Rotational surfaces of prescribed Gauss curvature in $\mathbb{R}^3$

Antonio Bueno; Irene Ortiz

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arXiv · 2201.07057

Rotational surfaces of prescribed Gauss curvature in $\mathbb{R}^3$

Antonio Bueno·Irene Ortiz

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Abstract

We study rotational surfaces in Euclidean 3-space whose Gauss curvature is given as a prescribed function of its Gauss map. By means of a phase plane analysis and under mild assumptions on the prescribed function, we generalize the classification of rotational surfaces of constant Gauss curvature; exhibit examples that cannot exist in the constant Gauss curvature case; and analyze the asymptotic behavior of strictly convex graphs. We also prove the existence of singular radial solutions intersecting orthogonally the axis of rotation.

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Antonio Bueno, Irene Ortiz. 2022-01-18. Rotational surfaces of prescribed Gauss curvature in $\mathbb{R}^3$. https://arxiv.org/abs/2201.07057

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