Non-holomorphic, stable minimal surfaces in flat tori

本文解决了n维欧几里德空间中关于极小图的Bernstein问题。在那之后，伯恩斯坦问题的许多变体被考虑。作为推广，我们知道在三维欧几里得空间中完备的、可定向的、稳定的极小曲面只是一个平面。如果我们在三维黎曼平面环面上考虑一个紧凑的、可观测的最小曲面，那么我们看到它是完全测地线的，因为在单位法向量场方向上的变形使得面积很小。Micallef证明了四维黎曼平面环面中紧致、可定向、稳定的最小曲面对于一个合适的正交复环面结构是同纯的。我们期望一个四维黎曼平面环面中紧致的、可定向的稳定最小曲面对于一个合适的环面正交复结构是全纯的。我们期望在低维空间中得到紧致的、可定向的、稳定的极小曲面，更多的黎曼平面环面是同纯的。最近，Arezzo和Micallef证明了((2g-2),（2g-4),(2g-6)）维黎曼平面环面上g属（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8维黎曼平面环面中存在一个非全纯的、紧致的、可定向的、稳定的g属最小曲面（不小于4）。另一方面，Ejiri研究了黎曼平面环面中紧致可定向最小曲面的稳定性与面积极小性的区别，给出了以下问题：黎曼平面环面中g属的紧致可定向稳定最小曲面是否在同一同伦类中面积极小？在本研究中，我们得到了以下事实：设n是一个自然数，M_g(n)是g属黎曼曲面（不小于4）的模M_g的子集，其中M_g(n)的元素在含有n个不同面积的黎曼同伦类的稳定极小曲面的7元黎曼平面环面中存在稳定极小浸入。然后我们得到如下结果。定理M_g(n)在M_g中是稠密的。因此，我们知道Arezzo和Micallef猜想是正确的，并得到了Ejiiri问题Less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