Spacelike CMC 1 surfaces with elliptic ends in de Sitter 3-Space
Shoichi Fujimori
备注：
23 pages, 6 figures. v2: Section 3 added to give a criterion for embeddedness of elliptic ends. Corollary 5.8 added to show the existence of uncountably many CMC 1 faces from CMC 1 immersions in [MU]. New references added. v3: revision according to the referee's suggestions
We show that an Osserman-type inequality holds for spacelike surfaces of constant mean curvature (CMC) 1 with singularities and with elliptic ends in de Sitter 3-space. An immersed end of a CMC 1 surface is an ``elliptic end'' if the monodromy representation at the end is diagonalizable with eigenvalues in the unit circle. We also give a necessary and sufficient condition for equality in the inequality to hold, and in the process of doing this we derive a condition for determining when elliptic ends are embedded.
Fold singular points play important roles in the theory of maximal surfaces. For example, if a maximal surface admits fold singular points, it can be extended to a timelike minimal surface analytically. Moreover, there is a duality between conelike singular points and folds. In this paper, we investigate fold singular points on spacelike surfaces with non-zero constant mean curvature (spacelike CMC surfaces). We prove that spacelike CMC surfaces do not admit fold singular points. Moreover, we show that the singular point set of any conjugate CMC surface of a spacelike Delaunay surface with con…
In this paper, we study the singularities of spacelike constant mean curvature one (CMC 1) surfaces in the de Sitter 3-space. We prove the duality between generalized conelike singular points and 5/2-cuspidal edges on spacelike CMC 1 surfaces. To describe the duality between $A_{k+3}$ singularities and cuspidal $S_k$ singularities, we introduce two invariants, called the $α$-invariant and $σ$-invariant, of spacelike CMC 1 surfaces at their singular points. Moreover, we give a classification of non-degenerate singular points on spacelike CMC 1 surfaces.
We prove that a spacelike spherical symmetric constant mean curvature (SSCMC) surface and a general spacelike constant mean curvature (CMC) surface with certain boundary condition at the future null-infinity in Schwarzschild spacetime are asymptotically hyperbolic in the sense of Wang \cite{Wang2001} and Chruściel-Herzlich \cite{ChruscielHerzlich} respectively. Near the future null-infinity ($s=0$), we derive that the boundary data of spacelike CMC surfaces can be expressed as those on $\mathbb{S}^{2}$ up to three order and obtain a compatibility condition for fourth order derivatives near $s=…
We prove several topological properties of linear Weingarten surfaces of Bryant type, as wave fronts in hyperbolic 3-space. For example, we show the orientability of such surfaces, and also co-orientability when they are not flat. Moreover, we show an explicit formula of the non-holomorphic hyperbolic Gauss map via another hyperbolic Gauss map which is holomorphic. Using this, we show the orientability and co-orientability of CMC-1 faces (i.e., constant mean curvature one surfaces with admissible singular points) in de Sitter 3-space. (CMC-1 faces might not be wave fronts in general, but belon…
We solve spacelike spherically symmetric constant mean curvature (SS-CMC) hypersurfaces in Schwarzschild spacetimes and analyze their asymptotic behavior near the coordinate singularity r = 2M. Furthermore, we join SS-CMC hypersurfaces in the Kruskal extension to obtain complete ones and discuss the smooth properties.
We first summarize the characterization of smooth spacelike spherically symmetric constant mean curvature (SS-CMC) hypersurfaces in the Schwarzschild spacetime and Kruskal extension. Then use the characterization to prove special SS-CMC foliation property, and verify part of the conjecture by Malec and Ó Murchadha in their 2003 paper.
The first author studied spacelike constant mean curvature one (CMC-1) surfaces in de Sitter 3-space when the surfaces have no singularities except within some compact subset and are of finite total curvature on the complement of this compact subset. However, there are many CMC-1 surfaces whose singular sets are not compact. In fact, such examples have already appeared in the construction of trinoids given by Lee and the last author via hypergeometric functions. In this paper, we improve the Osserman-type inequality given by the first author. Moreover, we shall develop a fundamental framework …
In 1841, Delaunay constructed the embedded surfaces of revolution with constant mean curvature (CMC); these unduloids have genus zero and are now known to be the only embedded CMC surfaces with two ends and finite genus. Here, we construct the complete family of embedded CMC surfaces with three ends and genus zero; they are classified using their asymptotic necksizes. We work in a class slightly more general than embedded surfaces, namely immersed surfaces which bound an immersed three-manifold, as introduced by Alexandrov.
We give a mathematical foundation for, and numerical demonstration of, the existence of mean curvature 1 surfaces of genus 1 with either two elliptic ends or two hyperbolic ends in de Sitter 3-space. An end of a mean curvature 1 surface is an ``elliptic end'' (resp. a ``hyperbolic end'') if the monodromy matrix at the end is diagonalizable with eigenvalues in the unit circle (resp. in the reals). Although the existence of the surfaces is numerical, the types of ends are mathematically determined.
With the developments of the last decade on complete constant mean curvature 1 (CMC 1) surfaces in the hyperbolic 3-space $H^3$, many examples of such surfaces are now known. However, most of the known examples have regular ends. (An end is irregular, resp. regular, if the hyperbolic Gauss map of the surface has an essential singularity, resp. at most a pole, there.) There are some known surfaces with irregular ends, but they are all either reducible or of infinite total curvature. (The surface is reducible if and only if the monodromy of the secondary Gauss map can be simultaneously diagonali…
We give necessary conditions on complete embedded \cmc surfaces with three or four ends subject to reflection symmetries. The respective submoduli spaces are two-dimensional varieties in the moduli spaces of general \cmc surfaces. We characterize fundamental domains of our \cmc surfaces by associated great circle polygons in the three-sphere.
We extend the study of the vacuum Einstein constraint equations on manifolds with ends of cylindrical type initiated by Chruściel and Mazzeo by finding a class of solutions to the fully coupled system on such manifolds. We show that given a Yamabe positive metric g, which is conformally asymptotically cylindrical on each end, and a 2-tensor K such that (tr K)^2 is bounded below away from zero and asymptotically constant, then we may find an initial data set (g',K') such that g' lies in the conformal class of g.
In a recent paper Jorge and Mercuri proved that the image of Gauss map of a complete non flat minimal surfaces in R3 with finite total curvature omits at most 2 points. In this work we follow their idea and prove 3a similar result for CMC-1 with finite total curvature in H and CMC-1 faces with finite type and regular ends in S31.
CMC-1 trinoids (i.e. constant mean curvature one immersed surface with three regular embedded ends) in hyperbolic 3-space H^3 are irreducible generically, and the irreducible ones have been classified. However, the reducible case has not yet been fully treated, so in this paper we give an explicit description of CMC-1 trinoids in H^3 that includes the reducible case.
This thesis lies in the field of constant mean curvature (cmc) hypersurfaces and specifically cmc $1/2$ surfaces in the three-manifold $\mathbb{H}^2\times\mathbb{R}$. The value $1/2$ is the critical mean curvature for $\mathbb{H}^2\times\mathbb{R}$, in that there do no exist closed cmc surfaces with mean curvature $1/2$ or less. Daniel and Hauswirth have constructed a one-parameter family of complete, cmc $1/2$ annuli that are symmetric about a reflection in the horizontal plane $\mathbb{H}^2 \times \{0\}$, the horizontal catenoids. In this thesis we prove that these catenoids converge to a si…
In this paper, we review results on the existence (and nonexistence) of constant mean curvature spacelike hypersurfaces in the cosmological setting, and discuss the connection to the spacetime splittng problem. It is a pleasure to dedicate this paper to Robert Bartnik, who has made fundamental contributions to this area.
In this paper, we study the space of translational limits T(M) of a surface M properly embedded in R^3 with nonzero constant mean curvature and bounded second fundamental form. There is a natural map T which assigns to any surface M' in T(M), the set T(M') in T(M). Among various dynamics type results we prove that surfaces in minimal T-invariant sets of T(M) are chord-arc. We also show that if M has an infinite number of ends, then there exists a nonempty minimal T-invariant set in T(M) consisting entirely of surfaces with planes of Alexandrov symmetry. Finally, when M has a plane of Alexandro…
We derive necessary conditions on the parameters of the ends of a CMC-1 trinoid in hyperbolic 3-space $H^{3}$ with symmetry plane by passing to its conjugate minimal surface. Together with Daniel's results, this yields a classification of generic symmetric trinoids. We also discuss the relation to other classification results of trinoids by Bobenko et al. and Umehara-Yamada. To obtain the result above, we show that the conjugate minimal surface of a catenoidal CMC-1 end in $H^{3}$ with symmetry plane is asymptotic to a suitable helicoid.
As is well known, constant mean curvature (CMC) spacelike hypersurfaces play an important role in solving the Einstein equations, both in solving the contraints and the evolution equations. In this paper we review the CMC existence result obtained by the authors in [10] and consider some new existence results motivated by a conjecture of Dilts and Holst [8]. We also address some issues concerning the conformal structure of cosmological spacetimes.
We study singularities of spacelike, constant (non-zero) mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space $L^3$. We show how to solve the singular Björling problem for such surfaces, which is stated as follows: given a real analytic null-curve $f_0(x)$, and a real analytic null vector field $v(x)$ parallel to the tangent field of $f_0$, find a conformally parameterized (generalized) CMC $H$ surface in $L^3$ which contains this curve as a singular set and such that the partial derivatives $f_x$ and $f_y$ are given by $\frac{\dd f_0}{\dd x}$ and $v$ along the curve. Within the clas…
We give an infinite dimensional generalized Weierstrass representation for spacelike constant mean curvature (CMC) surfaces in Minkowski 3-space $\real^{2,1}$. The formulation is analogous to that given by Dorfmeister, Pedit and Wu for CMC surfaces in Euclidean space, replacing the group $SU_2$ with $SU_{1,1}$. The non-compactness of the latter group, however, means that the Iwasawa decomposition of the loop group, used to construct the surfaces, is not global. We prove that it is defined on an open dense subset, after doubling the size of the real form $SU_{1,1}$, and prove several results co…
We prove the existence and uniqueness of the Dirichlet problem for spacelike, spherically symmetric, constant mean curvature equation with symmetric boundary data in the extended Schwarzschild spacetime. As an application, we completely solve the CMC foliation conjecture which is posted by Malec and O Murchadha in 2003.
We consider spacetimes with compact Cauchy hypersurfaces and with Ricci tensor bounded from below on the set of timelike unit vectors, and prove that the results known for spacetimes satisfying the timelike convergence condition, namely, foliation by CMC hypersurfaces, are also valid in the present situation, if corresponding further assumptions are satisfied. In addition we show that the volume of any sequence of spacelike hypersurfaces, which run into the future singularity, decays to zero provided there exists a time function covering a future end, such that the level hypersurfaces have non…
We present a global representation for surfaces in 3-dimensional hyperbolic space with constant mean curvature 1 (CMC-1 surfaces) in terms of holomorphic spinors. This is a modification of Bryant's representation. It is used to derive explicit formulas in hypergeometric functions for CMC-1 surfaces of genus 0 with three regular ends which are asymptotic to catenoid cousins (CMC-1 trinoids).
In this article we provide a general construction when $n\ge3$ for immersed in Euclidean $(n+1)$-space, complete, smooth, constant mean curvature hypersurfaces of finite topological type (in short CMC $n$-hypersurfaces). More precisely our construction converts certain graphs in Euclidean $(n+1)$-space to CMC $n$-hypersurfaces with asymptotically Delaunay ends in two steps: First appropriate small perturbations of the given graph have their vertices replaced by round spherical regions and their edges and rays by Delaunay pieces so that a family of initial smooth hypersurfaces is constructed. O…
We construct a new class of complete constant mean curvature surfaces in R^3. These are geometrically different than the surfaces constructed by Kapouleas' gluing technique. These are obtained by piecing together half-Delaunay surfaces to the truncations of minimal k-noids. The gluing techniques are new: the surfaces are obtained by matching Cauchy data from the infinite dimensional family of exact solutions of the problem on each of the component pieces. These surfaces are also proved to be nondegenerate in the CMC moduli space.
Let M = M_{g,k} denote the space of properly (Alexandrov) embedded constant mean curvature (CMC) surfaces of genus g with k (labeled) ends, modulo rigid motions, endowed with the real analytic structure described in [kmp]. Let $P = P_{g,k} = r_{g,k} \times R_+^k$ be the space of parabolic structures over Riemann surfaces of genus g with k (marked) punctures, the real analytic structure coming from the 3g-3+k local complex analytic coordinates on the Riemann moduli space r_{g,k}. Then the parabolic classifying map, Phi: M --> P, which assigns to a CMC surface its induced conformal structure and…
We prove that many complete, noncompact, constant mean curvature (CMC) surfaces $f:Σ\to \R^3$ are nondegenerate; that is, the Jacobi operator $Δ_f + |A_f|^2$ has no $L^2$ kernel. In fact, if $Σ$ has genus zero and $f(Σ)$ is contained in a half-space, then we find an explicit upper bound for the dimension of the $L^2$ jernel in terms of the number of non-cylindrical ends. Our main tool is a conjugation operation on Jacobi fields which linearizes the conjugate cousin construction. Consequences include partial regularity for CMC moduli space, a larger class of CMC surfaces to use in gluing constr…
In this work, complete constant mean curvature 1 (CMC-1) surfaces in hyperbolic 3-space with total absolute curvature at most 4 pi are classified. This classification suggests that the Cohn-Vossen inequality can be sharpened for surfaces with odd numbers of ends, and a proof of this is given.
We prove each embedded, constant mean curvature (CMC) surface in Euclidean space with genus zero and finitely many coplanar ends is nondegenerate: there is no nontrivial square-integrable solution to the Jacobi equation, the linearization of the CMC condition. This implies that the moduli space of such coplanar surfaces is a real-analytic manifold and that a neighborhood of these in the full CMC moduli space is itself a manifold. Nondegeneracy further implies (infinitesimal and local) rigidity in the sense that the asymptotes map is an analytic immersion on these spaces, and also that the copl…
For spacelike stationary (i.e. zero mean curvature) surfaces in 4-dimensional Lorentz space one can naturally introduce two Gauss maps and Weierstrass representation. In this paper we investigate their global geometry systematically. The total Gaussian curvature is related with the surface topology as well as the indices of the so-called good singular ends by a generalized Jorge-Meeks formula. On the other hand, as shown by a family of counter-examples to Osserman's theorem, finite total curvature no longer implies that Gauss maps extend to the ends. Interesting examples include the generaliza…
Until now, the only known maximal surfaces in Minkowski 3-space of finite topology with compact singular set and without branch points were either genus zero or genus one, or came from a correspondence with minimal surfaces in Euclidean 3-space given by the third and fourth authors in a previous paper. In this paper, we discuss singularities and several global properties of maximal surfaces, and give explicit examples of such surfaces of arbitrary genus. When the genus is one, our examples are embedded outside a compact set. Moreover, we deform such examples to CMC-1 faces (mean curvature one …
