Mathematics – Differential Geometry
Scientific paper
2002-11-27
Mathematics
Differential Geometry
42 pages
Scientific paper
A surface $\Sigma \subset S^5 \subset \mathbb{C}^3$ is called \emph{special Legendrian} if the cone $0 \times \Sigma \subset \mathbb{C}^3$ is special Lagrangian. The purpose of this paper is to propose a general method toward constructing compact special Legendrian surfaces of high genus. It is proved \emph{there exists a compact, orientable, Hamiltonian stationary Lagrangian surface of genus $1+\frac{k(k-3)}{2}$ in $ \mathbb{C}P^2$ for each integer $ k \geq 3$, which is a smooth branched surface except at most finitely many conical singularities.} If this surface is smooth, it is minimal and the Legendrian lift of the surface is the desired compact special Legendrian surface. We first establish the existence of a minimizer of area among Lagrangian disks in a relative homotopy class of a K\"ahler-Einstein surface without Lagrangian homotopy classes with respect to a configuration $ \Gamma$ that consists of the fixed point loci of K\"ahler involutions. $ \Gamma$ in addition must satisfy certain null relative homotopy conditions and angle criteria. The fundamental domain thus obtained is smooth along the boundary, and has finitely many interior singular points. We then apply successive \emph{reflection} of this fundamental domain along its boundary to obtain a complete or compact Lagrangian surface.
No associations
LandOfFree
Compact special Legendrian surfaces in $S^5$ does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Compact special Legendrian surfaces in $S^5$, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Compact special Legendrian surfaces in $S^5$ will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-420047