Isoparametric functions and exotic spheres (Q2861477)

A hypersurface \(M^n\) in a real space form \(N^{n+1}(c)\) is said to be isoparametric if it has constant principal curvatures. The subject of isoparametric hypersurfaces has been studied for a long time by many authors, starting with \textit{É. Cartan} [Ann. Mat. pura appl., Bologna, (4) 17, 177--191 (1938; JFM 64.1361.02); Math. Z. 45, 335--367 (1939; JFM 65.0792.01)]. These studies were continued by \textit{H. F. Münzner} [Math. Ann. 251, 57--71 (1980; Zbl 0417.53030); Math. Ann. 256, 215--232 (1981; Zbl 0438.53050)], \textit{T. E. Cecil} and \textit{P. J. Ryan} [Tight and taut immersions of manifolds. Boston-London-Melbourne: Pitman Advanced Publishing Program (1985; Zbl 0596.53002)], \textit{G. Thorbergsson} [in: Handbook of differential geometry. Volume I. Amsterdam: North-Holland. 963--995 (2000; Zbl 0979.53002)], and others. They mainly refer to the spherical case, with isoparametric functions (or Cartan polynomial), satisfying the Cartan-Münzner equations in the case isoparametric, and conversely the set of level hypersurfaces of an isoparametric function consists of a family of parallel isoparametric hypersurfaces. Recent results in the area belong to \textit{T. E. Cecil} et al. [Ann. Math. (2) 166, No. 1, 1--76 (2007; Zbl 1143.53058)], the first author and \textit{Y. Xie} [J. Funct. Anal. 258, No. 5, 1682--1691 (2010; Zbl 1191.53038)] and \textit{S. Immervoll} [Ann. Math. (2) 168, No. 3, 1011--1024 (2008; Zbl 1176.53057)]. The authors of the present paper also indicate the work of G. Thorbergsson [loc. cit.] as ``an excellent survey'' on this theme.NEWLINENEWLINENEWLINENEWLINEIn the present paper the authors start from an article by \textit{Q. Wang} [Math. Ann. 277, 639--646 (1987; Zbl 0638.53053)] to study the isoparametric functions on general Riemannian manifolds, especially on exotic spheres (manifolds homeomorphic but not diffeomorphic to usual spheres). In the part on the fundamental theory of isoparametric functions on general Riemannian manifolds (Section 2) the authors prove some interesting results. They are related to the so-called transnormal function. A non-constant smooth function \(f:N\rightarrow \mathbb{R}\) defined on a Riemannian manifold \(N\) is called transnormal if there is a smooth function \(b:\mathbb{R}\rightarrow \mathbb{R}\) such that \(|\nabla f|^2=b(f)\), where \(\nabla f\) is the gradient of \(f\). If moreover there is a continuous function \(a:\mathbb{R}\rightarrow \mathbb{R}\) such that \(\Delta f=a(f)\), where \(\Delta f\) is the Laplacian of \(f\), then \(f\) is called isoparametric. Among the results of Section 2, the authors mention in the introduction Theorem 2.2 which asserts that each component of the singular sets of a transnormal function is a submanifold which has codimension not less than 2 if and only if the singular sets are exactly the focal set of every regular level set. Moreover in this case, each level set is connected. If in addition the function is isoparametric on a closed manifold, then at least one level hypersurface is minimal. NEWLINENEWLINENEWLINEA first main result of the paper is the following theorem: NEWLINENEWLINENEWLINETheorem 1.1. Suppose \(\Sigma^4\) is a homotopy 4-sphere and it admits a properly transnormal function under some metric. Then \(\Sigma^4\) is diffeomorphic to \(S^4\). NEWLINENEWLINENEWLINE(A homotopy \(n\)-sphere is a smooth manifold with the same homotopy type as \(S^n\).) NEWLINENEWLINENEWLINENEWLINEThe proof of Theorem 1.1 is given in Section 3. NEWLINENEWLINENEWLINETheorem 1.1, together with the result of \textit{M. H. Freedman} [J. Differ. Geom. 17, 357--453 (1982; Zbl 0528.57011)] according to which any homotopy 4-sphere is homeomorphic to \(S^4\), says that there exists no properly transnormal function on any exotic 4-sphere (if there could exist exotic 4-spheres). NEWLINENEWLINENEWLINEA second important result of the paper (Theorem 4.8) refers to the existence problem of isoparametric functions on exotic 7-sphers, especially on the Gromoll-Meyer sphere. The Gromoll-Meyer sphere is an exotic 7-sphere with a metric of non-negative sectional curvature \(\Sigma^7\cong \mathrm{Sp}(2)/S^3\) [\textit{D. Gromoll} and \textit{W. Meyer}, Ann. Math. (2) 100, 401--406 (1974; Zbl 0293.53015)]. NEWLINENEWLINENEWLINETheorem 4.8. There exists a function \(f:\Sigma^7\rightarrow \mathbb{R}\) which under the induced metric is properly transnormal but not an isoparametric function with two points as the focal varieties, and the regular level hypersurfaces of \(f\) have non-constant mean curvature. NEWLINENEWLINENEWLINEIn Section 2 the authors prove the following three ways to construct examples of isoparametric functions: NEWLINENEWLINENEWLINE(1) For a Riemannian manifold \((N,ds^2)\) with an isoparametric function \(f\), take a special conformal deformation \(\widetilde{ds^2}=e^{2u(f)}ds^2\). Then \(f\) is also isoparametric on \((N,\widetilde{ds^2})\). NEWLINENEWLINENEWLINE(2) For a cohomogenity-one manifold \((N,G)\) with a \(G\)-invariant metric, taking composition of some smooth function on \(N/G\) with the projection \(\pi:N\rightarrow N/G\), we get isoparametric functions on \(N\). NEWLINENEWLINENEWLINE(3) For a Riemannian submersion \(\pi:E\rightarrow B\) with minimal fibres, if \(f\) is an isoparametric function on \(B\), then so is \(F:=f\circ \pi\) on \(E\). NEWLINENEWLINENEWLINEApplying the second and the third method, the authors give examples of isoparametric functions on Brieskorn varieties and on isoparametric hypersurfaces of spheres. Also, in Section 4 the authors construct several examples of isoparametric functions on the Milnor spheres (exotic 7-spheres obtained as some \(S^3\)-bundles over \(S^4\) [\textit{M. A. Kervaire} and \textit{J. W. Milnor}, Ann. Math. (2) 77, 504--537 (1963; Zbl 0115.40505)]).