Min–max theory for constant mean curvature hypersurfaces

Abstract

In this paper, we develop a min-max theory for the construction of constant mean curvature (CMC) hypersurfaces of prescribed mean curvature in an arbitrary closed manifold. As a corollary, we prove the existence of a nontrivial, smooth, closed, almost embedded, CMC hypersurface of any given mean curvature c. Moreover, if c is nonzero then our min-max solution always has multiplicity one.

Min-max theory for networks of constant geodesic curvature

“…Nevertheless, using our new characterisation for blowups, Pitts's work [13] does directly imply the geodesic network regularity of his weak solution. In fact, away from finitely many points, Pitts' weak solution has the good replacement property in small balls, so any tangent varifold satisfies the assumptions of our classification result (using an observation in [17,Lemma 5.10]), and hence is an integer multiple of a line. With this, one can proceed the same as Pitts to obtain the desired regularity.…”

“…The goal of this article is to show that on a closed surface, for any c > 0, our CMC min-max theory [17,18] (which is based on the Almgren-Pitts min-max theory for minimal hypersurfaces [3,13]) produces a solution given by a curve of constant geodesic curvature c which is almost embedded, except for finitely many points, at which the solution is a stationary junction with integer density. Moreover, each smooth constant geodesic curvature segment has multiplicity one.…”

“…In this paper, we carry out the process described above in the setting for c > 0, using the theory developed by us in [17,18].…”

“…In [17,18], the authors established an existence theory, which in this setting yields that there is a 1-varifold V associated with L c satisfying a list of useful properties that we will summarize in the following. In particular, the theory in [17,18] works in any closed Riemannian manifold (M n+1 , g) (using the corresponding n-dimensional c-weighted area functional), and when 3 ≤ n + 1 ≤ 7, we proved that V is induced by the boundary of some Caccioppoli set Ω 0 , whose boundary Σ 0 = ∂Ω is an almost embedded closed hypersurface of constant mean curvature c. However, since the classification of stable minimal hypercones by Simons [16] does not hold in dimension n = 1, we cannot directly obtain similar regularity results for V when n = 1. Instead, we will exploit some stronger properties of V that were obtained in [17] to achieve some partial regularity.…”

“…In particular, the theory in [17,18] works in any closed Riemannian manifold (M n+1 , g) (using the corresponding n-dimensional c-weighted area functional), and when 3 ≤ n + 1 ≤ 7, we proved that V is induced by the boundary of some Caccioppoli set Ω 0 , whose boundary Σ 0 = ∂Ω is an almost embedded closed hypersurface of constant mean curvature c. However, since the classification of stable minimal hypercones by Simons [16] does not hold in dimension n = 1, we cannot directly obtain similar regularity results for V when n = 1. Instead, we will exploit some stronger properties of V that were obtained in [17] to achieve some partial regularity. In fact, we will use certain good replacement properties in small disks instead of just in small annuli.…”