Among various basic local
differential geometric invariants of a given hypersurface in the euclidean
(n + 1)-space, Mn⊂ En+1, the mean curvature, i.e. the trace of the second
fundamental form, is certainly one of the simplest numerical invariants with
important geometric meaning, namely, the first variation of “area”. Therefore,
complete hypersurfaces of constant mean curvatures in En+1 naturally constitute a
nice family of simple global geometric objects that certainly deserve special attention.
Especially, those closed ones can be considered as natural generalizations of “soapbubbles” and the problem of such generalized soap bubbles in the euclidean spaces
has been attracting the attention of differential geometers since Euler and
Monge.
This paper is the second part of a systematic study of hypersurfaces of constant
mean curvatures in En+1 which are of generalized rotational types, succeeding a
previous paper with the same title.
