Quartic differentials and harmonic maps in conformal surface geometry

where Ai(p) and A2(p) are the principal curvatures of M at p. When H is constant, M is called a surface of constant mean curvature. A surface is said to have finite type if it is homeomorphic to a closed surface with a finite number of points removed. An important problem in classical differential geometry is the classification of properly embedded finite type surfaces M of constant mean curvature in R . If M is a closed embedded surface of constant mean curvature, then it follows from Alexandrov [1] that M must be a round sphere. The classical examples of properly embedded surfaces with zero mean curvature are the plane, the helicoid and the catenoid. Surfaces of zero mean curvature are usually called minimal surfaces. The remaining classical examples of properly embedded surfaces of constant mean curvature were found by Delaunay [4]. The Delaunay surfaces are surfaces of revolution. Recently Hoffman and Meeks [6, 7] have found examples of properly embedded minimal surfaces which are homeomorphic to closed surfaces of positive genus with 3 points removed. Callahan, Hoffman and Meeks [3] have found other examples with more ends. An annular end E of & properly embedded surface in R 3 is a properly embedded annulus E in M where E is homeomorphic to S x [0,1). When M has finite type, then every end of M is annular. Hoffman and Meeks have developed a theory to deal with global problems concerning the geometry of properly embedded minimal surfaces M and, in particular, they show that most annular ends of M converge at infinity in R to a flat plane or to the end of a catenoid. Recently N. Kapouleas [8] in his thesis has shown that for every positive integer k > 2, there exists a properly embedded surface M& of finite type with nonzero mean curvature and with k ends. He also has constructed highergenus examples. As in the case of minimal surfaces, the annular ends of a properly embedded surface of nonzero constant mean curvature have a special geometry and play an important role in global theorems.

Discretizing constant curvature surfaces via loop group factorizations: The discrete sine- and sinh-Gordon equations

Local space time constant mean curvature and constant expansion foliations

The surfaces of revolution with constant mean curvature in 3 are classified by Delaunay [2] in 1841. They are locally plane, catenoid, sphere, circular cylinder, unduloid, and nodoid up to isometries of 3. On these 10 years, new and interesting examples of non-zero constant mean curvature surfaces are discovered. In the global study of complete surfaces with constant mean curvature, unduloids and nodoids play important role as the models of ends of such surfaces (see [5], [6]). The work by Delaunay is now revived after 150 years of his discovery. The purpose of this paper is to study surfaces of revolution with periodic mean curvature in order to extend the theory of constant mean curvature surfaces. In general such a surface is not periodic, because the catenoid gives the counter-example. First we show the criterion for a periodic function to be the mean curvature of a periodic surface of revolution and second describe a method how to construct these periodic surfaces of revolution whose mean curvatures are periodic functions satisfying the criterion. I thank Professor Yusuke Sakane for his interest and plotting of the beautiful pictures by the computer of Osaka University which I used in the seminar talk of Granada University on March, 2001, and in the AMS meeting held in Hoboken on April, 2001. The figures of this paper are all programmed by the author using the symbolic manipulation program Mathematica. I also thank my graduate student Shinya Hirakawa for his help to simplify the programing.

It is shown that timelike surfaces of constant mean curvature ± in anti-de Sitter 3-space ℍ3 1(−1) can be constructed from a pair of Lorentz holomorphic and Lorentz antiholomorphic null curves in ℙSL2ℝ via Bryant type representation formulae. These Bryant type representation formulae are used to investigate an explicit one-to-one correspondence, the so-called Lawson–Guichard correspondence, between timelike surfaces of constant mean curvature ± 1 and timelike minimal surfaces in Minkowski 3-space E 3 1. The hyperbolic Gaus map of timelike surfaces in ℍ3 1(−1), which is a close analogue of the classical Gaus map is considered. It is discussed that the hyperbolic Gaus map plays an important role in the study of timelike surfaces of constant mean curvature ± 1 in ℍ3 1(−1). In particular, the relationship between the Lorentz holomorphicity of the hyperbolic Gaus map and timelike surface of constant mean curvature ± 1 in ℍ3 1(−1) is studied.

The aim of this work is to show that a compact smooth star-shaped hypersurface <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma Superscript n"> <mml:semantics> <mml:msup> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\Sigma ^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the Euclidean sphere <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper S Superscript n plus 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {S}^{n+1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose second function of curvature <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S 2"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">S_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a positive constant must be a geodesic sphere <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper S Superscript n Baseline left-parenthesis rho right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi> ρ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {S}^{n}(\rho )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This generalizes a result obtained by Jellett in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1853"> <mml:semantics> <mml:mn>1853</mml:mn> <mml:annotation encoding="application/x-tex">1853</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for surfaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma squared"> <mml:semantics> <mml:msup> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\Sigma ^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with constant mean curvature in the Euclidean space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R cubed"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as well as a recent result of the authors for this type of hypersurface in the Euclidean sphere <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper S Superscript n plus 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {S}^{n+1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with constant mean curvature. In order to prove our theorem we shall present a formula for the operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript r Baseline left-parenthesis g right-parenthesis equals d i v left-parenthesis upper P Subscript r Baseline nabla g right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>r</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>d</mml:mi> <mml:mi>i</mml:mi> <mml:mi>v</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mi mathvariant="normal"> ∇ </mml:mi> <mml:mi>g</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">L_{r}(g)=div\left (P_r\nabla g\right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> associated with a new support function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> defined over a hypersurface <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Superscript n"> <mml:semantics> <mml:msup> <mml:mi>M</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">M^n</mml:annotation>

We establish a general `gluing theorem', which states roughly that if two nondegenerate constant mean curvature surfaces are juxtaposed, so that their tangent planes are parallel and very close to one another, but oppositely oriented, then there is a new constant mean curvature surface quite near to this configuration (in the Hausdorff topology), but which is a topological connected sum of the two surfaces. Here nondegeneracy refers to the invertibility of the linearized mean curvature operator. This paper treats the simplest context for our result namely when the surfaces are compact with nonempty boundary, however the construction applies in the complete, noncompact setting as well. The surfaces we produce here are nondegenerate for generic choices of the free parameters in the construction.

In this work we will deal with disc type surfaces of constant mean curvature in the three dimensional hyperbolic space which are given as graphs of smooth functions over planar domains. From the various types of graphs that could be defined in the hyperbolic space we consider in particular the horizontal and the geodesic graphs. We proved that if the mean curvature is constant, then such graphs are equivalent in the following sense: suppose that M is a constant mean curvature surface in the 3-hyperbolic space such that M is a geodesic graph of a function rho that is zero at the boundary, then there exist a smooth function f that also vanishes at the boundary, such that M is a horizontal graph of f. Moreover, the reciprocal is also true.

We introduce two notions of symmetry for surfaces in ‫ޓ‬ 3 .The first, special spherical symmetry, generalizes the notion of rotational symmetry, and we classify all complete surfaces of constant mean curvature having this symmetry.These surfaces turn out to also be rotationally symmetric, so our characterization answers a question first posed by Hsiang in 1982 and also considered by several authors since.From this point of view, these are the Delaunay surfaces of ‫ޓ‬ 3 .Our second notion of symmetry, spherical symmetry, is a substantial, and we believe important, technical generalization of special spherical symmetry.We classify all compact surfaces of constant mean curvature having this symmetry.We show in particular that the only compact embedded minimal surfaces possessing this kind of symmetry are the great spheres and the Clifford torus.We derive from our classification theorem a special case of Lawson's conjecture that the only embedded minimal torus in ‫ޓ‬ 3 is the Clifford torus.

Equilibrium surfaces originate from the mechanics of liquids and gases as the interface between two media that are in equilibrium. The equilibrium condition arises from the condition of minimum potential energy of the corresponding mechanical system. Equilibrium surfaces include the classes of minimal surfaces, surfaces of constant mean curvature and equilibrium capillary surfaces. The study of the stability of equilibrium surfaces is closely related to the questions of the existence of a solution to the variational multidimensional problem for the minimum of the potential energy functional. In particular, unstable solutions of the corresponding differential equations are not realizable in nature. Stability is characterized by the positivity of the form of the second variation of the corresponding functional (for example, the area functional for minimal surfaces). In most cases, this property means a lower bound for a quantity similar to the fundamental frequency of a region on a surface. In this article, I follow the approach of Sh.T. Yau obtained lower bounds for the quantity that generalizes the fundamental frequency of the region. Based on these estimates, the stability conditions for minimal surfaces and surfaces of constant mean curvature are proved.

In these notes we present some aspects of the basic theory on the geometry of a three-dimensional simply-connected Lie group X X endowed with a left invariant metric. This material is based upon and extends some of the results of Milnor in Curvatures of left invariant metrics on Lie groups . We then apply this theory to study the geometry of constant mean curvature H ≥ 0 H\geq 0 surfaces in X X , which we call H H -surfaces . The focus of these results on H H -surfaces concerns our joint on going research project with Pablo Mira and Antonio Ros to understand the existence, uniqueness, embeddedness and stability properties of H H -spheres in X X . To attack these questions we introduce several new concepts such as the H H -potential of X X , the critical mean curvature H ( X ) H(X) of X X and the notion of an algebraic open book decomposition of X X . We apply these concepts to classify the two-dimensional subgroups of X X in terms of invariants of its metric Lie algebra, as well as classify the stabilizer subgroup of the isometry group of X X at any of its points in terms of these invariants. We also calculate the Cheeger constant for X X to be Ch ( X ) = trace ( A ) (X)=\mbox {trace}(A) , when X = R 2 ⋊ A R X=\mathbb {R}^2\rtimes _A\mathbb {R} is a semidirect product for some 2 × 2 2\times 2 real matrix; this result is a special case of a more general theorem by Peyerimhoff and Samiou. We also prove that in this semidirect product case, Ch ( X ) = 2 H ( X ) = 2 I ( X ) (X)=2H(X)=2I(X) , where I ( X ) I(X) is the infimum of the mean curvatures of isoperimetric surfaces in X X . In the last section, we discuss a variety of unsolved problems for H H -surfaces in X X .

It is still an open question whether a compact embedded hypersurface in the Euclidean space with constant mean curvature and spherical boundary is necessarily a hyperplanar ball or a spherical cap, even in the simplest case of a compact constant mean curvature surface in \mathbb{R}^3 bounded by a circle. In this paper we prove that this is true for the case of the scalar curvature. Specifically we prove that the only compact embedded hypersurfaces in the Euclidean space with constant scalar curvature and spherical boundary are the hyperplanar round balls (with zero scalar curvature) and the spherical caps (with positive constant scalar curvature). The same applies in general to the case of embedded hypersurfaces with constant r -mean curvature, with r \geq 2 .

In this paper we shall present a construction of complete surfaces M in R3 with finitely many ends and finite topology, and with nonzero constant mean curvature (CMC). This construction is parallel to the well-known original construction by Kapouleas [5], but we feel that ours is somewhat simpler analytically, and controls the resulting geometry more closely. On the other hand, the surfaces we construct have a rather different, and usually simpler, geometry than those of Kapouleas; in particular, all of the surfaces constructed here are noncompact, so we do not obtain any of his immersed compact examples. The method we use here closely parallels the one we developed recently [10] to study the very closely related problem of constructing Yamabe metrics on the sphere with k isolated singular points, just as Kapouleas’ construction parallels the earlier construction of singular Yamabe metrics by Schoen [18]. The original examples of noncompact CMC surfaces were those in the one-parameter family of rotationally invariant surfaces discovered by Delaunay in 1841 [2]. One extreme element of this family is the cylinder; the ‘Delaunay surfaces’ are periodic, and the embedded members of this family (which are called unduloids) interpolate between the cylinder and an infinite string of spheres arranged along a common axis. The family continues beyond this, but the elements now are immersed (and are called nodoids). The role of Delaunay surfaces in the theory of complete CMC surfaces is analogous to the role of catenoids (and planes) in the study of complete minimal surfaces of finite total curvature. For example, just as any complete minimal surface with two ends must be a catenoid [19], it was proved by Meeks [14] and Korevaar, Kusner and Solomon [8] that any Alexandrov embedded constant mean curvature surface with at most two ends is necessarily a Delaunay surface. A rather more remarkable theorem, paralleling the fact that any end of a complete minimal surface of finite total curvature must be asymptotic to a catenoid or a plane, is the fact that any embedded end of a CMC surface must be asymptotic to one of these rotationally symmetric Delaunay surfaces (and in particular, must be cylindrically bounded).

We investigate surfaces of constant mean curvature one in the hyperbolic three-space with irregular ends, and prove that their irregular ends must self-intersect, which answers affirmatively a conjecture of Umehara and Yamada. Moreover we also obtain an explicit representation of a constant mean curvature one surface and a new minimal surface in the Euclidean three-space.

In this paper, we study the intrinsic structures of high-dimensional data sets for analyzing their geometrical properties, where the core message of the high-dimensional data is hiding on some nonlinear manifolds. Using the manifold learning technique with a particular focus on the mean curvature, we develop new methods to investigate the uniqueness of constant mean curvature spacelike hypersurfaces in the Lorentzian warped product manifolds. Furthermore, we extend the uniqueness of stochastically complete hypersurfaces using the weak maximum principle. For the more general cases, we propose some non-existence results and a priori estimates for the constant higher-order mean curvature spacelike hypersurface.