标题： 曲面在黎曼乘积中的极小曲面 显示英文标题

标题： Minimal surfaces in the Riemannian product of surfaces

摘要： 在常曲率曲面的黎曼乘积中的极小曲面最近受到了关注，特别是因为这些乘积作为三维空间形式的有向测地线空间出现。本文考虑了更一般的曲面黎曼乘积，并探讨了极小曲面所面临的几何和拓扑限制。我们证明了通常情况下，黎曼乘积中的全测地曲面局部要么是切片，要么是测地线的乘积。如果因子的高斯曲率是负的，则证明不存在极小2球面，而极小2环面则相对于两种乘积辛结构都是拉格朗日的。如果曲面具有非零有界曲率，我们建立了极小2球面面积的精确下限，并探讨了一般紧致极小曲面的高斯曲率和法曲率的性质。 显示英文摘要

摘要： Minimal surfaces in the Riemannian product of surfaces of constant curvature have been considered recently, particularly as these products arise as spaces of oriented geodesics of 3-dimensional space-forms. This papers considers more general Riemannian products of surfaces and explores geometric and topological restrictions that arise for minimal surfaces. We show that generically, a totally geodesic surface in a Riemannian product is locally either a slice or a product of geodesics. If the Gauss curvatures of the factors are negative, it is proven that there are no minimal 2-spheres, while minimal 2-tori are Lagrangian with respect to both product symplectic structures. If the surfaces have non-zero bounded curvatures, we establish a sharp lower bound on the area of minimal 2-spheres and explore the properties of the Gauss and normal curvatures of general compact minimal surfaces.