| Abstract: | In this paper we study Willmore Legendrian surfaces (that is Legendrian surfaces which are critical points of the Willmore functional). We use an equality proved in Luo (arXiv:1211.4227v6) to get a relation between Willmore Legendrian surfaces and contact stationary Legendrian surfaces in (mathbb {S}^5), and then we use this relation to prove a classification result for Willmore Legendrian spheres in (mathbb {S}^5). We also get an integral inequality for Willmore Legendrian surfaces and in particular we prove that if the square length of the second fundamental form of a Willmore Legendrian surface in (mathbb {S}^5) belongs to [0, 2], then it must be 0 and L is totally geodesic or 2 and L is a flat minimal Legendrian tori, which generalizes the result of Yamaguchi et al. (Proc Am Math Soc 54:276–280, 1976). We also study variation of the Willmore functional among Legendrian surfaces in 5-dimensional Sasakian manifolds. Let (Sigma ) be a closed surface and ((M,alpha ,g_alpha ,J)) a 5-dimensional Sasakian manifold with a contact form (alpha ), an associated metric (g_alpha ) and an almost complex structure J. Assume that (f:Sigma mapsto M) is a Legendrian immersion. Then f is called a contact stationary Legendrian Willmore surface (in short, a csL Willmore surface) if it is a critical point of the Willmore functional under contact deformations. To investigate the existence of csL Willmore surfaces we introduce a higher order flow which preserves the Legendre condition and decreases the Willmore energy. As a first step we prove that this flow is well posed if ((M,alpha ,g_alpha ,J)) is a Sasakian Einstein manifold, in particular (mathbb {S}^5). |
