Regularity of area minimizing currents mod p

Abstract

We establish a first general partial regularity theorem for area minimizing currents $${\mathrm{mod}}(p)$$ mod ( p ) , for every p, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of an m-dimensional area minimizing current $${\mathrm{mod}}(p)$$ mod ( p ) cannot be larger than $$m-1$$ m - 1 . Additionally, we show that, when p is odd, the interior singular set is $$(m-1)$$ ( m - 1 ) -rectifiable with locally finite $$(m-1)$$ ( m - 1 ) -dimensional measure.

An existence theorem for Brakke flow with fixed boundary conditions

“…,Ẽ N } is a partion of U , we have L n+1 (E i (t) Ẽ i ) = 0 for all i. This proves (9), and finishes the proof of ( 1)-( 11) except for (7), which is independent and is proved once we prove Proposition 6.9. Proposition 6.9 For all t ≥ 0, it holds (clos (spt V t )) \ U = ∂ 0 .…”

“…It is worth noticing that analogous generic nonuniqueness holds true also for Plateau's problem: in that context, different definitions of the key words surfaces, area, spanning in its formulation lead to solutions with dramatically different regularity properties, thus making each model a better or worse predictor of the geometric complexity of physical soap films; see e.g. the survey papers [6,15] and the references therein, as well as the more recent works [7][8][9][22][23][24]27]. It is then interesting and natural to investigate different formulations for Brakke flow as well.…”

The regularity theory for the area functional (in geometric measure theory)

“…[90]) pointed out to the authors that it is possible to derive Theorem 11.6 directly from the theory developed in [124], starting from one observation in [38] concerning tangent cones in the top stratum S m−1 \ S m−2 and the verification of Simon's no hole condition. In [40] Theorem 11.6 will be further used to confirm the conjectural picture in codimension 1, namely to prove Theorem 11.7. Let Σ be an m-dimensional area minimizing current mod p in R m+1 .…”

“…In the papers [41,42] the author, Jonas Hirsch, Andrea Marchese, and Salvatore Stuvard developed a theory to bound the dimension of flat singular points of a general area-minimizing current Σ mod p (i.e. in any dimension and codimension), which implies that the Hausdorff dimension of the set of flat singular points of Σ is at most m − 1.…”

“…Indeed the known examples would suggest that the set of flat singular points of any area minimizing current mod p is at most m − 2. The work [38] and the forthcoming one [39], by the author, Hirsch, Marchese, Spolaor, and Stuvard give a first step towards the latter picture in codimension 1.…”