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**A remark on almost umbilical hypersurfaces.**
*(English)*
Zbl 1299.53012

Let \((M^n,g)\) be a compact, connected, oriented Riemannian manifold without boundary isometrically immersed in \(\mathbb {R}^{n+1}\). It is a classical result that if \(M\) is Einstein (with positive scalar curvature) then \(M\) is isometric to the round sphere \(\mathbb {S}^n\). The aim of the present article is to formulate weaker conditions related to the Einstein property under which \(M\) is quasi-isometric to \(\mathbb {S}^n\). These conditions involve the mean curvature \(H\) of \(M \subset \mathbb {R}^{n+1}\) and also higher-order mean curvatures \(H_r\), \(r \in \{1,\ldots ,n\}\), with \(H_1=H\). Recall that a diffeomorphism \(F: M \to \mathbb {S}^n\) is \(\theta \)-quasi-isometry if for any \(x \in M\) and a unit vector \(u \in T_xM\) we have \(| | T_xF(u)| ^2-1| \leq \theta \).

More specifically, there are explicitly defined values \(k_{p,r}\) (depending on the higher-order mean curvatures and the norm \(\| ..\| _{2p}\)) for which the author obtains the following results. We choose \(\theta \in (0,1)\) and denote the second fundamental form by \(B\), the umbilicity tensor by \(\tau := B-H\cdot \operatorname {Id}\), the volume of \(M\) by \(\mathrm{Vol}(M)\) and the Ricci tensor on \((M,g)\) by \(\operatorname {Ric}\).

\(\bullet\) An almost-Einstein condition: Let \(q>\frac {n}{2}\) and assume \(H_r>0\) for \(r>1\). There is a constant \(\varepsilon _1\), depending on \(n\), \(\| H \| \), \(\mathrm{Vol}\, l(M)\) and \(\theta \), such that if \((M,g)\) satisfies \(\| \operatorname {Ric} - (n-1) k_{p,r} g \| _q \leq \varepsilon _1\) then \(M\) is diffeomorphic and \(\theta \)-quasi-isometric to \(\mathbb {S}^n\left (\sqrt {\frac {1}{k_{p,r}}}\right )\).

\(\bullet \) An almost-umbilicity condition: For every positive \(k\) there is a constant \(\varepsilon _2\), depending on \(n\), \(k\) and \(\theta \), such that if \((M,g)\) satisfies \(\| B - k g \| _\infty \leq \varepsilon _2\) then \(M\) is diffeomorphic and \(\theta \)-quasi-isometric to \(\mathbb {S}^n(\frac {1}{k})\).

\(\bullet\) Another almost-umbilicity condition: Let \(q>\frac {n}{2}\) and assume \(H_r>0\) for \(r>1\). There are constants \(\varepsilon _3\) and \(\varepsilon _4\), both depending on \(n\), \(\| H \| _\infty \), \(\mathrm{Vol}(M)\) and \(\theta \), such that if \((M,g)\) satisfies \(\| \tau \| _{2q} \leq \varepsilon _3\) and \(\| H^2 - k_{p,r} \| _q \leq \varepsilon _4\) for \(p\geq 4\) then \(M\) is diffeomorphic and \(\theta \)-quasi-isometric to \(\mathbb {S}^n\left (\sqrt {\frac {1}{k_{p,r}}}\right)\).

More specifically, there are explicitly defined values \(k_{p,r}\) (depending on the higher-order mean curvatures and the norm \(\| ..\| _{2p}\)) for which the author obtains the following results. We choose \(\theta \in (0,1)\) and denote the second fundamental form by \(B\), the umbilicity tensor by \(\tau := B-H\cdot \operatorname {Id}\), the volume of \(M\) by \(\mathrm{Vol}(M)\) and the Ricci tensor on \((M,g)\) by \(\operatorname {Ric}\).

\(\bullet\) An almost-Einstein condition: Let \(q>\frac {n}{2}\) and assume \(H_r>0\) for \(r>1\). There is a constant \(\varepsilon _1\), depending on \(n\), \(\| H \| \), \(\mathrm{Vol}\, l(M)\) and \(\theta \), such that if \((M,g)\) satisfies \(\| \operatorname {Ric} - (n-1) k_{p,r} g \| _q \leq \varepsilon _1\) then \(M\) is diffeomorphic and \(\theta \)-quasi-isometric to \(\mathbb {S}^n\left (\sqrt {\frac {1}{k_{p,r}}}\right )\).

\(\bullet \) An almost-umbilicity condition: For every positive \(k\) there is a constant \(\varepsilon _2\), depending on \(n\), \(k\) and \(\theta \), such that if \((M,g)\) satisfies \(\| B - k g \| _\infty \leq \varepsilon _2\) then \(M\) is diffeomorphic and \(\theta \)-quasi-isometric to \(\mathbb {S}^n(\frac {1}{k})\).

\(\bullet\) Another almost-umbilicity condition: Let \(q>\frac {n}{2}\) and assume \(H_r>0\) for \(r>1\). There are constants \(\varepsilon _3\) and \(\varepsilon _4\), both depending on \(n\), \(\| H \| _\infty \), \(\mathrm{Vol}(M)\) and \(\theta \), such that if \((M,g)\) satisfies \(\| \tau \| _{2q} \leq \varepsilon _3\) and \(\| H^2 - k_{p,r} \| _q \leq \varepsilon _4\) for \(p\geq 4\) then \(M\) is diffeomorphic and \(\theta \)-quasi-isometric to \(\mathbb {S}^n\left (\sqrt {\frac {1}{k_{p,r}}}\right)\).

Reviewer: Josef Šilhan (Brno)

### MSC:

53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |

53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |

53C20 | Global Riemannian geometry, including pinching |

53C24 | Rigidity results |