Tohoku Mathematical Journal
2024
March
SECOND SERIES VOL. 76, NO. 1

Tohoku Math. J.
76 (2024), 87-103

Title ROTATIONAL SURFACES OF PRESCRIBED GAUSS CURVATURE IN $\Mathbb{R}^3$

Author Antonio Bueno and Irene Ortiz

(Received March 2, 2022, revised June 23, 2022)
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.

Mathematics Subject Classification. Primary 53A10; Secondary 53C42, 34C05, 34C40.
Key words and phrases. Prescribed Gauss curvature, radial solution, nonlinear autonomous system, asymptotic behavior.

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