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

generic mean curvature flow i generic singularities it has long been conjectured that starting at a generic smooth closed embedded surface in r 3 the mean curvature ow remains smooth until it arrives at a singularity in a neighborhood of which the ow looks like concentric spheres or cylinders that is the only singularities of a generic ow are spherical or cylindrical we will address this conjecture here and in a sequel the higher dimensional case will be addressed elsewhere the key to showing this conjecture is to show that shrinking spheres cylinders and planes are the only stable self shrinkers under the mean curvature ow we prove this here in all dimensions an easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away

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

robustness with respect to sampling for stabilization of riesz spectral systems we suppose that a continuous time feedback is input output stabilizing for an infinite dimensional system we address the question of whether the sampled data controller obtained by applying idealized sample and hold to this continuous time feedback is also input output stabilizing if the sampling time is small enough this question has been previously addressed for fairly general systems under various conditions in this note we restrict our attention to riesz spectral systems for which we generalize the existing results specifically we give two relatively simple conditions which combined are sufficient for the sampled data controller to be stabilizing the first condition is a spectrum decomposition for the open loop system generator which by itself is necessary but not sufficient for the system to be stabilizable by sampled data control the second is a summability condition relating the real part of the spectrum of the generator and the expansion coefficients for the input and feedback operators

recent developments on the stability of systems with aperiodic sampling an overview this article presents basic concepts and recent research directions about the stability of sampled data systems with aperiodic sampling we focus mainly on the stability problem for systems with arbitrary time varying sampling intervals which has been addressed in several areas of research in control theory systems with aperiodic sampling can be seen as time delay systems hybrid systems input output interconnections discrete time systems with time varying parameters etc the goal of the article is to provide a structural overview of the progress made on the stability analysis problem without being exhaustive which would be neither possible nor useful we try to bring together results from diverse communities and present them in a unified manner for each of the existing approaches the basic concepts fundamental results converse stability theorems when available and relations with the other approaches are discussed in detail results concerning extensions of lyapunov and frequency domain methods for systems with aperiodic sampling are recalled as they allow to derive constructive stability conditions furthermore numerical criteria are presented while indicating the sources of conservatism the problems that remain open and the possible directions of improvement at last some emerging research directions such as the design of stabilizing sampling sequences are briefly discussed

sampled data boundary feedback control of 1 d parabolic pdes abstract the paper provides results for the application of boundary feedback control with zero order hold zoh to 1 d linear parabolic systems on bounded domains it is shown that the continuous time boundary feedback applied in a sample and hold fashion guarantees closed loop exponential stability provided that the sampling period is sufficiently small two different continuous time feedback designs are considered the reduced model design and the backstepping design the obtained results provide stability estimates for weighted 2 norms of the state and robustness with respect to perturbations of the sampling schedule is guaranteed

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

compactness of self shrinkers in r3 with fixed genus abstract we prove the compactness of self shrinkers in r 3 with bounded entropy and fixed genus as a corollary we show that numbers of ends of such surfaces are uniformly bounded by the entropy and genus

inequivalent complexity criteria for free boundary minimal surfaces abstract we obtain a series of results in the global theory of free boundary minimal surfaces which in particular provide a rather complete picture for the way different complexity criteria such as area topology and morse index compare beyond the regime where effective estimates are at disposal

some properties of projective modules over discrete hodge algebras all rings in this paper are assumed to be commutative with 1 and all modules finitely generated let b be a discrete hodge algebra over a r ing a by definition this means that b r i where r a xi x and i is an ideal in r generated by monomials t vorst showed in 1 12 that thebehavior of projective modules over such b is very similar to the case of an ordinary polynomial ring in the following sense if all projective modules over any polynomial ring with ground ring a are extended from a then every projective module over a discrete hodge algebra over a is extended from a let b be a discrete hodge algebra over a noetherian ring a and p a projective module over b s mandal proved in 7 that if rank p dim b dim a 1 then

the local picture theorem on the scale of topology we prove a descriptive theorem on the extrinsic geometry of an embedded minimal surface of injectivity radius zero in a homogeneously regular riemannian threemanifold in a certain small intrinsic neighborhood of a point of almost minimal injectivity radius this structure theorem includes a limit object which we call a minimal parking garage structure on r 3 whose theory we also develop

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

sampled data relay control of diffusion pdes we consider a vector reaction advection diffusion equation on a hypercube the measurements are weighted averages of the state over different subdomains these measurements are asynchronously sampled in time subject to matched disturbances the discrete control signals are applied through shape functions and zero order holds the feature of this work is that we consider generalized relay control the control signals take their values in a finite set this allows for networked control through low capacity communication channels first we derive linear matrix inequalities lmis whose feasibility guarantees the ultimate boundedness with a limit bound proportional to the sampling period then we construct a switching procedure for the controller parameters that ensures semi global practical stability for an arbitrarily large domain of initial conditions the trajectories converge to a set whose size does not depend on the domain size for the disturbance free system this procedure guarantees exponential convergence to the origin the results are demonstrated by two examples 2d catalytic slab and a chemical reactor

multi valued graphs in embedded constant mean curvature disks in this paper we prove that an embedded constant mean curvature disk with gaussian curvature large at a point contains a multi valued graph around that point on the scale of a 2 this generalizes colding and minicozzi s result for minimal surfaces

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

embedded minimal disks proper versus nonproper global versus local we construct a sequence of compact embedded minimal disks in a ball in r 3 with boundaries in the boundary of the ball and where the curvatures blow up only at the center the sequence converges to a limit which is not smooth and not proper if instead the sequence of embedded disks had boundaries in a sequence of balls with radii tending to infinity then we have shown previously that any limit must be smooth and proper

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

structure theorems for singular minimal laminations in this paper we apply our local removable singularity theorem and local structure theorems for embedded minimal surfaces and minimal laminations in r 3 proven in 16 15 to obtain global structure theorems for certain possibly singular minimal laminations of r 3 we will use theorems 1 3 and theorem 1 6 below in 14 to prove that a complete embedded minimal surface in r 3 with finite genus and a countable number of ends is proper theorem 1 6 will also be applied in 13 to obtain bounds on the index and the topology of complete embedded minimal surfaces of fixed genus and finite topology in r 3

on the uniqueness of the helicoid and enneper s surface in the lorentz minkowski space mathbb r _ 1 3

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

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

event based control of linear hyperbolic systems of conservation laws in this article we introduce event based boundary controls for 1 dimensional linear hyperbolic systems of conservation laws inspired by event triggered controls developed for finite dimensional systems an extension to the infinite dimensional case by means of lyapunov techniques is studied the main contribution of the paper lies in the definition of two event triggering conditions by which global exponential stability and well posedness of the system under investigation is achieved some numerical simulations are performed for the control of a system describing traffic flow on a roundabout

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

generalized sampled data stabilization of well posed linear infinite dimensional systems we consider well posed linear infinite dimensional systems the outputs of which are sampled in a generalized sense using a suitable weighting function under certain natural assump tions on the system the weighting function and the sampling period we show that there exists a generalized hold function such that unity sampled data feedback renders the closed loop system exponentially stable in the state space sense as well as l 2 stable in the input output sense to illustrate our main result we describe an application to a structurally damped euler bernoulli beam

sampled data boundary feedback control of 1 d linear transport pdes with non local terms abstract the paper provides results for the application of boundary feedback control with zero order hold zoh to 1 d inhomogeneous linear transport partial differential equations on bounded domains with constant velocity and non local terms it is shown that the emulation design based on the recently proposed continuous time boundary feedback designed by means of backstepping guarantees closed loop exponential stability provided that the sampling period is sufficiently small it is also shown that contrary to the parabolic case a smaller sampling period implies a faster convergence rate with no upper bound for the achieved convergence rate the obtained results provide stability estimates for the sup norm of the state and robustness with respect to perturbations of the sampling schedule is guaranteed

stabilization of well posed infinite dimensional systems by dynamic sampled data feedback it is shown that a set of six natural conditions is necessary and sufficient for the existence of a finite dimensional stabilizing sampled data controller for a well posed infinite dimensional system the underlying stability concept for the sampled data system is reminiscent of the notion of input to state stability from nonlinear control theory

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

on automorphisms of modules over polynomial rings write f xfi bxf then f is in xp and f q 0 clearly 1 qf is in el p x and it is a lift of 1 pg in a similar way we can lift 1 pg when g is unimodular this completes the proof of 2 8 the following is a variant of a proposition of lindel l 2 71 see 4 3 for the statement like the proposition of lindel in his paper this proposi tion plays a key roll in the proof of our main theorems 3 1 and 3 2 we shall go into detailed discussions on lindel s proposition l 2 71 in our later sections sects 4 5 6 and now we shall state our proposition 2 9

stability analysis of a text 2 times text 2 linear hyperbolic system with a sampled data controller via backstepping method and looped functionals this paper is concerned with the global exponential stability of a 2 times 2 linear hyperbolic system with a sampled data boundary feedback control designed by means of the backstepping method for a nominal continuous input we show that there exists a sufficiently small intersampling time that encompasses both periodic and aperiodic sampling for which the global exponential stability of the closed loop system is guaranteed in addition we provide easily tractable sufficient stability conditions that can be used to find an upper bound of the maximum intersampling time the results rely on the combination of the lyapunov method and looped functionals the effectiveness of the proposed results is illustrated with a numerical example

brief paper a novel stability analysis of linear systems under asynchronous samplings this article proposes a novel approach to assess the stability of continuous linear systems with sampled data inputs the method which is based on the discrete time lyapunov theorem provides easy tractable stability conditions for the continuous time model sufficient conditions for asymptotic and exponential stability are provided dealing with synchronous and asynchronous samplings and uncertain systems an additional stability analysis is provided for the cases of multiple sampling periods and packet losses several examples show the efficiency of the method