Abstract. There exist two new embedded minimal surfaces, asymptotic to the helicoid. One is periodic, with quotient (by orientation-preserving translations) of genus one. The other is nonperiodic of genus one. We have constructed two minimal surfaces of theoretical interest. The first is a complete, embedded, singly periodic minimal surface (SPEMS) that is asymptotic to the helicoid, has infinite genus, and whose quotient by translations has genus one. The quotient of the helicoid by translations has genus zero and the helicoid itself is simply connected. Theorem 1. There exists an embedded singly periodic minimal surface W1, asymptotic to the helicoid and invariant under a translation T. The quotient surface W1/T has genus equal to one and two ends. W1 contains a vertical axis, as does the helicoid, and W1/T contains two horizontal lines. The second surface is a complete, properly embedded minimal surface of finite topology with infinite total curvature. It is the first such surface to be found since the helicoid, which was discovered in the eighteenth century. (See Figure

Abstract. In this paper, we prove effective estimates for the number of exceptional values and the totally ramified value number for the Gauss map of pseudo-algebraic minimal surfaces in Euclidean four-space and give a kind of unicity theorem. 1.

We derive global Weierstrass representations for complete minimal surfaces obtained by substituting the ends of the Costa surface by symmetry curves.

Enneper’s surface and the catenoid are the simplest minimal surfaces in R 3 that are complete, orientable and nonplanar. This is because a complete orientable minimal surface has the total curvature of −4kπ for some nonnegative integer k, while k = 1 for Enneper’s surface and the catenoid. Enneper’s surface has one end and is a minimal immersion of R 2 in R 3, whereas the catenoid has two ends and is a surface of revolution. Not only in R 3 but also in R n, n ≥ 4, the catenoid has been known to exist. It is a minimal hypersurface which is rotationally symmetric. The higher dimensional catenoid has been the only example that is a higher dimensional analogue of a 2-dimensional minimal surface. In this paper, however, we prove that there also exists an n-dimensional Enneper’s surface Σ n in R n+1 for n=3,4,5,6, which is a minimal immersion of R n in R n+1. For two-dimensional minimal surfaces in R 3 there is the Weierstrass representation. This representation makes it easy to write down an enormous number of complete minimal surfaces in R 3. Moreover, one can construct arbitrarily many minimal submanifolds of even codimension in R 2n, as every complex submanifold of R 2n is minimal. But in higher dimen-sion one does not even have a good way to construct examples of complete immersed minimal

Abstract. We show that mean curvature flow of a compact submanifold in a complete Riemannian manifold cannot form singularity at time infinity if the ambient Riemannian manifold has bounded geometry and satisfies certain curvature and volume growth conditions. 1.

We introduce the concepts of harmonic stability and harmonic index for a complete minimal hypersurface in R n+1 (n ≥ 3) and prove that the hypersurface has only finitely many ends if its harmonic index is finite. Furthermore, the number of ends is bounded from above by 1 plus the harmonic index. Each end has a representation of nonnegative harmonic function, and these functions form a partition of unity. We also give an explicit estimate of the harmonic index for a class of special minimal hypersurfaces, namely, minimal hypersurfaces with finite total scalar curvature. It is shown that for such a submanifold the space of bounded harmonic functions is exactly generated by the representation functions of the ends. 1.

Curvature estimates for graphs with prescribed mean curvature and flat normal bundle

Abstract. In this paper, we prove effective estimates for the number of exceptional values and the totally ramified value number for the Gauss map of pseudo-algebraic minimal surfaces in Euclidean 4-space and give a kind of unicity theorem. 1.

submanifolds of R n with finite topology

minimal examples