Almost-Einstein hypersurfaces in the Euclidean space. (English) Zbl 1213.53072

Vlachos, Theodoros

Ill. J. Math. 53, No. 4, 1221-1235 (2009).

The author shows that almost-Einstein hypersurfaces \((M^n,g), n\geq 3\) (in the sense that \(\text{Ric}_g-kg\) or \(\text{Ric}_g-\frac{s_g}{n}g\) are small in the \(L^{n/2}\)-norm for a constant \(k\geq 0\), the Ricci tensor \(\text{Ric}_g\) and the scalar curvature \(s_g\)) in the Euclidean space \(\mathbb R^{n+1}\) are homeomorphic to spheres \(S^n\). The idea for the proofs relies on universal lower bounds in terms of the Betti numbers for the \(L^{n/2}\)-norms of the Ricci and the traceless Ricci tensor of compact oriented \(n\)-dimensional hypersurfaces. Moreover, certain counterexamples show that the assumption on the codimension to be 1 is essential.

Reviewer: Mirjana Djoric (Belgrade)

MSC:

53C40	Global submanifolds
53C20	Global Riemannian geometry, including pinching
53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Keywords:

almost-Einstein hypersurface; Euclidean space; sphere; homeomorphic; \(L^{n/2}\)-norm

References:

[1]	J. Cheeger and D. Ebin, Comparison theorems in Riemannian geometry , North-Holland Mathematical Library, vol. 9, North-Holland, Amsterdam–Oxford; American Elsevier Publishing, New York, 1975. · Zbl 0309.53035
[2]	S. S. Chern and R. K. Lashof, On the total curvature of immersed manifolds , Amer. J. Math. 79 (1957), 306–318. JSTOR: · Zbl 0078.13901
[3]	S. S. Chern and R. K. Lashof, On the total curvature of immersed manifolds II , Michigan Math. J. 5 (1958), 5–12. · Zbl 0095.35803
[4]	A. Fialkow, Hypersurfaces of a space of constant curvature , Ann. of Math. (2) 39 (1938), 762–783. JSTOR: · Zbl 0020.06601
[5]	N. H. Kuiper, Minimal total absolute curvature for immersions , Invent. Math. 10 (1970), 209–238. · Zbl 0195.51102
[6]	J. Milnor, Morse theory. Based on lecture notes by M. Spivak and R. Wells , Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, NJ, 1963. · Zbl 0108.10401
[7]	J. Roth, Pinching of the first eigenvalue of the Laplacian and almost-Einstein hypersurfaces of the Euclidean space , Ann. Global Anal. Geom. 33 (2008), 293–306. · Zbl 1151.53010
[8]	J. Roth, Sphere rigidity in the Euclidean space ; available at arXiv :0710.5041.
[9]	P. J. Ryan, Homogeneity and some curvature conditions for hypersurfaces , Tôhoku Math. J. (2) 21 (1969), 363–388. · Zbl 0185.49904
[10]	K. Shiohama and H. Xu, Lower bound for \(L^n/2\) curvature norm and its application , J. Geom. Anal. 7 (1997), 377–386. · Zbl 0960.53022
[11]	T. Y. Thomas, On closed space of constant mean curvature , Amer. J. Math. 58 (1936), 702–704. JSTOR: · Zbl 0015.27303
[12]	N. R. Wallach, Minimal immersions of symmetric spaces into spheres , Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969–1970), Dekker, New York, pp. 1–40. Pure and Appl. Math., Vol. 8, 1972. · Zbl 0232.53027