Non-holomorphic, stable minimal surfaces in flat tori

解决了n维欧氏空间中极小图的伯恩斯坦问题。在那之后，考虑了伯恩斯坦问题的许多变化。作为推广，我们知道三维欧氏空间中完备的、可定向的、稳定的极小曲面只是一个平面。如果我们考虑一个3维黎曼平坦环面中的紧致的、可定向的极小曲面，那么我们看到它是全测地的，因为在单位法向量场方向上的变形使得面积很小。Micallef证明了四维黎曼平坦环面中的紧致、可定向、稳定的极小曲面对于适当的正交复环面结构是次纯的。我们期望，紧的，我们期望，紧的，可定向稳定的极小曲面在一个4维黎曼平坦环面是全纯的适当的正交复结构的环面。我们期望在低维空间中得到紧致、可定向、稳定的极小曲面 ...更多信息 所有黎曼平坦环面都是次纯的。最近，阿雷佐和Micallef证明了（2g-2），（2g-4），（2g-6）维黎曼平坦环面中亏格为g（不小于7，9，10）的非全纯、紧致、可定向、稳定极小曲面的存在性.他们期望在一个8维黎曼平坦环面中存在一个亏格为g（不小于4）的非全纯的、紧的、可定向的、稳定的极小曲面。另一方面，Ejiri研究了黎曼平坦环面中紧致可定向k极小曲面的稳定性与面积极小性的区别，给出了如下问题：2g维黎曼平坦环面中亏格为g的紧致可定向稳定极小曲面是否在同一同伦类中面积极小？设n为自然数，M_g（n）为亏格为g（不小于4）的黎曼曲面的模M_g的子集，其中M_g（n）的元素在包含n个面积不同的稳定极小曲面的7维黎曼平坦环面中有稳定极小浸入。然后我们得到以下结果。定理M_g（n）在M_g中是稠密的。因此，我们知道阿雷佐和Micallef猜想是正确的，并获得了Ejiiri问题的一个反例

The Bernstein Problem about minimal graphs in an n-dimensional Euclidean space have been solved. After that, many variations of the Bernstein problem were considered. As an generalization, we know that complete, an orientable, stable minimal surface in a 3-dimensional Euclidean space is only a plane. If we consider a compact, oerientable, minimal surface in a 3-dimensional Riemannian flat torus, then we see that it is totally geodesic, because the deformation in the direction of the unit normal vector field makes the area to be small. Micallef proved that a compact, orientabl, stable minimal surface in an in a 4-dimensional Riemannian flat torus is horomorphic for a suitable orthogonal complex structure of the torus. We expect that compact, We expect that compact, orientable stable minimal surface in a 4-dimensional riemannian flat torus is holomorphic for a suitable orthogonal complex structure of the torus. We expect that compact, orientable, stable minimal surface in lower dimension … More al Riemannian flat tori are horomorphic. Recently, Arezzo and Micallef proved the existence of non-holomorphic, compact, orientable, stable minimal surface of genus g (not less than 7,9,10) in a ((2g-2),(2g-4),(2g-6))-dimensional Riemannian flat torus. They expect the existence of a non-holomorphic, compact, orientable, stable minimal surface of genus g (not less than 4) in a 8-dimensional Riemannian flat torus. On the other hand, investigating the difference between stableness and area area-minimizingness of compact, orientablek minimal surface in Riemannian flat tori, Ejiri have given the following problem: Are compact, orientablestable minimal surfaces of genus g in a 2g-dimensional Riemannian flat torus area-minimizing in the same homotopy class? In this research, we obtain the following fact: Let n be a natural number and M_g(n) the subset of the moduli M_g of Riemann surfaces of genus g(not less than 4) where the element of M_g(n) admits a stable minimal immersion in a 7-demen Rimannian flat torus containing n stable minimal surfaces with different areas in the same homotopy class. Then we obtain the following. Theorem M_g(n) is dense in M_g.. Hence, we know that the Arezzo and Micallef conjecture is true and obtain a counter example for Ejiiri's problem Less