Non-holomorphic, stable minimal surfaces in flat tori

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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