classification of rotational special weingarten surfaces of minimal type in mathbb s 2 times mathbb r and mathbb h 2 times mathbb r in this paper we classify the complete rotational special weingarten surfaces in mathbb s 2 times mathbb r and mathbb h 2 times mathbb r i e rotational surfaces in mathbb s 2 times mathbb r and mathbb h 2 times mathbb r whose mean curvature h and extrinsic curvature ke satisfy h f h2 ke for some function f in mathcal c 1 0 infty such that f 0 0 and 4x f x 2 1 for any x 0 furthermore we show the existence of non complete examples of such surfaces

adding handles to the helicoid there exist two new embedded minimal surfaces asymptotic to the helicoid one is periodic with quotient by orientation preserving translations of genus one the other is nonperiodic of genus one

properly embedded minimal planar domains in 1997 collin proved that any properly embedded minimal surface in r 3 with nite topology and more than one end has

on the topology and index of minimal surfaces we show that for an immersed two sided minimal surface in r 3 there is a lower bound on the index depending on the genus and number of ends using this we show the nonexistence of an embedded minimal surface in r 3 of index 2 as conjectured by choe cho90 moreover we show that the index of an immersed two sided minimal surface with embedded ends is bounded from above and below by a linear function of the total curvature of the surface

index one minimal surfaces in spherical space forms abstract we prove that orientable index one minimal surfaces in spherical space forms with large fundamental group have genus at most two this confirms a conjecture of r schoen for an infinite class of 3 manifolds

on the growth of supersolutions of nonlinear pde s on exterior domains abstract we obtain a comparison principle on annuli with catenoid like functions for supersolutions of non linear elliptic pdes over exterior domains in a non positively curved manifold with a pole this result is applied to get an upper estimate on the growth of such supersolutions and in particular of exterior graphs of non negative mean curvature

the calabi yau conjectures for embedded surfaces in this talk i will discuss the proof of the calabi yau conjectures for embedded surfaces this is joint work with bill minicozzi cm9 the calabi yau conjectures about surfaces date back to the 1960s much work has been done on them over the past four decades in particular examples of jorge xavier from 1980 and nadirashvili from 1996 showed that the immersed versions were false we will show here that for embedded surfaces i e injective immersions they are in fact true their original form was given in 1965 in ca where e calabi made the following two conjectures about minimal surfaces see also s s chern page 212 of ch conjecture 1 prove that a complete minimal hypersurface in r must be unbounded calabi continued it is known that there are no compact minimal submanifolds of r or of any simply connected complete riemannian manifold with sectional curvature 0 a more ambitious conjecture is conjecture 2 a complete minimal hypersurface in r has an unbounded projection in every n 2 dimensional flat subspace

the structure of singly periodic minimal surfaces the only properly embedded minimal surface with more than one topological end and a cont inuous g roup of symmetries is the catenoid considering periodic minimal surfaces with more than one end a surface in n 3 is periodic if it is invariant under an infinite discrete g roup of isometrics that act freely on 1r3 the only classical examples are those discovered by riemann the riemann examples possess a quite special set of properties 1 they have an infinite number of flat annular ends 1 2 they are invariant under a nontrivial t ranslat ion t 3 the surfaces t h a v e genus 1 and 2 ends 4 the surfaces t have total curvature equal to 8 zr mot iva ted by the riemann examples we established the existence of an infinite family of proper ly embedded periodic minimal surfaces d k each with an infinite number of ends 1 these surfaces which are discussed below see sect 1 2 on rigidity also have an infinite number of flat annular ends and are invariant under a t ranslat ion t the quot ient of these surfaces by t is a surface of genus 2 k 1 with two ends and total curvature 8 r e k 1 k l we have been able to twis t each of these new surfaces that is we construct surfaces with the same properties that are invariant under screw motions t r where r is a rota t ion a round an axis in the direction of t see sect 1 3 below together with the catenoid and the examples of riemann these new surfaces comprise all the known properly embedded minimal surfaces with more than one topological end whose symmetry g roup contains an infinite discrete subgroup these examples share m a n y of the properties of the riemann examples a major goal of this paper is to show that this is not accidental

complete embedded minimal surfaces of finite total curvature with planar ends of smallest possible order using a method by traizet j differ geom 60 103 153 2002 which reduces the construction of minimal surfaces via the weierstras theorem and the implicit function theorem to solving algebraic equations in several complex variables we will show the existence of complete embedded minimal surfaces of finite total curvature with planar ends of least possible order

the spaces of index one minimal surfaces and stable constant mean curvature surfaces embedded in flat three manifolds it is proved that the spaces of index one minimal surfaces and stable constant mean curvature surfaces with genus greater than one in non fixed flat three manifolds are compact in a strong sense given a sequence of any of the above surfaces we can extract a convergent subsequence of both the surfaces and the ambient manifolds in the ck topology these limits preserve the topological type of the surfaces and the affine diffeomorphism class of the ambient manifolds some applications to the isoperimetric problem are given

the space of embedded minimal surfaces of fixed genus in a 3 manifold v fixed genus 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 itnonsimply 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 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 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 two parameter family of minimal surfaces known as the riemann examples

constant mean curvature graphs on exterior domains of the hyperbolic plane we prove an existence result for non rotational constant mean curvature ends in mathbb h 2 times mathbb r where mathbb h 2 is the hyperbolic real plane the value of the curvature is h in big 0 frac 1 2 big we use schauder theory and a continuity method for solutions of the prescribed mean curvature equation on exterior domains of mathbb h 2 we also prove a fine property of the asymptotic behavior of the rotational ends introduced by sa earp and toubiana

complete embedded minimal surfaces of finite total curvature we will survey what is known about minimal surfaces s r 3 which are complete embedded and have finite total curvature int s k da infty the only classically known examples of such surfaces were the plane and the catenoid the discovery by costa 14 15 early in the last decade of a new example that proved to be embedded sparked a great deal of research in this area many new examples have been found even families of them as will be described below the central question has been transformed from whether or not there are any examples except surfaces of rotation to one of understanding the structure of the space of examples

probability representations of a class of two way diffusions there has been little progress in the analysis of two way diffusion in the last few decades due to the difficulties brought by the interface section similar to a free boundary condition in this paper however the equivalent probability model is considered and the interface section is precisely described by an integral equation the solution of two way diffusion is then expressed in an integral form with the integrand being the solution of a classical first passage time model and the solution of a one dimensional integral equation which is relatively easier to solve the exact expression of the two way diffusion enables us to find the explicit solution of the model with infinite horizontal boundaries and without drifting

stationary isothermic surfaces and uniformly dense domains we establish a relationship between stationary isothermic surfaces and uniformly dense domains a stationary isothermic surface is a level surface of temperature which does not evolve with time a domain in the n dimensional euclidean space r n is said to be uniformly dense in a surface r n of codimension 1 if for every small r 0 the volume of the intersection of with a ball of radius r and center x does not depend on x for x we prove that the boundary of every uniformly dense domain which is bounded or whose complement is bounded must be a sphere we then examine a uniformly dense domain with unbounded boundary and we show that the principal curvatures of satisfy certain identities the case in which the surface coincides with is particularly interesting in fact we show that if the boundary of a uniformly dense domain is connected then i if n 2 it must be either a circle or a straight line and ii if n 3 it must be either a sphere a spherical cylinder or a minimal surface we conclude with a discussion on uniformly dense domains whose boundary is a minimal surface

unstable minimal surfaces and heegaard splittings in recent years minimal surfaces have played an important role in the study of three dimensional manifolds least area surfaces which minimize area in their isotopy 15 homotopy 23 or homology 10 classes have been used to make progress towards the understanding of three manifolds which contain such surfaces such surfaces can be shown to exist and to have strong regularity properties under the assumption that the homotopy isotopy or homology class in which they minimize area is non trivial however these techniques tend not to be applicable to the study of surfaces which are not essential in one of the above classes or to the study of manifolds which contain only such nonessential surfaces recently developed techniques open the way for the use of a new class of surfaces in three dimensional manifolds namely the class of unstable embedded minimal surfaces pitts and rubinstein have established the existence of unstable minimal submanifolds which arise from a heegaard splitting in such a way that their genus is less than or equal to that of the heegaard splitting 19 in this paper we extend their result to obtain greater topological control of the resulting minimal surface and apply the resulting theorem to purely topological problems relating to heegaard splittings of a three manifold a compact manifold is a handlebody if it is homeomorphic to a regular neighborhood of a graph in r 3 a heegaard splitting of a three manifold m is a triple f h1 h2 where f is an embedded connected orientable surface in m h1 and h 2 are handlebodies embedded in m m h 1 w h 2 and h c h2 f o h i o h 2 f is called a heegaard surface we say that two heegaard splittings of m f h1 h2 and f h i h are equivalent if there is a homeomorphism h m m such that f f f f h o h i and f h2 h by considering neighborhoods of triangulations one can show that any closed orientable three manifold admits a heegaard splitting of particular interest are the splittings with heegaard surface of smallest possible genus the genus of such a splitting is called the heegaard genus of the three manifold

processes of flats induced by higher dimensional processes stationary processes of k flats in mathbb e d can be thought of as point processes on the grassmannian mathcal l kd of k dimensional subspaces of mathbb e d if such a process is sampled by a d k j dimensional space f it induces a process of j flats in f in this work we will investigate the possibility of determining the original k process from knowledge of the intensity measures of the induced j processes we will see that this is impossible precisely when 1 k d 1 and j 0 2 r 2 1 where r is the rank of the manifold mathcal l kd we will show how the problem is equivalent to the study of the kernel of various integral transforms these will then be investigated using harmonic analysis on grassmannian manifolds

skew flat fibrations a fibration of mathbb r n by oriented copies of mathbb r p is called skew if no two fibers intersect nor contain parallel directions conditions on p and n for the existence of such a fibration were given by ovsienko and tabachnikov a classification of smooth fibrations of mathbb r 3 by skew oriented lines was given by salvai in analogue with the classification of oriented great circle fibrations of s 3 by gluck and warner we show that salvai s classification has a topological variation which generalizes to characterize all continuous fibrations of mathbb r n by skew oriented copies of mathbb r p we show that the space of fibrations of mathbb r 3 by skew oriented lines deformation retracts to the subspace of hopf fibration and therefore has the homotopy type of a pair of disjoint copies of s 2 we discuss skew fibrations in the complex and quaternionic setting and give a necessary condition for the existence of a fibration of mathbb c n respectively mathbb h n by skew oriented copies of mathbb c p respectively mathbb h p

a rigidity theorem for properly embedded minimal surfaces in bf r sp 3 we consider the question of when an intrinsic isometry of a properly embedded minimal surface is induced by an ambient isometry we prove it always extends when the surface has at least two ends there are several interesting theorems and conjectures on the rigidity of complete surfaces in r 3 which satisfy some natural geometric constraint perhaps the best known theorem of this type is the cohn vossen theorem which shows that a compact riemannian surface of positive curvature has a unique immersion into r a similar result is conjectured for complete surfaces of nonpositive or nonnegative curvature with the additional hypothesis that the surface has finite area it is also conjectured that tight surfaces are rigid in this paper we prove a rigidity theorem for properly embedded minimal surfaces in r 3 which have more than one end this theorem states that the inclusion map of the surface into r3 represents the unique isometric minimal immersion of such a surface up to a rigid motion of r 3 in particular it follows from our theorem that every intrinsic isometry of this type of surface extends to an isometry of r 3 in 1 we derive a geometric condition on an immersed minimal surface which guarantees that the surface is minimally rigid we prove that if the surface intersects a plane transversally along an immersed closed curve then any other isometric minimal immersion of the surface into r differs from the original immersion by an ambient isometry in 2 we obtain some asymptotic properties of solutions of the minimal surface equation over annular planar domains in 3 we prove that if a properly embedded minimal surface m in r3 has more than one end then m is transverse

the strong halfspace theorem for minimal surfaces d hoffman and w h meeks iii department of mathematics university of massachusetts amherst ma 01003 usa in 6 jorge and xavier constructed examples of complete nonplanar minimal surfaces contained between two parallel planes in 3 recently rosenberg and toubiana have found complete minimally immersed annuli that are proper in an open slab 1 12 on the basis of these and other results the question has arisen as to whether or not there could exist a properly immersed minimal surface in 3 that was contained between two parallel planes the answer is negative an immediate consequence of the following theorem

the classical theory of minimal surfaces we present here a survey of recent spectacular successes in classical minimal surface theory we highlight this article with the theorem that the plane the helicoid the catenoid and the one parameter family frtgt2 0 1 of riemann minimal examples are the only complete properly embedded minimal planar domains in r 3 the proof of this result depends primarily on work of colding

global smooth geodesic foliations of the hyperbolic space we consider foliations of the whole three dimensional hyperbolic space mathbb h 3 by oriented geodesics let mathcal l be the space of all the oriented geodesics of mathbb h 3 which is a four dimensional manifold carrying two canonical pseudo riemannian metrics of signature left 2 2 right we characterize in terms of these geometries of mathcal l the subsets mathcal m in mathcal l that determine foliations of mathbb h 3 we describe in a similar way some distinguished types of geodesic foliations of mathbb h 3 regarding to which extent they are in some sense trivial in some directions on the one hand foliations whose leaves do not lie in a totally geodesic surface not even at the infinitesimal level on the other hand those for which the forward and backward gauss maps varphi pm mathcal m rightarrow mathbb h 3 left infty right are local diffeomorphisms besides we prove that for this kind of foliations varphi pm are global diffeomorphisms onto their images the subject of this article is within the framework of foliations by congruent submanifolds and follows the spirit of the paper by gluck and warner where they understand the infinite dimensional manifold of all the great circle foliations of the three sphere

uncertainty quantification by geometric characterization of sensitivity spaces we propose a systematic procedure for both aleatory and epistemic uncertainty quantification of numerical simulations through geometric characteristics of global sensitivity spaces two mathematical concepts are used to characterize the geometry of these spaces and to identify possible impacts of variability in data or changes in the models or solution procedures the dimension of the maximal free generator subspace in vector spaces and the principal angles between subspaces we show how these characters can be used as indications on the aleatory and epistemic uncertainties in the case of large dimensional parameter spaces these characterizations are established for quantile based extreme scenarios and a multi point moment based sensitivity direction permits to propose a directional uncertainty quantification concept for directional extreme scenarios des the approach is non intrusive and exploits in parallel the elements of existing mono point gradient based design platforms the ingredients of the paper are illustrated on a model problem with the burgers equation with control and on a constrained aerodynamic performance analysis problem

minimal hypersurfaces with bounded index we prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index and volume if the ambient dimension is greater than three in a riemannian manifold m n g 3 le n le 7 can degenerate loosely speaking our results show that embedded minimal hypersurfaces with bounded index behave qualitatively like embedded stable minimal hypersurfaces up to controlled errors several compactness finiteness theorems follow from our local picture

heegaard splittings the virtually haken conjecture and property τ the behaviour of finite covers of 3 manifolds is a very important but poorly understood topic there are three increasingly strong conjectures in the field that have remained open for over twenty years the virtually haken conjecture the positive virtual b1 conjecture and the virtually fibred conjecture any of these would have profound ramifications for 3 manifold theory in this paper we explore the interaction of these conjectures with the following seemingly unrelated areas eigenvalues of the laplacian and heegaard splittings

the existence of embedded minimal surfaces and the problem of uniqueness it was in the early nineteen thirties that douglas and radd solved the plateau problem for a rectifiable jordan curve in euclidean space the minimal surfaces that they obtained are realized by conformal harmonic maps from the unit disk in 1948 morrey generalized the theorem of douglas and radd to minimal surfaces in a homogeneously regular riemannian manifold in both cases the solutions are defective as they may possess branch points it was not until 1968 that osserman 24 was able to prove that interior branch points do not exist on the douglas radd morrey solution osserman s theorem in this full generality was proved by alt 2 3 and gulliver 8 if the jordan curve is real analytic gulliver and lesley 9 showed that the solution surface is free of branch points even on its boundary hence at least in this case we know that the douglas radd morrey solution is an immersion and represents a classical surface however the douglas radd morrey solution may in general have selfintersections it was then a classical problem to find conditions on the jordan curve to guarantee that the solution is an embedded disk in 1932 radd was the first to produce a general condition of this type he proved that if the curve can be projected onto a convex curve in the plane then it bounds a unique embedded minimal disk which is a graph over this plane it was not until the seventies that there was further progress on the problem of embeddedness osserman had conjectured that if a jordan curve is extreme that is lies on the boundary of its convex hull then the douglas rad6 solution is embedded gulliver and spruck 10 proved this conjecture under the additional assumption that the absolute total curvature of the jordan curve is less than 4 z right after this work tomi and tromba 32 showed that an extreme curve must bound an embedded minimal disk which need not be stable in 1976 almgren and simon 1 established the existence for such a curve of an embedded stable minimal disk which need not be a douglas radd solution finally in the same year the authors 18 proved that any douglas rad6 solution must be embedded

minimal surfaces with the area growth of two planes the case of infinite symmetry we prove that a connected properly immersed minimal surface in with infinite symmetry group and area growth constant less than is a plane a catenoid or a scherk singly periodic minimal surface as a consequence the scherk minimal surfaces are the only connected periodic minimal desingularizations of the intersections of two planes

triply periodic constant mean curvature surfaces abstract given a closed flat 3 torus n for each h 0 and each non negative integer g we obtain area estimates for closed surfaces with genus g and constant mean curvature h embedded in n this result contrasts with the theorem of traizet 31 who proved that every flat 3 torus admits for every positive integer g with g 2 connected closed embedded minimal surfaces of genus g with arbitrarily large area

the asymptotic behavior of properly embedded minimal surfaces of finite topology while advances have been made in recent years in the study of properly embedded minimal surfaces of finite topology in r3 3 5 8 9 11 12 13 23 progress has depended in an essential manner on the special structure of such surfaces with finite total curvature a properly immersed minimal surface with finite total curvature is conformally a compact riemann surface punctured in a finite number of points and its gauss map extends to the compact surface as a meromorphic function 22 in particular all the topological ends are conformally equivalent to a punctured disk and there is a well defined limit tangent plane at each end outside of a sufficiently large compact set such an end is a multisheeted graph over the limit tangent plane and if the end is embedded it is asymptotic to either a plane or a half catenoid see e g schoen 24 for a survey of results on properly immersed surfaces of finite total curvature see 9 1 without the assumption of finite total curvature these results are not true in general as an indication of the relative lack of information consider that it is unknown whether or not the helicoid is the only simply connected minimal surface that is properly embedded and nonplanar a question raised by osserman the helicoid has infinite total curvature and a single annular end if a surface has finite topology then all of its ends are annular any general theory of properly embedded minimal surfaces of finite topology must include an understanding of the behavior of these ends in this paper we prove

the blaschke conjecture and great circle fibrations of spheres we construct an explicit diffeomorphism taking any fibration of a sphere by great circles into the hopf fibration we use elementary differential geometry and no surgery or k theory to carry out the construction indeed the diffeomorphism is a local differential invariant algebraic in derivatives this result is new only for 5 dimensional spheres but our new method of proof is elementary

a schoen theorem for minimal surfaces in h 2 r abstract in this paper we prove that a complete minimal surface immersed in h 2 r with finite total curvature and two ends each one asymptotic to a vertical geodesic plane must be a horizontal catenoid moreover we give a geometric description of minimal ends of finite total curvature in h 2 r we also prove that a minimal complete end e with finite total curvature is properly immersed and that the gaussian curvature of e is locally bounded in terms of the geodesic distance to its boundary

half range analysis of a counter current separator abstract chemical separations are usually accomplished in counter current reactors because in these reactors the chemical equilibrium continually shifts toward increasing the purity of the product stream as it flows through the reactor however as a result of the counter current design these devices are often governed by forward backward two way diffusion equations in this article we analyze the mathematical model of a single interface counter current separator introduced by fitt ockendon and shillor for relatively long reactors for such reactors the mathematical model which is a forward backward diffusion system can be solved by eigenfunction expansions we construct these expansions using complex variable methods developed for kinetic theory from this analysis we obtain explicit formulas for the power of the reactor both in terms of the average purity of the output stream and in terms of the equivalent number of theoretical plates

branch structure of j holomorphic curves near periodic orbits of a contact manifold let m be a three dimensional contact manifold and d 0 m x r a finite energy pseudoholomorphic map from the punctured disc in c that is asymptotic to a periodic orbit of the contact form this article examines conditions under which smooth coordinates may be defined in a tubular neighbourhood of the orbit such that resembles a holomorphic curve invoking comparison with the theory of topological linking of plane complex algebroid curves near a singular point examples of this behaviour which are studied in some detail include pseudoholomorphic maps into e p q r where e p q denotes a rational ellipsoid contact structure induced by the standard complex structure on c 2 as well as contact structures arising from non standard circle fibrations of the three sphere

qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area we prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed riemannian manifolds in terms of their index and area restricting to the case where the hypersurface has dimension less than seven in particular we prove that if we are given a sequence of closed minimal hypersurfaces of bounded area and index the total curvature along the sequence is quantised in terms of the total curvature of some limit surface plus a sum of total curvatures of complete properly embedded minimal hypersurfaces in euclidean space all of which are finite thus we obtain qualitative control on the topology of minimal hypersurfaces in terms of index and area as a corollary

partial range completeness and existence of solutions to two way diffusion equations separating variables to solve a two way diffusion equation leads to a nonstandard eigenvalue problem in one variable it is shown that the eigenfunctions having negative eigenvalues are complete on the part of the domain where initial conditions are imposed while those with positive eigenvalues are complete where final conditions are imposed the corresponding exponentially growing and exponentially decaying solutions may be used to expand arbitrary solutions on semi infinite intervals in the time variable a natural iterative procedure for obtaining solutions on finite intervals is shown to converge in some cases a linearly growing solution must also be taken into account

the space of embedded minimal surfaces of fixed genus in a 3 manifold iii planar domains this paper is the third in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed 3 manifold in cm3 cm5 we describe the case where the surfaces are topologically disks on any fixed small scale although the focus of this paper general planar domains is more in line with cm6 we will prove a result here namely corollary iii 3 5 below which is needed in cm5 even for the case of disks roughly speaking there are two main themes in this paper the first is that stability leads to improved curvature estimates this allows us to find large graphical regions these graphical regions lead to two possibilities

ensemble kalman filters and geometric characterization of sensitivity spaces for uncertainty quantification in optimization abstract we present an original framework for uncertainty quantification uq in optimization it is based on a cascade of ingredients with growing computational complexity for both forward and reverse uncertainty propagation the approach is merely geometric it starts with a complexity based splitting of the independent variables and the definition of a parametric optimization problem geometric characterization of global sensitivity spaces through their dimensions and relative positions by the principal angles between global search subspaces bring a first set of information on the impact of uncertainties on the functioning parameters on the optimal solution joining the multi point descent direction and the quantiles on the optimization parameters permits to define the notion of directional extreme scenarios des without sampling of large dimension design spaces one goes beyond des with ensemble kalman filters enkf after the multi point optimization algorithm is cast into an ensemble simulation environment this formulation accounts for the variability in large dimension the uq cascade ends with the joint application of the enkf and des leading to the concept of ensemble directional extreme scenarios edes which provides more exhaustive possible extreme scenarios knowing the probability density function of our optimization parameters a final interest of the approach is that it provides an indication of the size of the ensemble which must be considered in the enkf these ingredients are illustrated on an history matching problem

some uniqueness and nonexistence theorems for embedded minimal surfaces thanks to the works of callahan costa hoffman karcher meeks rosenberg wei etc see for example 1 2 3 5 8 11 12 16 20 we dispose now of a large number of properly embedded minimal surfaces in the euclidean space ir other than the classical examples all those surfaces can be viewed as minimal surfaces with finite total curvature properly embedded in complete flat three manifolds the most basic invariants associated to a surface of this type are its topology and its periodicity it is an interesting problem to decide if the simplest examples like the catenoid the helicoid the scherk s surfaces or the riemann example can be characterized in terms of some of these invariants in the non periodic case there are two important uniqueness theorems in this direction the first one obtained by schoen 17 characterizes the catenoid as the only complete minimal surface embedded

the space of complete embedded maximal surfaces with isolated singularities in the 3 dimensional lorentz minkowski space we prove that a complete embedded maximal surface in 3 dx12 dx22 dx32 with a finite number of singularities is an entire maximal graph with conelike singularities over any spacelike plane and so it is asymptotic to a spacelike plane or a half catenoid we show that the moduli space of entire maximal graphs over x3 0 in with n 1 2 singular points and vertical limit normal vector at infinity is a 3n 4 dimensional differentiable manifold the convergence in means the one of conformal structures and weierstrass data and it is equivalent to the uniform convergence of graphs on compact subsets of x3 0 moreover the position of the singular points in 3 and the logarithmic growth at infinity can be used as global analytical coordinates with the same underlying topology we also introduce the moduli space of marked graphs with n 1 singular points a mark in a graph is an ordering of its singularities which is a n 1 sheeted covering of we prove that identifying marked graphs differing by translations rotations about a vertical axis homotheties or symmetries about a horizontal plane the corresponding quotient space is an analytic manifold of dimension 3n 1 this manifold can be identified with a spinorial bundle associated to the moduli space of weierstrass data of graphs in

the classification of complete minimal surfaces with total curvature greater than 12 we classify complete orientable minimal surfaces with finite total curvature 87r introduction the classification of complete minimal surfaces with finite total curvature in r3 has been an important problem in the classical differential geometry some basic properties of these surfaces were studied by r osserman see 8 9 who showed the first nontrivial result about this subject concretely he characterized the catenoid and enneper surface as the unique complete orientable minimal surfaces of total curvature 47 however until recent years no more relevant results have been obtained w h meeks 6 gave the classification of nonorientable complete minimal surfaces with total curvature greater than 87 this paper is concerned with the total classification of orientable complete minimal surfaces with total curvature 87 chen and gackstatter 1 discovered the first example of a complete minimal surface properly of genus 1 see theorem 1 the picture of chen gackstatter surface is obtained by joining a handle on enneper s surface this genus one minimal surface has total curvature 87 and no other examples of such surfaces were found so it is expected that no other genus one orientable minimal surface of total curvature 87 does exist in this paper we give a proof of this fact more precisely we prove that chen gackstatter surface is the only genus one orientable complete minimal surface with finite total curvature 87 of course it is not difficult to find genus zero minimal surfaces with total curvature 8i a geometrically interesting example described by jorge and meeks 5 is the trinoid this surface has three embedded catenoid ends moreover its normal vectors at these ends are placed symmetrically in an equator of s2 received by the editors april 27 1990 and in revised form july 20 1990 1980 mathematics subject classification 1985 revision primary 53a10 research partially supported by dgicyt grant ps87 01 15 c03 02

least area incompressible surfaces in 3 manifolds let m be a riemannian manifold and let f be a closed surface a map f f m is called least area if the area of f is less than the area of any homotopic map from f to m note that least area maps are always minimal surfaces but that in general minimal surfaces are not least area as they represent only local stationary points for the area function the existence of least area immersions in a homotopy class of maps has been established when the homotopy class satisfies certain injectivity conditions on the fundamental group 18 17 in this paper we shall consider the possible singularities of such immersions our results show that the general philosophy is that least area surfaces intersect least meaning that the intersections and self intersections of least area immersions are as small as their homotopy classes allow when measured correctly one should note that evidence supporting this view had been found by meeks yau in their embedding theorems for minimal disks and 2 spheres 13 143 our main result asserts that if a least area immersion is homotopic to an embedding then it has no self intersections which clearly exemplifies the above philosophy the precise result is the following

the embedded singly periodic scherk costa surfaces we give a positive answer to m traizet s open question about the existence of complete embedded minimal surfaces with scherk ends without planar geodesics in the singly periodic case these examples get close to an extension of traizet s result concerning asymmetric complete minimal submanifolds of open image in new window with finite total curvature

constant mean curvature surfaces with boundary on a sphere in this article we study the shape of a compact surface of constant mean curvature of euclidean space whose boundary is contained in a round sphere we consider the case that the boundary is prescribed or that the surface meets the sphere with a constant angle we study under what geometric conditions the surface must be spherical our results apply in many scenarios in physics where in absence of gravity a liquid drop is deposited on a round solid ball and the air liquid interface is a critical point for area under all variations that preserve the enclosed volume

stationary isothermic surfaces in euclidean 3 space let omega be a domain in mathbb r 3 with partial omega partial mathbb r 3 setminus overline omega where partial omega is unbounded and connected and let u be the solution of the cauchy problem for the heat equation partial t u delta u over mathbb r 3 where the initial data is the characteristic function of the set omega c mathbb r 3 setminus omega we show that if there exists a stationary isothermic surface gamma of u with gamma cap partial omega varnothing then both partial omega and gamma must be either parallel planes or co axial circular cylinders this theorem completes the classification of stationary isothermic surfaces in the case that gamma cap partial omega varnothing and partial omega is unbounded to prove this result we establish a similar theorem for uniformly dense domains in mathbb r 3 a notion that was introduced by magnanini et al trans am math soc 358 4821 4841 2006 in the proof we use methods from the theory of surfaces with constant mean curvature combined with a careful analysis of certain asymptotic expansions and a surprising connection with the theory of transnormal functions