the immersed interface method for axis symmetric problems and application to the hele shaw flow many physical application problems are axis symmetric using axis symmetric properties many three dimensional problems can be solved efficiently using two dimensional axis symmetric coordinates in this paper the immersed interface method iim in axis symmetric coordinates is developed for elliptic interface problems that have a discontinuous coefficient solution or flux a staggered grid is used to overcome the pole singularity other challenges include deriving the jump relations in axis symmetric coordinates designing the numerical algorithm when the interface is close to the pole r 0 computing interface quantities such as the normal and tangential directions surface derivatives curvature information etc the numerical algorithm is based on a finite difference discretization and uniform grid in the axis symmetric coordinates the finite difference scheme is the standard one away from the interface but is modified at grid points near and on the interface the method is shown to be second order accurate in the infinity norm the developed new iim is applied to the hele shaw flow and compared with the results from the literature

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

constant mean curvature surfaces in m 2 times mathbf r the subject of this paper is properly embedded h surfaces in riemannian three manifolds of the form m 2 x r where m 2 is a complete riemannian surface when m 2 r 2 we are in the classical domain of h surfaces in r 3 in general we will make some assumptions about m 2 in order to prove stronger results or to show the effects of curvature bounds in m 2 on the behavior of h surfaces in m 2 x r

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 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

the geometry of minimal surfaces of finite genus ii nonexistence of one limit end examples we demonstrate that a properly embedded minimal surface in 3 with finite genus cannot have one limit end

singularities in hele shaw flows we study analytically and numerically singularities in an interface flow driven by surface tension and a sink in a two dimensional hele shaw cell our analysis proves that singularity formation is inevitable in generic situations our numerical simulations suggest that the hele shaw solution develops singularities via the interface reaching the sink before all the fluid is sucked out a corner is formed right at the tip of the interface that touches the sink

the space of embedded minimal surfaces of fixed genus in a 3 manifold ii multi valued graphs in disks this paper is the second in a series where we give a description of the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed 3 manifold the key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an embedded minimal disk in a ball in r 3 we show here that if the curvature of such a disk becomes large at some point then it contains an almost flat multi valued graph nearby that continues almost all the way to the boundary this will be proved by showing the existence of small multi valued graphs near points of large curvature and then using the extension result for multi valued graphs proved in the first paper in this series there are two local models for embedded minimal disks by an embedded disk we mean a smooth injective map from the closed unit ball in r 2 into r 3 one model is the plane or more generally a minimal graph the other is a piece of a helicoid in the first four papers of this series we will show that every embedded minimal disk is either a graph of a function or is a double spiral staircase like a helicoid recall that a double spiral staircase consists of two spiral staircases that spiral together around a common axis one inside the other this will be done by showing that if the curvature is large at some point and hence the surface is not a graph then it is a double spiral staircase to prove that it is a double spiral staircase we will first prove that it is built out of n valued graphs where n is a fixed number these n valued graphs are like a single spiral staircase connecting n floors the existence of the n valued graphs was initiated in the first paper and will be completed here the third and fourth papers of this series will deal with how the multi valued graphs fit together and in particular prove regularity of the set of points of large curvature the axis of the double spiral staircase

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

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

computation of axisymmetric suction flow through porous media in the presence of surface tension abstract the effect of small surface tension on a class of axisymmetric flows with suction is studied numerically the dynamic evolution of a blob of incompressible viscous fluid surrounded by air and drawn into an interior sink is considered the velocity field of the viscous fluid is assumed to satisfy darcy s law and thus the motion is that of a flow through porous media the fluid interface motion is computed using a highly accurate boundary integral method this method combines recent numerical techniques to achieve efficient and high order space and time discretizations for this type of flow through accurate computations it is shown that in the presence of small surface tension the dynamic behavior of the axisymmetric flows is very similar to that of the two dimensional hele shaw counterparts a long finger develops in the fluid interface and forms a cone singularity when it reaches the sink before all the fluid is sucked out the finger bulges and develops a neck for sufficiently small surface tension the bulge neck formation is enhanced by the additional component of the mean curvature in the three dimensional flow however its effect is not strong enough to cause the interface to pinch off at the neck for the data considered here

embedded minimal ends of finite type we prove that the end of a complete embedded minimal surface in r 3 with infinite total curvature and finite type has an explicit weierstrass representation that only depends on a holomorphic function that vanishes at the puncture reciprocally any choice of such an analytic function gives rise to a properly embedded minimal end e provided that it solves the corresponding period problem furthermore if the flux along the boundary vanishes then the end is c 0 asymptotic to a helicoid we apply these results to proving that any complete embedded one ended minimal surface of finite type and infinite total curvature is asymptotic to a helicoid and we characterize the helicoid as the only simply connected complete embedded minimal surface of finite type in r 3

a cauchy integral approach to hele shaw problems with a free boundary the case of zero surface tension in this paper we study a nonlinear and nonlocal free boundary dynamics the hele shaw problem without surface tension when the fluid domain is either bounded or unbounded the key idea is to use a global quantity the cauchy integral of the free boundary to capture the motion of the boundary this cauchy integral is shown to be linear in time the free boundary at a fixed time is then recovered from its cauchy integral at that time the main tool in our analysis ischerednichenko s theorem concerning the inverse properties of the cauchy integrals as products of our approach we establish the short time existence and uniqueness of classical solutions for analytic initial boundaries we also show the non existence of classical solutions for all smooth but non analytic initial boundaries when there is a sink at either a finite point or at infinity when the fluid domain is bounded all solutions except the circular one break down before all the fluid is sucked out from the sink regularity results are also obtained when there is a source at a finite point or at infinity

the space of embedded minimal surfaces of fixed genus in a 3 manifold i estimates off the axis for disks this paper is the first in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed riemannian 3 manifold the key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an embedded minimal disk in a ball in r3 with the flat metric this study is undertaken here and completed in cm6 these local results are then applied in cm7 where we describe the general structure of fixed genus surfaces in 3 manifolds there are two local models for embedded minimal disks by an embedded disk we mean a smooth injective map from the closed unit ball in r2 into r3 one model is the plane or more generally a minimal graph the other is a piece of a helicoid in the first four papers of this series we will show that every embedded minimal disk is either a graph of a function or is a double spiral staircase where each staircase is a multi valued graph this will be done by showing that if the curvature is large at some point and hence the surface is not a graph then it is a double spiral staircase to prove that such a disk

singularities in hele shaw flows driven by a multipole we study analytically and numerically singularity formation in an interface flow driven by a multipole for a two dimensional hele shaw cell with surface tension our analysis proves that singularity formation is inevitable in the case of a dipole for a multipole of a higher order we show that the solution does not tend to any stationary solution as time goes to infinity if its initial center of mass is not at the multipole it is therefore very likely that this solution will develop finite time singularities our extensive numerical studies suggest that a solution develops finite time singularities via the interface reaching the multipole while forming a corner at the tip of the finger that touches the multipole in addition it is observed that the interface approaches the multipole from directions which can be predicted beforehand we also estimate as a function of time the distance between the finger tip and multipole and the results are in excellent agreement with numerical computations

singular perturbation of quantum stochastic differential equations with coupling through an oscillator mode we consider a physical system which is coupled indirectly to a markovian resevoir through an oscillator mode this is the case for example in the usual model of an atomic sample in a leaky optical cavity which is ubiquitous in quantum optics in the strong coupling limit the oscillator can be eliminated entirely from the model leaving an effective direct coupling between the system an the resevoir here we provide a mathematically rigorous treatment of this limit as a weak limit of the time evolution and observables on a suitably chosen exponential domain in fock space the resulting effective model may contain emission and absorption as well as scattering interactions

the geometry of minimal surfaces of finite genus i curvature estimates and quasiperiodicity let mathcal m be the space of properly embedded minimal surfaces in 3 with genus zero and two limit ends normalized so that every surface m mathcal m has horizontal limit tangent plane at infinity and the vertical component of its flux equals one we prove that if a sequence m i i mathcal m has the horizontal part of the flux bounded from above then the gaussian curvature of the sequence is uniformly bounded this curvature estimate yields compactness results and the techniques in its proof lead to a number of consequences concerning the geometry of any properly embedded minimal surface in 3 with finite genus and the possible limits through a blowing up process on the scale of curvature of a sequence of properly embedded minimal surfaces with locally bounded genus in a homogeneously regular riemannian 3 manifold

the space of embedded minimal surfaces of fixed genus in a 3 manifold iv locally simply connected this paper is the fourth in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed 3manifold the key is to understand the structure of an embedded minimal disk in a ball in r3 this was undertaken in cm3 cm4 and the global version of it will be completed here see the discussion around figure 12 for the local case and cm15 for some more details our main results are theorem 0 1 the lamination theorem and theorem 0 2 the one sided curvature estimate the lamination theorem is stated in the global case where the lamination is in fact a foliation the first four papers of this series show that every embedded minimal disk is either a graph of a function or is a double spiral staircase where each staircase is a multivalued graph this is done by showing that if the curvature is large at some point and hence the surface is not a graph then it is a double spiral staircase like the helicoid to prove that such a disk is a double spiral staircase we showed in the first three papers of the series that it is built out of n valued graphs where n is a fixed number in this paper we will deal with how the multi valued graphs fit together and in particular prove regularity of the set of points of large curvature the axis of the double spiral staircase the first theorem is the global version of the statement that every embedded minimal disk is a double spiral staircase