On bending of a convex surface to a convex surface with prescribed spherical image
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We prove the following theorem. LetF be a regular convex surface homeomorphic to the disk. Suppose the Gaussian curvature ofF is positive and the geodesic curvature of its boundary is positive as well. LetG be a convex domain on the unit sphere bounded by a smooth curve and strictly contained in a hemisphere. LetP be an arbitrary point on the boundary ofF andP* be an arbitrary point on the boundary ofG. If the area ofG is equal to the integral curvature of the surfaceF, then there exists a continuous bending of the surfaceF to a convex surfaceF′ such that the spherical image ofF′ coincides withG andP* is the image of the point inF′ corresponding to the pointP ∈ F under the isometry.
KeywordsUnit Sphere Arbitrary Point Smooth Curve Gaussian Curvature Integral Curvature
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