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We show the area-minimality property of all homogeneous area-minimizing hypercones in Euclidean spaces (classified by Lawlor) following Lawson's original idea in his 72' Trans. A.M.S. paper 'The equivariant Plateau problem and interior regularity'. Moreover, each of them enjoys (coflat) calibrations singular only at the origin. [ABSTRACT FROM AUTHOR]

Using a standard fact in hyperbolic geometry, we give a simple proof of the uniqueness of PL minimal surfaces, thus filling in a gap in the original proof of Jaco and Rubinstein. Moreover, in order to clarify some ambiguity, we sharpen the definition of PL minimal surfaces, and prove a technical lemma on the Plateau problem in the hyperbolic space. [ABSTRACT FROM AUTHOR]

The heat flow for the minimal surface under Plateau boundary condition is defined to be a parabolic variational inequality, and then the existence, uniqueness, regularity, continuous dependence on the initial data and the asymptotics are studied. It is applied as a deformation of the level sets in the critical point theory. [ABSTRACT FROM AUTHOR]