Toponogov-type area comparison theorem of 2-dimensional manifolds (Q2853991)

The authors treat the question whether the area of a triangle on a 2-dimensional manifold \(M\) (with curvature \(\geq k\)) can be compared with that of its comparison triangle on \(S_k^2\). The authors give first the following definition for the area of a triangle: ``We say that the triangle \(\triangle pqr\) on \(M\) is areable if there is a simply connected domain bounded by \(\triangle pqr\) and the area of \(\triangle pqr\) is defined by the minimum of the areas of such domains.'' Then, the main result of the paper is: ``Let \(\triangle p_1p_2p_3\) be a geodesic triangle on \(M\) (a complete \(2\)-dimensional Riemannian manifold of curvature \(\geq k\)) and let \(\widetilde{\triangle} p_1p_2p_3\) be its comparison triangle on \(S_k^2\) (a complete and simply connected 2-dimensional manifold of constant curvature \(k\)). If \(\triangle p_1p_2p_3\) is areable, then its area is not less than that of \(\widetilde{\triangle} p_1p_2p_3\).''