the exterior algebra λkrn and area minimization abstract the structure of the exterior algebra k r n is studied in low dimensions and consequences are drawn for k dimensional area minimizing surfaces in r n for a general form o 2 r 4 section 3 gives explicit formulas for the comass o and the face of the grassmannian exposed by o section 4 classifies the faces of the grassmannian of the 3 planes in r 6 and hence the associated geometries of area minimizing surfaces there are four types section 5 establishes an equality involving the comass norm in low dimension and draws implications on when the cartesian product of area minimizing surfaces is area minimizing new examples of area minimizing integral currents with singularities follow

on hamiltonian stationary lagrangian spheres in non einstein kahler surfaces hamiltonian stationary lagrangian spheres in kahler einstein surfaces are minimal we prove that in the family of non einstein kahler surfaces given by the product 1 2 of two complete orientable riemannian surfaces of different constant gauss curvatures there is only a non minimal hamiltonian stationary lagrangian sphere this example defined when the surfaces 1 and 2 are spheres is unstable

efficient estimation of class a noise parameters via the em algorithm the class a middleton noise model is a commonly used statistical physical parametric model for non gaussian interference superimposed on a gaussian background in this study the problem of efficient estimation of the class a parameters for small sample sizes is considered the proposed estimator is based on the em algorithm a two step iterative estimation technique that is ideally suited for the class a estimation problem since the observations can be readily treated as incomplete data for the single parameter estimation problem a closed form expression for the estimator is obtained furthermore for the single parameter estimation problem it is shown that the sequence of estimates obtained via the em algorithm converges and a characterization of the point to which the sequence converges is given in particular it is shown that if the limit point of this convergent sequence is an interior point of the parameter set of interest then it must be a stationary point of the traditional likelihood function in addition for both the single parameter and two parameter estimation problems the small sample size performance of the proposed em algorithm is examined via an extensive simulation study

implicit lyapunov functions and isochrones of linear systems lyapunov s direct method is based on a lyapunov function that explicitly depends on the system s state vector this article describes implicit lyapunov functions and corresponding stability theorems that are counterparts of the explicit case we also define isochrones of linear systems and describe their relationship to implicit lyapunov functions determining settling times is considered as an application

minimal surfaces in finite volume non compact hyperbolic 3 manifolds we prove there exists a compact embedded minimal surface in a complete finite volume hyperbolic 3 manifold mathcal n we also obtain a least area incompressible properly embedded finite topology 2 sided surface we prove a properly embedded minimal surface of bounded curvature has finite topology this determines its asymptotic behavior some rigidity theorems are obtained

stability and eigenvalue estimates of linear weingarten hypersurfaces in a sphere abstract let m be an n dimensional compact hypersurface without boundary in a unit sphere s n 1 1 m is called a linear weingarten hypersurface if c r d h e 0 where c d and e are constants with c 2 d 2 0 r and h denote the scalar curvature and the mean curvature of m respectively by the gauss equation we can rewrite the condition c r d h e 0 as n 1 e h 2 a h b where h 2 is the 2nd mean curvature a b and e are constants such that a 2 e 2 0 when e 0 it reduces to the constant mean curvature case in this paper we obtain some stability results about linear weingarten hypersurfaces which generalize the stability results about the hypersurfaces with constant mean curvature or with constant scalar curvature we show that linear weingarten hypersurfaces satisfying n 1 h 2 a h b where a and b are constants can be characterized as critical points of the functional m a n h d v for volume preserving variations we prove that such a linear weingarten hypersurface is stable if and only if it is totally umbilical and non totally geodesic we also obtain optimal upper bounds for the first and second eigenvalues of the jacobi operator of linear weingarten hypersurfaces

positivity of the universal pairing in 3 dimensions associated to a closed oriented surface s is the complex vector space with basis the set of all compact oriented 3 manifolds which it bounds gluing along s defines a hermitian pairing on this space with values in the complex vector space with basis all closed oriented 3 manifolds the main result in this paper is that this pairing is positive i e that the result of pairing a nonzero vector with itself is nonzero this has bearing on the question of what kinds of topological information can be extracted in principle from unitary 2 1 dimensional tqfts the proof involves the construction of a suitable complexity function c on all closed 3 manifolds satisfying a gluing axiom which we call the topological cauchy schwarz inequality namely that c ab max c aa c bb for all a b which bound s with equality if and only if a b the complexity function c involves input from many aspects of 3 manifold topology and in the process of establishing its key properties we obtain a number of results of independent interest for example we show that when two finite volume hyperbolic 3 manifolds are glued along an incompressible acylindrical surface the resulting hyperbolic 3 manifold has minimal volume only when the gluing can be done along a totally geodesic surface this generalizes a similar theorem for closed hyperbolic 3 manifolds due to agol storm thurston

technical communique a fast nonlinear control method for linear systems with input saturation we present a novel fast saturating nonlinear feedback law for single input systems with linear dynamics and input saturation it is fast in the sense that it yields a better performance than a saturating linear control law the control law is based on implicit soft variable structure control a convex optimization procedure for the controller synthesis based on linear matrix inequalities lmis is derived at the price of some conservatism as an example we consider the control of a submarine

on the singular structure of three dimensional area minimizing surfaces a sufficient condition is given for the union of two three dimensional planes through the origin in r to be area minimizing the condition is in terms of the three angles 0 7 their singular sets have hausdorff dimension at most m 1 larger singular sets occur in higher codimensions for example complex analytic varieties which are automatically area minimizing provide real m dimensional surfaces with 7 2 dimensional singular sets recent work of almgren 1 seems to show that the dimension of a singular set never exceeds m 2 little is known about the structure of the singular set only for two dimensional area minimizing surfaces is even the first order structure of singularities understood 5 cf also 6 this paper studies the first order structure of singularities in three dimensional area minimizing surfaces at each point in the surface there is a tangent cone not known to be unique itself area minimizing which records first order behavior if a tangent cone is a plane with multiplicity one the surface is locally a smooth manifold since the next simplest cone is a sum i e union of planes we address the following basic question question when is the sum of two oriented 3 planes through the origin in r area minimizing the geometric relationship between two oriented 3 planes in r is characterized by three angles 0 y y2 w 2 y2 y3 s m y2 see lemma 1 and remarks we prove a sufficient condition in terms of these three angles theorem 7 y3 s y y2 the sum of the planes is area minimizing we conjecture that the converse holds as well except for the trivial case of the

a nonlinear controller for input amplitude and rate constrained linear systems this technical note presents a nonlinear state feedback controller for systems with linear dynamics subject to an amplitude and rate limited actuator the nonlinear state feedback facilitates faster convergence compared to linear state feedback while guaranteeing domains of attraction of similar size the parametrization of the controller is done solving a convex optimization problem based on linear matrix inequalities lmis finally an example illustrates that the controller achieves large stability regions and short settling times

the angle criterion we complete the proof of the angle conjecture on when a pair of orientedm planes is area minimizing the nonzero sum oriented union p1 p2 is area minimizing if and only if the characterizing angles between them satisfy the inequality beta m leqq beta 1 beta m 1

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

rigidity of min max minimal spheres in three manifolds in this paper we consider min max minimal surfaces in three manifolds and prove some rigidity results for instance we prove that any metric on a three sphere which has scalar curvature greater than or equal to 6 and is not round must have an embedded minimal sphere of area strictly smaller than 4 and index at most one if the ricci curvature is positive we also prove sharp estimates for the width

surfaces of bounded mean curvature in riemannian manifolds consider a sequence of closed orientable surfaces of fixed genus g in a riemannian manifold m with uniform upper bounds on the norm of mean curvature and area we show that on passing to a subsequence we can choose parametrisations of the surfaces by inclusion maps from a fixed surface of the same genus so that the distance functions corresponding to the pullback metrics converge to a pseudo metric and the inclusion maps converge to a lipschitz map we show further that the limiting pseudo metric has fractal dimension two as a corollary we obtain a purely geometric result namely we show that bounds on the mean curvature area and genus of a surface f subset of m together with bounds on the geometry of m give an upper bound on the diameter of f our proof is modelled on gromov s compactness theorem for j holomorphic curves

min max minimal hypersurface in manifolds with convex boundary and hbox ric ge 0 let m n 1 partial m g be a compact manifold with non negative ricci curvature convex boundary and 2 le n le 6 we show that the min max minimal hypersurface with respect to one parameter families of hypersurfaces in m partial m is orientable of index one and multiplicity one

minimal hypersurfaces of least area in this paper we study closed embedded minimal hypersurfaces in a riemannian n 1 manifold 2 n 6 that minimize area among such hypersurfaces we show they exist and arise either by minimization techniques or by min max methods they have index at most 1 we apply this to obtain a lower area bound for such minimal surfaces in some hyperbolic 3 manifolds

decision boundary evaluation of optimum and suboptimum detectors in class a interference the middleton class a mca model is one of the most accepted models for narrow band impulsive interference superimposed to additive white gaussian noise awgn the mca density consists of a weighted linear combination of infinite gaussian densities which leads to a non tractable form of the optimum detector to reduce the receiver complexity one can start with a two term approximation of the mca model which has only two noise states gaussian and impulsive state our objective is to introduce a simple method to estimate the noise state at the receiver and accordingly reduce the complexity of the optimum detector furthermore we show for the first time how the decision boundaries of binary signals in mca noise should look like in this context we provide a new analysis of the behavior of many suboptimum detectors such as a linear detector a locally optimum detector lod and a clipping detector based on this analysis we insert a new clipping threshold for the clipping detector which significantly improves the bit error rate performance

the existence of least area surfaces in 3 manifolds this paper presents a new and unified approach to the existence theorems for least area surfaces in 3 manifolds introduction a surface f smoothly embedded or immersed in a riemannian manifold m is minimal if it has mean curvature zero at all points it is a least area surface in a class of surfaces if it has finite area which realizes the infimum of all possible areas for surfaces in this class the connection between these two ideas is that a surface which is of least area in a reasonable class of surfaces must be minimal the converse is false minimal surfaces are in general only critical points for the area function there are close analogies between these two concepts and the theory of geodesics in a riemannian manifold minimal surfaces correspond to geodesics and least area surfaces correspond to geodesic arcs or closed geodesics which have shortest length in some class of paths any geodesic a in a riemannian manifold m has the property that it is locally shortest i e if p and q are nearby points on a then the subarc of a which joins p and q is the shortest path in m from p to q it can also be proved that minimal surfaces are locally of least area but the proof is difficult and involves substantial knowledge of the theory of partial differential equations there are now a large number of theorems asserting the existence of surfaces of least area in various classes surfaces of this type have become an important tool in 3 dimensional topology in this paper we present a new approach to the proofs of these existence theorems this yields a simplified and unified method for the proof of the existence of minimal surfaces in 3 dimensional riemannian manifolds in 1930 douglas do and rado ra independently showed that a simple closed curve in ri which bounds a disk of finite area bounds a disk of least possible area this result was extended by morrey moi in 1948 to a general class of riemannian manifolds the homogeneously regular manifolds a class which includes all closed manifolds work of osserman 0 and gulliver g later showed that least area disks in 3 manifolds were immersed in their interiors more recently there has been a series of new existence results for closed surfaces of least area in manifolds of any dimension it follows from work of sacks and uhlenbeck s ui that if m is closed and 7r2 m is nonzero then there is an essential map of the 2 sphere into m which has least area among all essential maps sacks and uhlenbeck s uii and schoen and yau s y independently showed that if f f m is a map of a closed orientable surface f not the 2 sphere into a closed riemannian manifold m such received by the editors june 23 1986 1980 mathematics subject classification 1985 revision primary 53a10 secondary 57m35 the first author was supported by nsf postdoctoral fellowship dms84 14097 1988 american mathematical society 0002 9947 88 1 00 25 per page

min max theory for free boundary minimal hypersurfaces ii general morse index bounds and applications for any smooth riemannian metric on an n 1 dimensional compact manifold with boundary m partial m where 3 le n 1 le 7 we establish general upper bounds for the morse index of free boundary minimal hypersurfaces produced by min max theory in the almgren pitts setting we apply our morse index estimates to prove that for almost every in the c infty baire sense riemannan metric the union of all compact properly embedded free boundary minimal hypersurfaces is dense in m if partial m is further assumed to have a strictly mean convex point we show the existence of infinitely many compact properly embedded free boundary minimal hypersurfaces whose boundaries are non empty our results prove a conjecture of yau for generic metrics in the free boundary setting

controller design for a class of nonlinear systems with input saturation using convex optimization abstract in this paper we propose a new design strategy for nonlinear systems with input saturation the resulting nonlinear controllers are locally asymptotically stabilizing the origin the proposed methodology is based on exact feedback linearization which is used to reformulate the nonlinear system as a linear system having state dependent input saturation linear saturating state feedback controllers and soft variable structure controllers are developed based on this system formulation the resulting convex optimization problems can be written in terms of linear matrix inequalities and sum of squares conditions for which efficient solvers exist polynomial approximation based on legendre polynomials is used to extend the methodology to a more general class of nonlinear systems to demonstrate the benefit of this design method a stabilizing controller for a single link manipulator with flexible joint is developed

shrinkwrapping and the taming of hyperbolic 3 manifolds we introduce a new technique for finding cat 1 surfaces in hyperbolic 3 manifolds we use this to show that a complete hyperbolic 3 manifold with finitely generated fundamental group is geometrically and topologically tame

on stabilizing linear systems with input saturation via soft variable structure control laws abstract a large number of control methods deal with stabilizing single input linear systems with input saturation the linear full state feedback is one of the most used yielding either nonsaturating or high gain controls whereby the saturating effects of the latter are reduced by means of anti windup structures among the nonlinear controls the time optimal control in this case a bang bang type control yields the fastest time response but its switching surface is generally not characterizable related to the speed of the time response is the convergence rate which can be determined using invariant sets the most used ones are the ellipsoidal sets since they can be analyzed using powerful tools such as the lyapunov equation and designed via convex optimization for this reason they are also used for designing soft variable structure controls the paper presents a nonconservative design method for a stabilizing control of this type employing implicit lyapunov functions ilf a nonsaturating control law is given including some infinitely densely nested and contractive invariant sets of the equilibrium state the control law is then optimized by maximizing the ilf based lower bound of the convergence rate the maximal convergence control is shown to be of bang bang type with a parameter dependent switching scheme to overcome possible difficulties of a switching controller a saturating high gain control is also presented

area minimizing projective planes in 3 manifolds let m g be a compact riemannian manifold of dimension 3 and let denote the collection of all embedded surfaces homeomorphic to input amssym bbb r bbb p 2 we study the infimum of the areas of all surfaces in this quantity is related to the systole of m g it makes sense whenever is nonempty in this paper we give an upper bound for this quantity in terms of the minimum of the scalar curvature of m g moreover we show that equality holds if and only if m g is isometric to input amssym bbb r bbb p 3 up to scaling the proof uses the formula for the second variation of area and hamilton s ricci flow 2010 wiley periodicals inc

embedded minimal surfaces exotic spheres and manifolds with positive ricci curvature let n be a three dimensional riemannian manifold let e be a closed embedded surface in n then it is a question of basic interest to see whether one can deform in its isotopy class to some canonical embedded surface from the point of view of geometry a natural canonical surface will be the extremal surface of some functional defined on the space of embedded surfaces the simplest functional is the area functional the extremal surface of the area functional is called the minimal surface such minimal surfaces were used extensively by meeks yau my21 in studying group actions on three dimensional manifolds in my2 the theory of minimal surfaces was used to simplify and strengthen the classical dehn s lemma loop theorem and the sphere theorem in the setting there one minimizes area among all immersed surfaces and proves that the extremal object is embedded in this paper we minimize area among all embedded surfaces isotopic to a fixed embedded surface in the category of these surfaces we prove a general existence theorem theorem 1 a particular consequence of this theorem is that for irreducible manifolds an embedded incompressible surface is isotopic to an embedded incompressible surface with minimal area we also prove that there exists an embedded sphere of least area enclosing a fake cell provided the complementary volume is not a standard ball and provided there exists no embedded one sided rp2 by making use of the last result and a cutting and pasting argument we are able to settle a well known problem in the theory of three dimensional manifolds we prove that the covering space of any irreducible orientable three dimensional manifold is irreducible it is possible to exploit our existence theorem to study

a regularity theorem for minimizing hypersurfaces modulo it is proved that an n 1 dimensional area minimizing flat chain modulo v in rw with smooth extremal boundary of at most v 2 components has an interior singular set of hausdorff dimension at most n 8 similar results hold for more general integrands

size minimizing rectifiable currents 1 introduct ion 333 2 on the existence and regularity of size minimizers 336 3 isoperimetric inequalities and size bounds 345 4 bernstein theorems 346 references 348

lower bounds on volumes of hyperbolic haken 3 manifolds we prove a volume inequality for 3 manifolds having c o metrics bent along a surface and satisfying certain curvature conditions the result makes use of perelman s work on the ricci flow and geometrization of closed 3 manifolds corollaries include a new proof of a conjecture of bonahon about volumes of convex cores of kleinian groups improved volume estimates for certain haken hyperbolic 3 manifolds and a lower bound on the minimal volume of orientable hyperbolic 3 manifolds an appendix compares estimates of volumes of hyperbolic 3 manifolds drilled along a closed embedded geodesic with experimental data

calibrations modulo v abstract the theory of calibrations is extended to show that certain surfaces are area minimizing modulo v for example a complex algebraic variety in c n of degree d is area minimizing modulo v for all v 2 d

a new conformal invariant and its applications to the willmore conjecture and the first eigenvalue of compact surfaces let m be a compact riemannian manifold with a fixed conformal structure then we introduce the concept of conformal volume of m in the following manner for each branched conformal immersion q9 of m into the unit sphere s n we consider the set of all branched conformal immersions obtained by composi t ion of qo with the conformal automorphisms of s we let vc n qg be the max imum volume of these branched immersions the conformal volume of m is defined to be the infimum of v n q0 where qo ranges over all branched conformal immersions of m into the unit sphere s in this paper we study the case when m is a compact surface and we call the conformal volume of m to be the conformal area of m we demonst ra te that this conformal invariant is non trivial in fact we prove that if there exists a minimal immersion of m into s where coordinate functions are first eigenfunctions then the conformal area of m is given by the area of m with respect to the induced metric this enables us to compute the conformal area for several surfaces for example the conformal area of r p z is 6n and the conformal area of the square torus is 27c 2 we believe that the computa t ion of the conformal area for general surfaces will be very impor tant in studying the geometry of compact surfaces we demonstrate this claim by applying the concept of conformal area to two different branches of surface theory the first application is to study the total curvature of a compact surface in r this problem has a long history fenchel and fary 9 proved that for a closed curve o in r slk 27c where k is its curvature then milnor 12

maximizing neumann fundamental tones of triangles we prove sharp isoperimetric inequalities for neumann eigenvalues of the laplacian on triangular domains the first nonzero neumann eigenvalue is shown to be maximal for the equilateral triangle among all triangles of given perimeter and hence among all triangles of given area similar results are proven for the harmonic and arithmetic means of the first two nonzero eigenvalues

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

on otsuki tori and their willmore energy abstract otsuki tori in the three dimensional unit sphere are studied from calculations of the willmore energy the lower and upper bound estimates for the otsuki tori are established invariance of the otsuki tori under antipodal maps is also considered and in such cases better lower bound estimates of their willmore energies are obtained

the willmore functional on lagrangian tori its relation to area and existence of smooth minimizers in this paper we prove an existence and regularity theorem for la grangian tori minimizing the willmore functional in euclidean four space r4 with the standard metric and symplectic structure technical difficulties arise because the euler lagrange equation for this problem is a sixth order nonlinear partial differential equation this research was motivated by a study of the seemingly unrelated plateau problem for lagrangian tori and in this paper we illustrate this connection department of mathematics stanford university stanford california 94305 2125 courant institute of mathematical sciences 251 mercer street new york new york 10012 e mail address minicozz0ccims nyu edu this content downloaded from 157 55 39 27 on wed 07 sep 2016 06 38 19 utc all use subject to http about jstor org terms

embeddedness of the solutions to the h plateau problem abstract we generalize meeks and yau s embeddedness result on the solutions of the plateau problem to constant mean curvature disks we show that any minimizing h disk in an h 0 convex domain is embedded for any h 0 h 0 in particular for the unit ball b in r 3 this implies that for any h 0 1 any jordan curve in b bounds an embedded h disk in b

majoration de la seconde valeur propre d un operateur de schrodinger sur une variete compacte et applications abstract in the first part of this paper we obtain an optimal upper bound for the second eigenvalue of schrodinger operators through using a conformal invariant i e conformal volume in the second part we obtain some applications of this estimate concerning the morse index of constant mean curvature or minimal hypersurfaces the last part deals with some isospectrality problems

examples of unoriented area minimizing surfaces a comprehensive study is made of constructions of area minimizing flat chains modulo two many have singularities for instance any bounded area mini mizing submanifold of rn occurs as the singular set of some area minimizing flat chain modulo two in some rn introduction given a k l dimensional boundary in r one can seek to minimize area in some class of surfaces we want to take a large class that includes nonorientable as well as orientable surfaces without restrictions on the singularities or topological type such a class appears in geometric measure theory as the flat chains modulo two the existence and regularity of area minimizing flat chains modulo two are already well established cf 7 4 2 26 9 unfortunately examples have been rare this paper presents several methods for constructing area minimizing flat chains modulo two the first five theorems include the following five examples 1 a mobius strip 2 the graph over a convex region of any function satisfying the minimal surface equation

a statistical and physical mechanisms based interference and noise model for array observations a statistical noise model is developed from mathematical modeling of the physical mechanisms that generate noise in communication receivers employing antenna arrays such models have been lacking for cases where the antenna observations may be statistically dependent from antenna to antenna the model is developed by generalizing an approach for single antenna cases suggested by middleton 1967 1974 1976 1977 the model derived here is applicable to a wide variety of physical situations the focus is primarily on problems defined by middleton to be class a interference the number of noise sources in a small region of space is assumed to be poisson distributed and the emission times are assumed to be uniformly distributed over a long time interval finally an additive gaussian background component is included to represent the thermal noise that is always present in real receivers

the smale conjecture for seifert fibered spaces with hyperbolic base orbifold let m be a closed orientable 3 manifold admitting an mathbb h 2 times mathbb r or widetilde mathrm sl 2 mathbb r geometry or equivalently a seifert fibered space with a hyperbolic base 2 orbifold our main result is that the connected component of the identity map in the diffeomorphism group mathrm diff m is either contractible or homotopy equivalent to s 1 according as the center of pi 1 m is trivial or infinite cyclic apart from the remaining case of non haken infranilmanifolds this completes the homeomorphism classifications of mathrm diff m and of the space of seifert fiberings mathrm sf m for compact orientable aspherical 3 manifolds we also prove that when m has an mathbb h 2 times mathbb r or widetilde mathrm sl 2 mathbb r geometry and the base orbifold has underlying manifold the 2 sphere with three cone points the inclusion mathrm isom m to mathrm diff m is a homotopy equivalence

on the friedlander nadirashvili invariants of surfaces let m be a closed smooth manifold in 1999 friedlander and nadirashvili introduced a new differential invariant i 1 m using the first normalized nonzero eigenvalue of the lalpace beltrami operator delta g of a riemannian metric g they defined it taking the supremum of this quantity over all riemannian metrics in each conformal class and then taking the infimum over all conformal classes by analogy we use k th eigenvalues of delta g to define the invariants i k m indexed by positive integers k in the present paper the values of these invariants on surfaces are investigated we show that i k m i k mathbb s 2 unless m is a non orientable surface of even genus for orientable surfaces and k 1 this was earlier shown by petrides in fact friedlander and nadirashvili suggested that i 1 m i 1 mathbb s 2 for any surface m different from mathbb rp 2 we show that surprisingly enough this is not true for non orientable surfaces of even genus for such surfaces one has i k m i k mathbb s 2 we also discuss the connection between the friedlander nadirashvili invariants and the theory of cobordisms and conjecture that i k m is a cobordism invariant

scheduled controllers for linear systems with bounded actuators state and output feedback controllers are designed for disturbance attenuation in linear systems with bounded actuators the controller is scheduled according to the proximity to the origin of the state of the plant in the state feedback case and the compensator state in the case of output feedback this procedure yields a linear parameter varying structure for the controller that allows higher gain and hence higher performance controllers as the states move closer to the origin the main results give sufficient conditions for the satisfaction of a parameter dependent performance measure without violating the saturation bounds linear splines are used to obtain solutions that can be obtained by standard lmi software examples including an open loop unstable system highlight the application of the results

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

on the geometric and topological rigidity of hyperbolic 3 manifolds theorem 0 1 let n be a closed hyperbolic 3 manifold containing an embedded hyperbolic tube of radius log 3 2 549306 about a closed geodesic then i iff m n is a homotopy equivalence where m is an irreducible 3 manifold then f is homotopic to a homeomorphism ii if f g m s n are homotopic homeomorphisms then f is isotopic to g iii the space of hyperbolic metrics on n is path connected

analysis aspects of willmore surfaces a new formulation for the euler lagrange equation of the willmore functional for immersed surfaces in m is given as a nonlinear elliptic equation in divergence form with non linearities comprising only jacobians letting vec h be the mean curvature vector of the surface our new formulation reads mathcal l vec h 0 where mathcal l is a well defined locally invertible self adjoint elliptic operator several consequences are studied in particular the long standing open problem asking for a meaning to the willmore euler lagrange equation for immersions having only l 2 bounded second fundamental form is now solved the regularity of weak willmore immersions with l 2 bounded second fundamental form is also established its proof relies on the discovery of conservation laws which are preserved under weak convergence a weak compactness result for willmore surfaces with energy less than 8 the li yau condition ensuring the surface is embedded is proved via a point removability result established for wilmore surfaces in m thereby extending to arbitrary codimension the main result in ks3 finally from this point removability result the strong compactness of willmore tori below the energy level 8 is proved both in dimension 3 this had already been settled in ks3 and in dimension 4

boundary rigidity and filling volume minimality of metrics close to a flat one we say that a riemannian manifold m g with a non empty boundary m is a minimal orientable filling if for every compact orientable m g with m m the inequality d g x y d g x y for all x y m implies vol m g vol m g we show that if a metric g on a region m c r n with a connected boundary is sufficiently c 2 close to a euclidean one then it is a minimal filling by studying the equality case vol m g vol m g we show that if d g x y d g x y for all x y m then m g is isometric to m g this gives the first known open class of boundary rigid manifolds in dimensions higher than two and makes a step towards a proof of michel s conjecture

generic uniqueness of area minimizing disks for extreme curves we show that for a generic nullhomotopic simple closed curve gamma in the boundary of a compact orientable mean convex 3 manifold m with h 2 m bf z 0 there is a unique area minimizing disk d embedded in m with partial d gamma we also show that the same is true for nullhomologous curves in the absolutely area minimizing surface case

second eigenvalue of paneitz operators and mean curvature for n 7 we give the optimal estimate for the second eigenvalue of paneitz operators for compact n dimensional submanifolds in an n p dimensional space form in terms of the mean curvature and the q curvature

non proper helicoid like limits of closed minimal surfaces in 3 manifolds we show that there exists a metric with positive scalar curvature on s 2 s 1 and a sequence of embedded minimal cylinders that converges to a minimal lamination that in a neighborhood of a strictly stable 2 sphere is smooth except at two helicoid like singular ities on the 2 sphere the construction is inspired by a recent example by d hoffman and b white

boundary slopes of immersed surfaces in 3 manifolds author s hass joel rubinstein j hyam wang shicheng abstract this paper presents some finiteness results for the number of boundary slopes of immersed essential surfaces of given genus g in a compact 3 manifold with torus boundary in the case of hyperbolic 3 manifolds we obtain uniform quadratic bounds in g for the number of possible slopes independent of the 3 manifold we also look at some related quantities such as how many times the slopes of two such surfaces of specified genus can intersect

threshold detection in narrow band non gaussian noise the middleton class a narrow band non gaussian noise model 9 12 is examined it is shown that this noise model which is known to fit closely a variety of non gaussian noises can itself be closely approximated by a computationally much simpler noise model it is then shown by numerical examples that for the problem of locally optimum detection the simplest form of this approximation yields nearly optimal asymptotic performance the performance of other simple suboptimal threshold detectors in class a noise is also examined finally a useful relationship between the class a model and the e mixture model is developed

on improving the performance with bounded continuous feedback laws we present controller design methods to smooth the discontinuity resulting from a piecewise linear control law which was proposed to improve the convergence performance for systems with input constraints the continuous control laws designed in this paper are explicit functions of the state and are easily implementable we also show that the convergence performance can be further improved by using a saturated high gain feedback law the efficiency of the proposed methods is illustrated with the puma 560 robot model

harmonic maps and 2 cycles realizing the thurston norm let m 3 be an oriented 3 manifold we investigate when one of the fibers or a combination of fiber components f best of a harmonic map f m 3 s 1 with morse type singularities delivers the thurston norm x f best of its homology class f best h 2 m 3 z in particular for a map f with connected fibers and any well positioned oriented surface e c m in the homology class of a fiber we show that the thurston number x satisfies an inequality x x f best f var x f here the variation var x f is can be expressed in terms of the x invariants of the fiber components and the twist f measures the complexity of the intersection of with a particular set f r of bad fiber components this complexity is tightly linked with the optimal f height of e being lifted to the f induced cyclic coverm 3 m 3 based on these invariants for any morse map f we introduce the notion of its twist x f we prove that for a harmonic f x f best x f best if and only if x f 0

the geometric complexity of special lagrangian t2 cones we prove a number of results on the geometric complexity of special lagrangian slg t2 cones in 3 every slg t2 cone has a fundamental integer invariant its spectral curve genus we prove that the spectral curve genus of an slg t2 cone gives a lower bound for its geometric complexity i e the area the stability index and the legendrian index of any slg t2 cone are all bounded below by explicit linearly growing functions of the spectral curve genus we prove that the cone on the clifford torus which has spectral curve genus zero in s5 is the unique slg t2 cone with the smallest possible legendrian index and hence that it is the unique stable slg t2 cone this leads to a classification of all rigid index 1 slg cone types in dimension three for cones with spectral curve genus two we give refined lower bounds for the area the legendrian index and the stability index one consequence of these bounds is that there exist s1 invariant slg torus cones of arbitrarily large area legendrian and stability indices we explain some consequences of our results for the programme due to joyce to understand the most common three dimensional isolated singularities of generic families of slg submanifolds in almost calabi yau manifolds

on the geometry and topology of initial data sets with horizons we study the relationship between initial data sets with horizons and the existence of metrics of positive scalar cur vature we dene a cauchy domain of outer communications cdoc to be an asymptotically at initial set m g k such that the boundary m of m is a collection of marginally outer or inner trapped surfaces motss and or mitss and such that mn m contains no motss or mitss this denition is meant

piecewise linear lq control for systems with input constraints abstract a piecewise linear control plc law based on lq theory is derived ths law raises an lq gain as the controlled error converges towards the origin it can be computed off line and it guarantees that input bounds are never exceeded without causing input saturation a succession of positively invariant sets are defined of diminishing size and associated with each is the corresponding highest lq gain possible in the presence of the input bounds the formulation gives rise to an iteration function whose solution is a fixed point fp the fp defines the optimal gain and the invariant set which comprise ellipsoids fp existence uniqueness and convergence properties are proven we explain the algorithms for the generation and subsequent implementation of the plc law the notion of a switching function is introduced consideration is given to the computational aspects of the fp the plc scheme is demonstrated for the linearized dynamics of the simple pendulum and the puma 560 robot a discrete time plc law can also be derived using the discrete time algebraic riccati equation

minimal immersions of surfaces by the first eigenfunctions and conformal area let 0 m s be a minimal immersion of a compact surface into a unit sphere then the linear functions of 0 are eigenfunctions for the laplacian of m corresponding to the eigenvalue 2 2 the main purpose of this paper is to study those minimal surfaces for which 2 is exactly the first non zero eigenvalue of its laplacian this kind of immersions have a peculiar behaviour among all compact minimal surfaces of the sphere and they appear naturally when one considers different geometric problems as li and yau have shown in 6 the methods that we use in this paper are based for the most part on 6 it is known that the only metric on a 2 dimensional sphere admitting a minimal immersion into s by the first eigenfunctions is the standard one this follows for example from the fact that the multiplicity of the first eigenvalue for such a metric is at most three see the cheng work 3 our first result shows that it is possible to extend this property for an arbitrary compact surface in the following way

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

volume of minimal hypersurfaces in manifolds with nonnegative ricci curvature we establish a min max estimate on the volume width of a closed riemannian manifold with nonnegative ricci curvature more precisely we show that every closed riemannian manifold with nonnegative ricci curvature admits a pl morse function whose level set volume is bounded in terms of the volume of the manifold as a consequence of this sweep out estimate there exists an embedded closed possibly singular minimal hypersurface whose volume is bounded in terms of the volume of the manifold

injectivity radius and gonality of a compact riemann surface we obtain a sharp lower bound for the volumes of purely 1 dimensional complex analytic subvarieties in a geodesic tubular neighborhood of the diagonal of the cartesian product of a compact riemann surface with itself this leads to a lower bound of the seshadri number of the canonical line bundle of the cartesian product with respect to the diagonal as a consequence we obtain an upper bound for the hyperbolic injectivity radii of compact riemann surfaces of a fixed gonality in particular we obtain the limiting behavior of the gonalities of a tower of compact riemann surfaces we also give an application of our results to an invariant related to the ample cone of the symmetric product of a riemann surface