satisfying certain natural conditions, that allows to apply similar results of C. de Lellis and F. Pellandini [J. Reine Angew. Math. 644, 47–99 (2010; Zbl 1201.53009)], defining the width \(W(M,\Lambda)\) associated with \(\Lambda\).

If \(M\) has a connected boundary \(\partial M\) of positive mean curvature, and assuming the existence of a saturated set \(\Lambda\) with width \(>|\partial M|\), by the following similar pull-tight procedure developed by T.H. Colding and C. De Lellis [in: S.-T. Yau (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, 2002. Somerville, MA: International Press. Surv. Differ. Geom. 8, 75–107 (2003; Zbl 1051.53052)], the authors prove that there is a mini-max sequence of slices \(\{\Sigma^n_{t_n}\}\in \Lambda\) such that the squares of the two-dimensional Hausdorff measures satisfy \(\mathcal{H}^2(\Sigma_{t_n}^n)\to W(M, \Lambda)\), and that it converges to a stationary varifold, and regularity holds. A suitable vector field is defined in the radial direction defined by the distance function from \(\partial M\). This generates a one-parameter family of diffeomorphisms that deform the initial family to obtain a new family that, for \(t>t_0\), lies in the interior of \(M\), reducing the problem to these cases, obtaining a sequence that converges to a minimal surface \(\Sigma\) contained in the interior of \(M\) and of area \( W(M, \Lambda)\).

Assuming \(M\) closed, the authors prove that a sweepout of oriented surfaces of a saturated collection \(\Lambda\) with a distinguished element at \(t=0\), with area \(W(M,\Lambda,g)\) and of strictly larger area \(|\Sigma_0|>|\Sigma_t|=:f(t)\), and satisfying \(f''(0)<0\), then \(\Sigma_0\) is an embedded minimal surface of index one. The proof consists in assuming that the index is greater than one leads to a contradiction using the Jacobi operator and suitable Jacobi fields. The authors say that \((M,g)\) satisfies the \((*)_h\)-condition, where \(h\) is a nonnegative integer, if \(M\) does not contain embedded nonorientable surfaces and no minimal embedded surface of genus less or equal to \(h\) is stable. Taking the Heegaard genus \(h\) of \(M\), i.e., the lowest possible genus of a surface \(\Sigma\) that splits \(M\backslash \Sigma\) into two connected components that are both diffeomorphic to a solid ball with handles attached, they prove that there exist an orientable embedded minimal surface \(\Sigma_0\) of index one and genus \(h\) with area equal to \(W(M, \Lambda^h)\), where \(\Lambda^h\) is the union af all saturated sets defined by certain families associated to this Heegaard genus \(h\), and contained in a sweepout with \(f(t)\) satisfying the above properties. This leads to the conclusion that if \(M\) is closed and every embedded surface is oriented, then there exists an embedded minimal surface of index \(\leq 1\) and genus \(\leq \tilde{h}\), where \(\tilde{h}\) is the Heegaard genus of \(M\). In case \(M\) admits an nonorientable embedded surface they take \(\tilde{h}\) as the lowest possible genus among all nonorientable embedded surfaces and obtain the same result using a similar argument as in [H. Bray et al., Commun. Pure Appl. Math. 63, No. 9, 1237–1247 (2010; Zbl 1200.53053)]. As a corollary, they conclude that, if \(M= S^3\), there exists a minimal sphere of index at most one.

In order to obtain further results on \(S^3\) with a metric \(g\) of scalar curvature \(R\geq 6\), the authors consider on \(M\) a family of metrics \(g(t)\) defined by the Ricci flow. Then, they show that the width of \(M\) of a saturated family \(\Lambda\) w.r.t. \(g(t)\) defines a Lipschitz continuous function on \(t\), and some estimation is given. They prove that, if the scalar curvature of \(M\) w.r.t \(g\) is greater or equal to \(6\) and the Ricci is positive, then there exists an embedded minimal surface \(\Sigma\) of index less or equal to one with area \(|\Sigma|\leq 4\pi\). Using a maximum principle and short-time existence for the Ricci flow, they show that the infimum of \(|\Sigma|\), with \(\Sigma\) running over all minimal surfaces with index \(\leq 1\), equals \(4\pi\) if and only if \(M=S^3\) and \(g\) has constant sectional curvature. If \(g\) is any metric on \(S^3\) of positive Ricci curvature and \(R\geq 6\), then there exists an embedded minimal sphere \(\Sigma\) of index one and area \(W(S^3,g)\) less or equal to \(4\pi\), and equality holds if and only if \(g\) has constant scalar curvature. Dropping the assumption of positiveness of the Ricci tensor and using an argument of stability for minimal spheres, they prove that, if \(g\) is not of constant sectional curvature 1, such embedding exists with area strictly smaller than \(4\pi\) and of index zero or one.

Similar results are obtained for the more general ambient space \(M=S^3/\Gamma\) of positive Ricci curvature and scalar curvature \(R\geq 6\).

Reviewer: Isabel Salavessa (Lisboa)

### MSC:

53C24 | Rigidity results |

53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |

35J15 | Second-order elliptic equations |

35J20 | Variational methods for second-order elliptic equations |

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\textit{F. C. Marques} and \textit{A. Neves}, Duke Math. J. 161, No. 14, 2725--2752 (2012; Zbl 1260.53079)

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