Tohoku Mathematical Journal
2026
June
SECOND SERIES VOL. 78, NO. 2

Tohoku Math. J.
78 (2026), 213-232

Title ON LEGENDRIAN SURFACES OF THE TANGENT SPHERE BUNDLE OF $\mathbb{R}^3$

Author Mingyan Li

(Received April 24, 2024, revised September 20, 2024)
Abstract. In this paper we study CMC and PMC Legendrian surfaces in tangent sphere bundle $T_1\mathbb{R}^3$. Two holomorphic Hopf differentials for CMC Legendrian surfaces are constructed. Using this, complete CMC Legendrian surfaces are classified. For PMC Legendrian surfaces in $T_1\mathbb{R}^3$, we also prove a non-existence result when the surface is closed or complete with non-negative Gauss curvature.

Mathematics Subject Classification. Primary 53C42; Secondary 53C24.
Key words and phrases. Legendrian surface, tangent sphere bundle, constant mean curvature, parallel mean curvature vector, classification.

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