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- Title
Two-dimensional solutions to the c-Plateau problem in $$\mathbb {R}^{3}$$.
- Authors
Rosales, Leobardo
- Abstract
We give several regularity results for two-dimensional solutions to the c-Plateau problem in $$\mathbb {R}^{3}:$$ given $$c>0$$ and an integer one-rectifiable current $$\varGamma $$ without boundary in $$\mathbb {R}^{3},$$ we study integer two-rectifiable currents which minimize c-isoperimetric mass $$\mathbf {M}^{c}(T):=\mathbf {M}(T)+c \mathbf {M}(\partial T)^{2},$$ where $$\mathbf {M}$$ is the usual mass on currents, amongst all integer two-rectifiable currents T with boundary of the form $$\partial T = \varGamma + \varSigma $$ where $$\varGamma ,\varSigma $$ have disjoint supports. We show the following three results for $$\mathbf {T}_{c}$$ a solution to the c-Plateau problem with $$\partial \mathbf {T}_{c} = \varGamma +\varSigma _{c}$$ : if $$\varSigma _{c}$$ is a smooth closed embedded curve, then $$\varSigma _{c}$$ parameterized by arc-length must have at some point large third derivative; $$\mathbf {T}_{c}$$ cannot have a tangent cone at a singular point of $$\varSigma _{c}$$ supported in a plane but with constant orientation; $$\varSigma _{c}$$ is regular wherever we can write the support of $$\mathbf {T}_{c}$$ as a finite union of $$C^{1}$$ surfaces-with-boundary.
- Subjects
PLATEAU'S problem; ISOPERIMETRICAL problems; BOUNDARY element methods; ARC length; MATHEMATICAL analysis
- Publication
Annals of Global Analysis & Geometry, 2016, Vol 50, Issue 2, p129
- ISSN
0232-704X
- Publication type
Article
- DOI
10.1007/s10455-016-9505-5