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Minimal surfaces from circle patterns: geometry from combinatorics
 Ann. of Math
"... The theory of polyhedral surfaces and, more generally, the field of discrete differential geometry are presently emerging on the border of differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete geometry studie ..."
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Cited by 48 (10 self)
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The theory of polyhedral surfaces and, more generally, the field of discrete differential geometry are presently emerging on the border of differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete geometry studies geometric
An embedded genusone helicoid
, 2005
"... There exists a properly embedded minimal surface of genus one with one end. The end is asymptotic to the end of the helicoid. This genus one helicoid is constructed as the limit of a continuous oneparameter family of screwmotion invariant minimal surfacesalso asymptotic to the helicoidthat ha ..."
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Cited by 12 (3 self)
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There exists a properly embedded minimal surface of genus one with one end. The end is asymptotic to the end of the helicoid. This genus one helicoid is constructed as the limit of a continuous oneparameter family of screwmotion invariant minimal surfacesalso asymptotic to the helicoidthat have genus equal to one in the quotient.
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 ..."
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Cited by 6 (6 self)
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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. ..."
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Cited by 5 (1 self)
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We derive global Weierstrass representations for complete minimal surfaces obtained by substituting the ends of the Costa surface by symmetry curves.
Materializing 3D QuasiFuchsian Fractals
, 2005
"... Abstract. This paper reports experiments of materializing recent new discovered mathematical surfaces, 3D quasiFuchsian fractals. Three different models in glass, plastic, and metal are created to realize rich mathematical properties including selfsimilarity, 3embeddable, simplyconnected, and com ..."
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Abstract. This paper reports experiments of materializing recent new discovered mathematical surfaces, 3D quasiFuchsian fractals. Three different models in glass, plastic, and metal are created to realize rich mathematical properties including selfsimilarity, 3embeddable, simplyconnected, and complicated surface consisting of infinite number of cusps. Different models can complementary provide mathematicians and anyone unprofessional better understanding of the mathematical properties of the new discovery. 1.
Geometrization Program of Semilinear Elliptic Equations
"... Abstract. Understanding the entire solutions of nonlinear elliptic equations in RN such as (0.1) Δu + f(u) =0 inR N, is a basic problem in PDE research. This is the context of various classical results in literature like the GidasNiNirenberg theorems on radial symmetry, Liouville type theorems, or ..."
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Abstract. Understanding the entire solutions of nonlinear elliptic equations in RN such as (0.1) Δu + f(u) =0 inR N, is a basic problem in PDE research. This is the context of various classical results in literature like the GidasNiNirenberg theorems on radial symmetry, Liouville type theorems, or the achievements around De Giorgi’s conjecture. In those results, the geometry of level sets of the solutions turns out to be a posteriori very simple (planes or spheres). On the other hand, problems of the form (0.1) do have solutions with more interesting patterns, and the structure of their solution sets has remained mostly a mystery. A major aspect of our research program is to bring ideas from Differential Geometry into the analysis and construction of entire solutions for two important equations: (1) the AllenCahn equation and (2) the nonlinear Schrodinger equation (NLS). Though simplelooking, they are typical representatives of two classes of semilinear elliptic problems. The structure of entire solutions is quite rich. In this survey, we shall establish an intricate correspondence between the study of entire solutions of some scalar equations and the theories of minimal surfaces and constant mean curvature surfaces (CMC). 1. Part I: Geometrization Program of AllenCahn Equation In this section, we survey the studies on entire solutions of AllenCahn equation. 1.1. Background. The AllenCahn equation in R N is the semilinear elliptic problem (1.1) Δu + u − u 3 =0 inR N. Originally formulated in the description of biphase separation in fluids [16] and ordering in binary alloys [2], Equation (1.1) has received extensive mathematical study. It is a prototype for the modeling of phase transition phenomena in a variety of contexts. Introducing a small positive parameter ε and writing v(x):=u(ε −1 x), we get the scaled version of (1.1), (1.2) ε 2 Δv + v − v 3 =0 inR N. c ○ 2012 American Mathematical Society and International Press 831 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 832 J. WEI On every bounded domain Ω ⊂ RN, (1.1) is the EulerLagrange equation for the action functional ε Jε(v) =