Smallest area surface evolving with unit areal speed Constantin Udri¸ ste, Ionel T ¸ evy Abstract. The theory of smallest area surfaces evolving with unit areal speed is a particular case of the theory of surfaces of minimum area subject to various constraints. Based on our recent results, such problems can be solved using the two-time maximum principle in a controlled evolution. Section 1 studies a controlled dynamics problem (smallest area surface evolving with unit areal speed) via the two-time maximum principle. The evolution PDE is of 2-flow type and the adjoint PDE is of divergence type. Section 2 analyzes the smallest area surfaces evolving with unit areal speed, avoiding an obstacle. Section 3 reconsiders the same problem for touching an obstacle, detailing the results for the cylinder and the sphere. M.S.C. 2010: 49J20, 49K20, 49N90, 90C29. Key words: multitime maximum principle, smallest area surface, minimal surfaces, controlled dynamics with obstacle. 1 Smallest area surface evolving with unit areal speed, passing through two points The minimal surfaces are characterized by zero mean curvature. They include, but are not limited to, surfaces of minimum area subject to various constraints. Minimal surfaces have become an area of intense mathematical and scientific study over the past 15 years, specifically in the areas of molecular engineering and materials sciences due to their anticipated nanotechnology applications (see [1]-[5]). Let Ω 0τ be a bidimensional interval fixed by the diagonal opposite points 0,τ ∈ R 2 + . Looking for surfaces x i (t)= x i (t 1 ,t 2 ), (t 1 ,t 2 ) ∈ Ω 0τ , i =1, 2, 3, that evolve with unit areal speed and relies transversally on two curves Γ 0 and Γ 1 , let us show that a minimum area surface (2-sheet) is a solution of a special PDE system, via the optimal control theory (multitime maximum principle, see [11]-[22]). An example is a planar quadrilateral (totally geodesic surface in R 3 ) fixed by the origin x i (0) = x i 0 on Γ 0 and passing through the diagonal terminal point x i (τ )= x i 1 on Γ 1 . * Balkan Journal of Geometry and Its Applications, Vol.16, No.1, 2011, pp. 155-169. c  Balkan Society of Geometers, Geometry Balkan Press 2011.