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192
The space of embedded minimal surfaces of fixed genus in a 3manifold V; Fixed genus
, 2005
"... This paper is the fifth and final in a series on embedded minimal surfaces. Following our earlier papers on disks, we prove here two main structure theorems for nonsimply connected embedded minimal surfaces of any given fixed genus. The first of these asserts that any such surface without small nec ..."
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Cited by 66 (10 self)
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This paper is the fifth and final in a series on embedded minimal surfaces. Following our earlier papers on disks, we prove here two main structure theorems for nonsimply connected embedded minimal surfaces of any given fixed genus. The first of these asserts that any such surface without small necks can be obtained by gluing together two oppositely–oriented double spiral staircases; see Figure 1. The second gives a pair of pants decomposition of any such surface when there are small necks, cutting the surface along a collection of short curves; see Figure 2. After the cutting, we are left with graphical pieces that are defined over a disk with either one or two sub–disks removed (a topological disk with two sub–disks removed is called a pair of pants). Both of these structures occur as different extremes in the twoparameter family of minimal surfaces known as the Riemann examples. The results of [CM3]–[CM6] have already been used by many authors; see, e.g., the surveys [MeP], [P], [Ro] and the introduction in [CM6] for some of these applications. There is much current research on minimal surfaces with infinite topology. Some of the results of the present paper were announced previously and have already been widely used to study
Minimal surfaces in M × R
 Illinois Journal of Math
"... recent developments in the theory of minimal ..."
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The uniqueness of the helicoid
 ANNALS OF MATHEMATICS, 161 (2005), 723–754
, 2005
"... In this paper we will discuss the geometry of finite topology properly embedded minimal surfaces M in R3. M of finite topology means M is homeomorphic to a compact surface M ̂ (of genus k and empty boundary) minus a finite number of points p1,..., pj ∈ M ̂ , called the punctures. A closed neighbor ..."
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Cited by 52 (9 self)
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In this paper we will discuss the geometry of finite topology properly embedded minimal surfaces M in R3. M of finite topology means M is homeomorphic to a compact surface M ̂ (of genus k and empty boundary) minus a finite number of points p1,..., pj ∈ M ̂ , called the punctures. A closed neighborhood E of a puncture in M is called an end of M. We will choose the ends sufficiently small so they are topologically S1 × [0, 1) and hence, annular. We remark that M ̂ is orientable since M is properly embedded in R3. The simplest examples (discovered by Meusnier in 1776) are the helicoid and catenoid (and a plane of course). It was only in 1982 that another example was discovered. In his thesis at Impa, Celso Costa wrote down the Weierstrass representation of a complete minimal surface modelled on a 3punctured torus. He observed the three ends of this surface were embedded: one top catenoidtype end1, one bottom catenoidtype end, and a middle planartype end2 [8]. Subsequently, Hoffman and Meeks [15] proved this example is embedded and they constructed for every finite positive genus k embedded examples of genus
Schrödinger operators associated to a holomorphic map
 Proceedings Conference on Global Analysis and Global Differential Geometry
, 1990
"... In this work we will expose certain ideas and results concerning a kind of Schrödinger operators which can be considered on a compact Riemann surface. These operators will be constructed by using as a potential the energy density of a holomorphic map from the surface to the twosphere. Besides the ..."
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Cited by 41 (7 self)
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In this work we will expose certain ideas and results concerning a kind of Schrödinger operators which can be considered on a compact Riemann surface. These operators will be constructed by using as a potential the energy density of a holomorphic map from the surface to the twosphere. Besides the interest that their study has from an analytical point of view, we will see that they appear, in a natural way, in different geometrical situations such as the study of the index of complete minimal surfaces with finite total curvature and the study of the critical points of the Willmore functional. This paper is, in fact, an expanded version of an invited lecture given by the first author in the Global Differential Geometry and Global Analysis Conference held at the Technische Universität of Berlin in June, 1990. Introduction and preliminaries Let Σ be a compact Riemann surface and φ: Σ → S2 a holomorphic map from this surface to the unit two–sphere S2. Consider any metric ds2 on Σ compatible with the complex structure and let ∆ and ∇ be its Laplacian and gradient respectively. Having
BERNSTEIN AND DE GIORGI TYPE PROBLEMS: NEW RESULTS VIA A GEOMETRIC APPROACH
"... Abstract. We use a Poincaré type formula and level set analysis to detect onedimensional symmetry of stable solutions of possibly degenerate or singular elliptic equation of the form div a(∇u(x))∇u(x) + f(u(x)) = 0. Our setting is very general and, as particular cases, we obtain new proofs of a ..."
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Cited by 38 (15 self)
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Abstract. We use a Poincaré type formula and level set analysis to detect onedimensional symmetry of stable solutions of possibly degenerate or singular elliptic equation of the form div a(∇u(x))∇u(x) + f(u(x)) = 0. Our setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in R 2 and R 3 and of the Bernstein problem on the flatness of minimal area graphs in R 3. A onedimensional symmetry result in the halfspace is also obtained as a byproduct of our analysis. Our approach is also flexible to nonelliptic operators: as an application, we prove onedimensional symmetry for 1Laplacian type operators. 1.
The Geometry of Periodic Minimal Surfaces
, 1993
"... this paper we shall demonstrate a surprising relationship between the topology of a properly embedded periodic minimal surface in R ..."
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Cited by 34 (1 self)
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this paper we shall demonstrate a surprising relationship between the topology of a properly embedded periodic minimal surface in R
Estimates for parametric elliptic integrands
 Int. Math. Res. Not. 2002
"... Let M3 be a Riemannian 3manifold. Given a function φ ≥ 1 on the unit sphere bundle ofM, we define a functional Φ on an immersed oriented surface Σ inM by ..."
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Cited by 32 (6 self)
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Let M3 be a Riemannian 3manifold. Given a function φ ≥ 1 on the unit sphere bundle ofM, we define a functional Φ on an immersed oriented surface Σ inM by
The minimal lamination closure theorem
 Duke Math. J
"... We prove that the closure of a complete embedded minimal surface M in a Riemannian threemanifold N has the structure of a minimal lamination, when M has positive injectivity radius. When N is R3, we prove that such a surface M is properly embedded. Since a complete embedded minimal surface of fini ..."
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Cited by 30 (7 self)
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We prove that the closure of a complete embedded minimal surface M in a Riemannian threemanifold N has the structure of a minimal lamination, when M has positive injectivity radius. When N is R3, we prove that such a surface M is properly embedded. Since a complete embedded minimal surface of finite topology in R3 has positive injectivity radius, the previous theorem implies a recent theorem of Colding and Minicozzi: A complete embedded minimal surface of finite topology in R3 is proper. More generally, we prove that if M is a complete embedded minimal surface of finite topology and N has nonpositive sectional curvature (or is the Riemannian product of a homogeneously regular Riemannian surface with R), then the closure of M has the structure of a minimal lamination.
Heat Kernel and Essential Spectrum of Infinite Graphs
"... ABSTRACT. We study the existence and uniqueness of the heat kernel on infinite, locally finite, connected graphs. For general graphs, a uniqueness criterion, shown to be optimal, is given in terms of the maximal valence on spheres about a fixed vertex. A sufficient condition for nonuniqueness is al ..."
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Cited by 30 (3 self)
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ABSTRACT. We study the existence and uniqueness of the heat kernel on infinite, locally finite, connected graphs. For general graphs, a uniqueness criterion, shown to be optimal, is given in terms of the maximal valence on spheres about a fixed vertex. A sufficient condition for nonuniqueness is also presented. Furthermore, we give a lower bound on the bottom of the spectrum of the discrete Laplacian and use this bound to give a condition ensuring that the essential spectrum of the Laplacian is empty. 1.