TY - JOUR

T1 - Area density and regularity for soap film-like surfaces spanning graphs

AU - Gulliver, Robert

AU - Yamada, Sumio

PY - 2006/3

Y1 - 2006/3

N2 - For a given boundary Γ in R n consisting of arcs and vertices, with two or more arcs meeting at each vertex, we treat the problem of estimating the area density of a soap film-like surface ∑ spanning Γ. ∑ is assumed locally to minimize area, or more generally, to be strongly stationary for area with respect to Γ. We introduce a notion of total curvature [InlineMediaObject not available: see fulltext.](Γ) for such graphs, or nets, Γ. We show that 2π times the area density of ∑ at any point is less than or equal to [InlineMediaObject not available: see fulltext.](Γ). For n=3, these density estimates imply, for example, that if [InlineMediaObject not available: see fulltext.](Γ)≤3.649π, then the only possible singularities of a piecewise smooth (M,0,δ)-minimizing set ∑ are curves, along which three smooth sheets of ∑ meet with equal angles of 120°.

AB - For a given boundary Γ in R n consisting of arcs and vertices, with two or more arcs meeting at each vertex, we treat the problem of estimating the area density of a soap film-like surface ∑ spanning Γ. ∑ is assumed locally to minimize area, or more generally, to be strongly stationary for area with respect to Γ. We introduce a notion of total curvature [InlineMediaObject not available: see fulltext.](Γ) for such graphs, or nets, Γ. We show that 2π times the area density of ∑ at any point is less than or equal to [InlineMediaObject not available: see fulltext.](Γ). For n=3, these density estimates imply, for example, that if [InlineMediaObject not available: see fulltext.](Γ)≤3.649π, then the only possible singularities of a piecewise smooth (M,0,δ)-minimizing set ∑ are curves, along which three smooth sheets of ∑ meet with equal angles of 120°.

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U2 - 10.1007/s00209-005-0903-9

DO - 10.1007/s00209-005-0903-9

M3 - Article

AN - SCOPUS:33645526649

SN - 0025-5874

VL - 253

SP - 315

EP - 331

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 2

ER -