Title: Total curvatures and minimal areas of complete surfaces
Abstract: Minimal areas for certain classes of finitely connected complete open surfaces are obtained by using a Bonnesen-style isoperimetric inequality for large balls on the surfaces. In particular, the minimal area of Riemannian planes whose Gaussian curvatures are bounded above by 1 is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4 pi"> <mml:semantics> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>π<!-- π --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">4\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.