Consider the family of generalized parabolas {y=−axr+c|a,r,c,x>0,risafixedconstant} that pass through a given point in first quadrant (and hence, depend on one parameter only). Find values for which piece corresponding parabola either encloses minimum area, or has length. We find sufficient condition under fixed point, area minimizing curve and length coincide. The problem led us to certain ...

we define minimal cn-dominating graph $mathbf {mcn}(g)$, commonality minimal cn-dominating graph $mathbf {cmcn}(g)$ and vertex minimal cn-dominating graph $mathbf {m_{v}cn}(g)$, characterizations are given for graph $g$ for which the newly defined graphs are connected. further serval new results are developed relating to these graphs.

SuÆcient conditions for which a minimal graph over a nonconvex domain is area-minimizing are presented. The conditions are shown to hold for subsurfaces of Enneper's surface, the singly periodic Scherk surface, and the associated surfaces of the doubly periodic Scherk surface which previously were unknown to be area-minimizing. In particular these surfaces are graphs over (angularly accessible)...

The set of the three dimensional polyominoes of minimal area and of volume n contains a polyomino which is the union of a quasicube j × (j + δ) × (j + θ), δ, θ ∈ {0, 1}, a quasisquare l × (l + ), ∈ {0, 1}, and a bar k. This shape is naturally associated to the unique decomposition of n = j(j +δ)(j +θ)+ l(l+ )+k as the sum of a maximal quasicube, a maximal quasisquare and a bar. For n a quasicub...

We describe the asymptotic behavior of minimal area submanifolds in product spacetimes of an asymptotically hyperbolic space times a compact internal manifold. In particular, we find that unlike the case of a minimal area submanifold just in an asymptotically hyperbolic space, the internal part of the boundary submanifold is constrained to be itself a minimal area submanifold. For applications ...

⋆ Partially supported by the National Science Foundation; Alfred P. Sloan Research Fellow. † Permanent address: Center for Theoretical Physics, MIT, Cambridge, Mass. 02139. Supported in part D.O.E. contract DE-AC02-76ER03069 and NSF grant PHY91-06210. We study the metric of minimal area on a punctured Riemann surface under the condition that all nontrivial homotopy closed curves be longer than ...

Recently a curvature theory for polyhedral surfaces has been established which associates with each face a mean curvature value computed from areas and mixed areas of that face and its corresponding Gauss image face. Therefore a study of minimal surfaces requires studying pairs of polygons with vanishing mixed area. We show that the mixed area of two edgewise parallel polygons equals the mixed ...