Sphere theorem by means of the ratio of mean curvature functions

Abstract

Abstract. It is well known that a compact embedded hypersurface of the Euclidean space without boundary is a round sphere if one of mean curvature functions is constant. In this note, we show that a compact embedded hypersurface of the Euclidean space (and other constant curvature spaces) without boundary is a round sphere if the ratio of some two mean curvature functions is constant.

Weighted Hsiung-Minkowski formulas and rigidity of umbilical hypersurfaces

“…Theorem 2 contains the case where H k H l = η(r) for some monotone decreasing function η and k > l. We notice that the same result also applies to the space forms R n , H n and S n + (open hemisphere) without the star-shapedness assumption (Theorem 7). Our result extends [20,Theorem B] We next prove, in Section 4, a rigidity theorem for self-expanding solitons to the weighted generalized inverse curvature flow in Euclidean space R n≥3 :…”

Integral Formulae for Spacelike Hypersurfaces in Conformally Stationary Spacetimes and Applications

“…Related to this, Koh [22] has recently proved the same kind of result under the hypothesis that the ratio H s /H r is constant, 1 r < s n.…”

Addendum to the paper: Sphere theorem by means of the ratio of mean curvature functions

“…For Theorem A in [1] is not true in this case as Wente's disproving the Hopf's conjecture [3] shows. In this note, however, we prove the theorem above for an immersion by slightly changing the argument of [1]. Theorem 2.…”

“…As H 0 is defined to be 1, H k =H 0 ¼ H k . Thus, if H k =H 0 is constant, the theorem above does not hold for the same reason that the proof of [1] does not apply directly. The first-named author would like to thank the referee of [1] for suggesting Theorem 2.…”

“…If we assume that is an immersion, the proof in [1] does not apply directly. For Theorem A in [1] is not true in this case as Wente's disproving the Hopf's conjecture [3] shows.…”