A pinching theorem for the first eigenvalue of the Laplacian on hypersurfaces of the Euclidean space. (English) Zbl 1112.53003

Let \(\phi\) be an isometric immersion of a compact orientable \(n\)-dimensional Riemannian manifold \((M,g)\) into Euclidean space \(\mathbb R^{m+1}\). Let \(dv\) be the Riemannian measure on \(M\), let \(H\) be the mean curvature vector, and let \(\lambda_1(M)\) be the first eigenvalue. Normalize the volume of \(M\) to be \(1\). R. Reilly [Comment. Math. Helv. 52, 525–533 (1977; Zbl 0382.53038)] showed \(\lambda_1(M)\leq n\int_M| H| ^2\,dv\); equality holds iff \(M\) is a geodesic hypersphere. The authors establish the following pinching theorem:
Theorem 1.1. Let \(x_0\) be the center of mass. For any \(p\geq2\) and for any \(\varepsilon>0\), there exists \(C=C(n,\varepsilon,| H| _\infty)\) so that if \(n| H| _{2p}^2-C<\lambda_1(M)\), the Hausdorff distance \(d_H\) from \(M\) to the sphere \(S(x_0,\sqrt{n\lambda_1(M)^{-1}})\) is less than \(\varepsilon\).
The authors show that the pinching constant \(C(n,\varepsilon,| H| _\infty)\rightarrow0\) when \(| H| _\infty\rightarrow\infty\) or \(\varepsilon\rightarrow0\). The authors also show that if the pinching is strong enough with a control on \(n\) and on \(| H| _\infty\), then \(M\) is diffeomorphic to a sphere and even almost isometric with a round sphere.


53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions


Zbl 0382.53038
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