Results 1  10
of
13
Adding handles to the helicoid
 Bulletin of the AMS, New Series
, 1993
"... Abstract. There exist two new embedded minimal surfaces, asymptotic to the helicoid. One is periodic, with quotient (by orientationpreserving 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, ..."
Abstract

Cited by 19 (7 self)
 Add to MetaCart
(Show Context)
Abstract. There exist two new embedded minimal surfaces, asymptotic to the helicoid. One is periodic, with quotient (by orientationpreserving 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
The Gauss map of pseudoalgebraic minimal surfaces
"... 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 pseudoalgebraic minimal surfaces in Euclidean fourspace and give a kind of unicity theorem. 1. ..."
Abstract

Cited by 16 (11 self)
 Add to MetaCart
(Show Context)
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 pseudoalgebraic minimal surfaces in Euclidean fourspace and give a kind of unicity theorem. 1.
A NOTE ON SINGULAR TIME OF MEAN CURVATURE FLOW
, 810
"... 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. ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
(Show Context)
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.
A FAMILY OF TRIPLY PERIODIC COSTA SURFACES
 PACIFIC JOURNAL OF MATHEMATICS
, 2003
"... We derive global Weierstrass representations for complete minimal surfaces obtained by substituting the ends of the Costa surface by symmetry curves. ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
We derive global Weierstrass representations for complete minimal surfaces obtained by substituting the ends of the Costa surface by symmetry curves.
Properly embedded minimal planar domains with infinite topology are Riemann minimal examples
, 2009
"... ..."
Curvature estimates for graphs with prescribed mean curvature and flat normal bundle
, 2008
"... ..."
ON SUBMANIFOLDS WITH TAMED SECOND FUNDAMENTAL FORM
, 805
"... Abstract. Based on the ideas of BessaJorgeMontenegro [4] we show that a complete submanifold M with tamed second fundamental form in a complete Riemannian manifold N with sectional curvature KN ≤ κ ≤ 0 are proper, (compact if N is compact). In addition, if N is Hadamard then M has finite topology. ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
Abstract. Based on the ideas of BessaJorgeMontenegro [4] we show that a complete submanifold M with tamed second fundamental form in a complete Riemannian manifold N with sectional curvature KN ≤ κ ≤ 0 are proper, (compact if N is compact). In addition, if N is Hadamard then M has finite topology. We also show that the fundamental tone is an obstruction for a Riemannian manifold to be realized as submanifold with tamed second fundamental form of a Hadamard manifold with sectional curvature bounded below. 1.
A note on the Gauss map of complete nonorientable minimal surfaces
"... We construct complete nonorientable minimal surfaces whose Gauss map omits two points of RP 2. This result proves that Fujimoto’s theorem is sharp in nonorientable case. 1. Introduction and Preliminaries. The study of the Gauss map of complete orientable minimal surfaces in R3 has achieved many impo ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
We construct complete nonorientable minimal surfaces whose Gauss map omits two points of RP 2. This result proves that Fujimoto’s theorem is sharp in nonorientable case. 1. Introduction and Preliminaries. The study of the Gauss map of complete orientable minimal surfaces in R3 has achieved many important advances and also has given rise to many problems in recent decades. The most interesting question is to determine the size of the spherical image of such a surface under its Gauss map.
On the Existence of Higher Dimensional Enneper’s Surface
"... 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 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
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 2dimensional minimal surface. In this paper, however, we prove that there also exists an ndimensional 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 twodimensional 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 dimension one does not even have a good way to construct examples of complete immersed minimal
ON MINIMAL HYPERSURFACES WITH FINITE HARMONIC INDICES
"... 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. Eac ..."
Abstract
 Add to MetaCart
(Show Context)
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.