Results 1  10
of
19
Properly embedded and immersed minimal surfaces in the Heisenberg group
 Bull. Austral. Math. Soc
"... We study properly embedded and immersed p(pseudohermitian)minimal surfaces in the 3dimensional Heisenberg group. From the recent work of Cheng, Hwang, Malchiodi, and Yang, we learn that such surfaces must be ruled surfaces. There are two types of such surfaces: band type and annulus type according ..."
Abstract

Cited by 29 (7 self)
 Add to MetaCart
(Show Context)
We study properly embedded and immersed p(pseudohermitian)minimal surfaces in the 3dimensional Heisenberg group. From the recent work of Cheng, Hwang, Malchiodi, and Yang, we learn that such surfaces must be ruled surfaces. There are two types of such surfaces: band type and annulus type according to their topology. We give an explicit expression for these surfaces. Among band types there is a class of properly embedded pminimal surfaces of so called helicoid type. We classify all the helicoid type pminimal surfaces. This class of pminimal surfaces includes all the entire pminimal graphs (except contact planes) over any plane. Moreover, we give a necessary and sufficient condition for such a pminimal surface to have no singular points. For general complete immersed pminimal surfaces, we prove a half space theorem and give a criterion for the properness.
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
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
ENTIRE SOLUTIONS OF THE ALLENCAHN EQUATION AND COMPLETE EMBEDDED MINIMAL SURFACES OF FINITE TOTAL CURVATURE IN R³
"... We consider minimal surfaces M which are complete, embedded and have finite total curvature in R³, and bounded, entire solutions with finite Morse index of the AllenCahn equation ∆u+f(u) = 0 in R³. Here f = −W ′ with W bistable and balanced, for instance W (u) = 1 4 (1 − u2) 2. We assume that M h ..."
Abstract

Cited by 13 (9 self)
 Add to MetaCart
(Show Context)
We consider minimal surfaces M which are complete, embedded and have finite total curvature in R³, and bounded, entire solutions with finite Morse index of the AllenCahn equation ∆u+f(u) = 0 in R³. Here f = −W ′ with W bistable and balanced, for instance W (u) = 1 4 (1 − u2) 2. We assume that M has m ≥ 2 ends, and additionally that M is nondegenerate, in the sense that its bounded Jacobi fields are all originated from rigid motions (this is known for instance for a Catenoid and for the CostaHoffmanMeeks surface of any genus). We prove that for any small α> 0, the AllenCahn equation has a family of bounded solutions depending on m−1 parameters distinct from rigid motions, whose level sets are embedded surfaces lying close to the blownup surface Mα: = α−1M, with ends possibly diverging logarithmically from Mα. We prove that these solutions are L∞nondegenerate up to rigid motions, and find that their Morse index coincides with the index of the minimal surface. Our construction suggests parallels of De Giorgi conjecture for general bounded solutions
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.
The space of complete minimal surfaces with finite total curvature as Lagrangian submanifold
 Trans. Amer. Math. Soc
"... Abstract. The space M of nondegenerate, properly embedded minimal surfaces in R 3 with finite total curvature and fixed topology is an analytic lagrangian submanifold of C n,wherenis the number of ends of the surface. In this paper we give two expressions, one integral and the other pointwise, for t ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
(Show Context)
Abstract. The space M of nondegenerate, properly embedded minimal surfaces in R 3 with finite total curvature and fixed topology is an analytic lagrangian submanifold of C n,wherenis the number of ends of the surface. In this paper we give two expressions, one integral and the other pointwise, for the second fundamental form of this submanifold. We also consider the compact boundary case, and we show that the space of stable nonflat minimal annuli that bound a fixed convex curve in a horizontal plane, having a horizontal end of finite total curvature, is a locally convex curve in the plane C. 1.
SOLITON SPHERES
, 905
"... Abstract. Soliton spheres are immersed 2–spheres in the conformal 4–sphere S 4 = HP 1 that allow rational, conformal parametrizations f: CP 1 → HP 1 obtained via twistor projection and dualization from rational curves in CP 2n+1. Soliton spheres can be characterized as the case of equality in the qu ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
(Show Context)
Abstract. Soliton spheres are immersed 2–spheres in the conformal 4–sphere S 4 = HP 1 that allow rational, conformal parametrizations f: CP 1 → HP 1 obtained via twistor projection and dualization from rational curves in CP 2n+1. Soliton spheres can be characterized as the case of equality in the quaternionic Plücker estimate. A special class of soliton spheres introduced by Taimanov are immersions into R 3 with rotationally symmetric Weierstrass potentials that are related to solitons of the mKdV–equation via the ZS–AKNS linear problem. We show that Willmore spheres and Bryant spheres with smooth ends are further examples of soliton spheres. The possible values of the Willmore energy for soliton spheres in the 3–sphere are proven to be W = 4πd with d ∈ N\{0, 2, 3, 5, 7}. The same quantization was previously known individually for each of the three special classes of soliton spheres mentioned above. 1.
V.: A limitmethod for solving period problems on minimal surfaces
 Math. Z
"... We introduce a new technique to solve period problems on minimal surfaces called “limitmethod”. If a family of surfaces has Weierstraßdata converging to the data of a known example, and this presents a transversal solution of periods, then the original family contains a subfamily with closed perio ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
We introduce a new technique to solve period problems on minimal surfaces called “limitmethod”. If a family of surfaces has Weierstraßdata converging to the data of a known example, and this presents a transversal solution of periods, then the original family contains a subfamily with closed periods. 1.