Esistenza e regolarità delle ipersuperfici di curvatura media assegnata in Rn

Global properties of constant mean curvature surfaces in ℍ2× ℝ

“…Furthermore, = ∂ Q \ S is regular at interior points, that is, at points of ∩ Int W h , and has mean curvature H at regular points. The existence follows from [Massari 1974, Theorem 1.1]; Massari states the result in Euclidean space, but, as his functional is the same as our F, the proofs are analogous.…”

A note on the isoperimetric constant

“…The second proof (section 4) is elementary, based on standard comparison arguments. The first one (section 3) uses a result from geometric measure theory: It is shown in [1] or follows indirectly though more accessibly from [19] (see also remark 3.3 below) that /z(M), as mentioned in the introduction, is a minimum, obtained for an open submanifold of M whose boundary X is a rectifiable current ( [13], p. 355) with the following.…”

Some Remarks on Uniqueness and Regularity of Cheeger Sets

“…p REMARK 2.6. If V is an open set and C is a minimizer of (2), by classical regularity results [25] we know that (@C n S) V is analytic, where S is a closed singular set of dimension at most N À 8. Moreover, if V is of class C 1;1 , then C is a minimizer of a prescribed curvature problem with curvature in L I [8], hence @C n S is of class W 2;p for all p5I (see also [29] for the case N 3).…”