Alexandrov’s theorem revisited

Abstract

We show that among sets of finite perimeter balls are the only volumeconstrained critical points of the perimeter functional.

Soap films with gravity and almost-minimal surfaces

“…In the case M j is the boundary of an open set (and thus, necessarily, Γ = ∅), and M j has almost-constant (non-zero) mean curvature, then the occurrence of bubbling is unavoidable, and its description has been undertaken in various papers, see e.g. [BC84, Str84, CM17, DMMN17, KM17,DM17]. From this point of view, the fact that we can avoid bubbling under somehow generic assumptions on the boundary data Γ is a remarkable rigidity feature of Plateau's problem.…”

Embeddedness of liquid-vapour interfaces in stable equilibrium

“…It turns out that, allowing virtual volume-preserving deformations "coalescence" and "break up" does indeed permit to rule out (a'); unlike for (a) and (b), however, it will be the stability assumption, rather than stationarity, that will do so (stationarity alone would not suffice). In fact we will prove the following proposition in Section 5 (the formal notions of "coalescence" and "break-up" deformations are given in Definitions 2, 3, 4 and the notions of stationarity and stability for these deformations are given in Definition 5 below): 10 For further comments on bounds on the full curvature, see Section 6.…”

“…It is not immediately clear why the curvature should play a role, since E Ω selects the mean curvature (and not the full curvature) as the key quantity that drives towards an equilibrium. The mean curvature of a tiny neck, obtained by a smoothing of two touching spherical caps, may attain any value, in particular the neck could be part of a Delaunay surface, which is stationary for g ≡ 0 (in stark contrast with (a), where smoothing leads to very high mean curvature) 10 .…”

“…The example produced in[6] shows that the L ∞ condition is too weak for this purpose 7. In the special case g ≡ 0 and assuming that E is bounded and Ω = R n+1 ,[10] strikingly proves that, under only this stationarity condition, E must equal a collection of balls with equal radii; the arguments in[10] exploit however global properties. A local regularity result (that goes beyond Allard's theorem) under this stationarity assumption alone does not seem within reach at the moment, and at any rate it would have to allow for non-embedded points, as we will also see below.…”

The Asymptotics of the Area-Preserving Mean Curvature and the Mullins-Sekerka Flow in Two Dimensions

“…There has been a lot of recent research on generalizations and quantifications of the Alexandrov theorem. We refer to [10] for an overview of this challenging problem, and mention the works [14,15,16] on the characterization of critical sets of the isoperimetric problem and [11,12,27] on quantification of the Alexandrov theorem.…”