Related papers: Compact special Legendrian surfaces in $S^5$

We prove a Ros-Rosenberg theorem in the setting of Special Weingarten surfaces. We show that a compact, connected, embedded, Special Weingarten surface in $\mathhb{R}^3$ with planar convex boundary is a topological disk under mild suitable…

We classify weakly complete constant Gaussian curvature $-1<K<0$ surfaces in the hyperbolic three-space in terms of holomorphic quadratic differentials. For this purpose, we first establish a loop group method for constant Gaussian…

Using the Kodaira dimension and the fundamental group of X, we succeed in classifying algebraic surfaces which are dominable by C^2 except for certain cases in which X is an algebraic surface of Kodaira dimension zero and the case when X is…

The discrete Laplacian on Euclidean triangulated surfaces is a well-established notion. We introduce discrete Laplacians on spherical and hyperbolic triangulated surfaces. On the one hand, our definitions are close to the Euclidean one in…

Let $(X, \omega)$ be a compact symplectic manifold and $L$ be a Lagrangian submanifold. Suppose $(X, L)$ has a Hamiltonian $S^1$ action with moment map $\mu$. Take an invariant $\omega$-compatible almost complex structure, we consider…

In this paper we describe the cohomogeneity one special Lagrangian 3-folds in the cotangent bundle of the 3-sphere, also known in the physics literature as a deformed conifold. Our main result gives a global foliation of the deformed…

In this paper we classify all singular irreducible symplectic surfaces, i.e., compact, connected complex surfaces with canonical singularities that have a holomorphic symplectic form $\sigma$ on the smooth locus, and for which every finite…

We study area-stationary, or maximal, surfaces in the space ${\mathbb L}({\mathbb H}^3)$ of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral K\"ahler structure. We prove that every holomorphic curve in ${\mathbb…

For each $k\geq2$, we construct two families of surfaces with constant mean curvature $H$ for $H\in[0,1/2]$ in $\Sigma(\kappa)\times\R$ where $\kappa+4H^2\leq0$. The surfaces are invariant under $2\pi/k$-rotations about a vertical fiber of…

For each integer m>1 and l>0 we construct a pair of compact embedded minimal surfaces of genus 1+4m(m-1)l. These surfaces desingularize the m Clifford tori meeting each other along a great circle at the angle of \pi/m. They are invariant…

For a given minimal Legendrian submanifold $L$ of a Sasaki-Einstein manifold we construct two families of eigenfunctions of the Laplacian of $L$ and we give a lower bound for the dimension of the corresponding eigenspace. Moreover, in the…

Suppose $M_{1}$ and $M_{2}$ are two special Lagrangian submanifolds of $\Rtn$ with boundary that intersect transversally at one point $p$. The set $M_{1} \cup M_{2}$ is a singular special Lagrangian variety with an isolated singularity at…

In this paper we develop the compactness theorem for $\lambda$-surface in $\mathbb R^3$ with uniform $\lambda$, genus, and area growth. This theorem can be viewed as a generalization of Colding-Minicozzi's compactness theorem for…

We study a surface discovered by Stover which is the surface with minimal Euler number and maximal automorphism group among smooth arithmetic ball quotient surfaces. We study the natural map $\wedge^{2}H^{1}(S,\mathbb{C})\to…

We construct a Riemannian metric $g$ on $\mathbb{R}^4$ (arbitrarily close to the euclidean one) and a smooth simple closed curve $\Gamma\subset \mathbb R^4$ such that the unique area minimizing surface spanned by $\Gamma$ has infinite…

We exhibit infinitely many, explicit special Lagrangian isolated singularities that admit no asymptotically conical special Lagrangian smoothings. The existence/ nonexistence of such smoothings is an important component of the current…

Given a compact surface $\Gamma$ embedded in $\mathbb R^3$ with boundary $\partial \Gamma$, our goal is to construct a set of representatives for a basis of the relative cohomology group $H^1(\Gamma, \partial \Gamma^c)$, where $\Gamma^c$ is…

We define a notion of isotropic surfaces in $\mathbb{O}$, i.e. on which some canonical symplectic forms vanish. Using the cross-product in $\mathbb{O}$ we define a map $\rho\colon Gr\_2(\mathbb{O})\to S^6$ from the Grassmannian of…

Let $M$ be a K\"ahler surface and $\Sigma$ be a closed symplectic surface which is smoothly immersed in $M$. Let $\alpha$ be the K\"ahler angle of $\Sigma$ in $M$. We first deduce the Euler-Lagrange equation of the functional…

Motivated by a recent work of Chen-Zheng [8] on Strominger space forms, we prove that a compact Hermitian surface with pointwise constant holomorphic sectional curvature with respect to a Gauduchon connection $\nabla^t $ is either K\"ahler,…