conformal deformation of a riemannian metric to a scalar flat metric with constant mean curvature on the boundary one of the most celebrated theorems in mathematics is the riemann mapping theorem it says that an open simply connected proper subset of the plane is conformally diffeomorphic to the disk in higher dimensions very few regions are conformally diffeomorphic to the ball however we can still ask whether a domain is conformally diffeomorphic to a manifold that resembles the ball in two ways namely it has zero scalar curvature and its boundary has constant mean curvature in this paper we generalize the riemann mapping theorem to higher dimensions in that sense

curvature diameter homotopy groups and cohomology rings we establish two topological results a if m is a 1 connected compact n manifold and q 2 then the minimal number of generators for the qth homotopy group q m mng q m is bounded above by a number depending only on mng h m and q where h m is the homology group c let mathscr m n y be the collection of compact orientable n manifolds whose oriented bundles admit so n invariant fibrations over y with fiber compact nilpotent manifolds such that the induced so n actions on y are equivalent then q m finitely generated m mathscr m n y contains only finite isomorphism classes depending only on n y q together with the results of cg and gr1 from a we conclude that i if m is a complete n manifold of nonnegative curvature then mng q m is bounded above by a number depending only on n and q 2 together with the results of ch and cfg from c we conclude that ii if m is a compact n manifold whose sectional curvature and diameter satisfy secm and diamm d then q m has a finite number of possible isomorphism classes depending on n d q 2 provided q m is finitely generated we also show that b if m is a compact n manifold with secm and diam m d then the cohomology ring h m may have infinitely many isomorphism classes in particular b answers some questions raised by k grove gro

constant scalar curvature metrics with isolated singularities we extend the results and methods of 6 to prove the existence of constant positive scalar curvature metrics g which are complete and conformal to the standard metric on s where is a disjoint union of submanifolds of dimensions between 0 and n 2 2 the existence of solutions with isolated singularities occupies the majority of the paper their existence was previously established by schoen 12 but the proof we give here based on the techniques of 6 is more direct and provides more information about their geometry when is discrete we also establish that these solutions are smooth points in the moduli spaces of all such solutions introduced and studied in 7 and 8

curvature evolution of nonconvex lens shaped domains we study the curvature flow of planar nonconvex lens shaped domains con sidered as special symmetric networks with two triple junctions we show that the evolv ing domain becomes convex in finite time then it shrinks homothetically to a point as proved in 22 our theorem is the analog of the result of grayson 13 for curvature flow of closed planar embedded curves

a new approach to adaptive nonlinear regulation this paper shows how the theory of adaptive observers can be effectively used in the design of internal models for nonlinear output regulation the main result obtained in this way is a new method for the synthesis of adaptive internal models which substantially enhances the existing theory of adaptive output regulation by allowing nonlinear internal models and more general classes of controlled plants

conformal deformation to constant negative scalar curvature on noncompact riemannian manifolds si m g est une variete de riemann complete de courbure scalaire non positive s satisfaisant s x e 0 pour x m m 0 ou m 0 est un ensemble compact alors il y a une metrique conforme complete g de courbure scalaire s 1

on yau s problem of evolving one curve to another convex case abstract a revised yau s curvature difference flow is considered to deform one convex curve x 0 to another one x it is proved that this flow exists globally on time interval 0 and the evolving curve preserving its convexity and bounded area a converges to a fixed limiting curve x congruent to a a x as time tends to infinity where a is the area bounded by the target curve x

path connectedness of the moduli spaces of metrics with positive isotropic curvature on four manifolds in the paper we prove the path connectedness of the moduli spaces of metrics with positive isotropic curvature on certain compact four dimensional manifolds

collapsing irreducible 3 manifolds with nontrivial fundamental group let m be a closed orientable irreducible non simply connected 3 manifold we prove that if m admits a sequence of riemannian metrics which volume collapses and whose sectional curvature is locally controlled then m is a graph manifold this is the last step in perelman s proof of thurston s geometrisation conjecture

termination proofs by multiset path orderings imply primitive recursive derivation lengths it is shown that a termination proof for a term rewriting system using multiset path orderings i e recursive path orderings with multiset status only yields a primitive recursive bound on the length of derivations measured in the size of the starting term confirming a conjecture of plaisted pla78 this result holds for a great variety of path orderings including path of subterms ordering recursive decomposition ordering and the path ordering of kapur narendran sivakumar if lexicographic status is not incorporated the result is essentially optimal as such derivation lengths can be found in each level of the grzegorczyk hierarchy even for string rewriting systems

brief robust adaptive compensation of biased sinusoidal disturbances with unknown frequency asymptotically stable observable linear systems of order n which are not required to be minimum phase and are affected by an additive noisy biased sinusoidal disturbance with unknown bias magnitude phase and frequency are considered the problem of designing an output feedback compensator which regulates the output to zero for any initial condition and for any biased sinusoidal disturbance with no noise is addressed under the assumption that the system parameters are known this problem is solved by a 2n 6 order compensator which generates asymptotically convergent estimates of the biased sinusoidal disturbance and of its parameters including frequency the robustness of the closed loop system with respect to sufficiently small additive unmodelled noise is characterized in terms of input to state stability

3 manifolds with nonnegative ricci curvature for a complete noncompact 3 manifold with nonnegative ricci curvature we prove that either it is diffeomorphic to 3 or the universal cover splits this confirms milnor s conjecture in dimension 3

inverse curvature flows in riemannian warped products abstract the long time existence and umbilicity estimates for compact graphical solutions to expanding curvature flows are deduced in riemannian warped products of a real interval with a compact fibre notably we do not assume the ambient manifold to be rotationally symmetric nor the radial curvature to converge nor a lower bound on the ambient sectional curvature the inverse speeds are given by powers 0 p 1 of a curvature function satisfying few common properties

immortal homogeneous ricci flows we show that for an immortal homogeneous ricci flow solution any sequence of parabolic blow downs subconverges to a homogeneous expanding ricci soliton this is established by constructing a new lyapunov function based on curvature estimates which come from real geometric invariant theory

eigenvalues and energy functionals with monotonicity formulae under ricci flow in this note we construct families of functionals of the type of mathcal f functional and mathcal w functional of perelman we prove that these new functionals are nondecreasing under the ricci flow as applications we give a proof of the theorem that compact steady ricci breathers must be ricci flat using these new functionals we also give a new proof of perelman s no non trivial expanding breather theorem furthermore we prove that compact expanding ricci breathers must be einstein by a direct method in this note we also extend cao s methods of eigenvalues in math ann 337 2 2007 and improve their results

disturbance attenuation and h sub infinity control via measurement feedback in nonlinear systems a solution to the problem of disturbance attenuation via measurement feedback with internal stability is presented for an affine nonlinear system it is shown that the concept of disturbance attenuation in the sense of truncated l sub 2 norms can be given an interpretation in terms of the response to periodic inputs in the sense of rms amplitude even in the nonlinear setup in the case of state feedback a family of controllers is also provided the proofs of all these results are simple and provide deeper insight even in the analysis of the corresponding linear control problem

stable inversion for nonlinear systems abstract there have been two recent major developments in output tracking for nonlinear systems and our first main contribution is to relate these under appropriate assumptions we show that the bounded solution of the partial differential equation of isidori and byrnes for each trajectory of an exosystem must be given by an integral representation formula of devasia chen and paden under restrictive hypotheses devasia chen and paden develop a picard process that converges to the solution of the integral equation this solution to the integral equation is also a bounded solution to a dynamical equation driven by the desired outputs in aircraft applications our nonlinear systems are perturbations of pure feedback systems with outputs and we find a solution to the stable inversion problem i e finding bounded controls and bounded state trajectories in response to bounded output signals in two steps the first step ignores the perturbation error and computes the major part of the desired control and corresponding state trajectory the second step computes the remaining control and states by finding a noncausal and stable solution to an error driven dynamical equation in other words the method of devasia chen and paden is applied to an error system and not to the full system this two step procedure is the second main contribution of this paper throughout this paper it is assumed that our nonlinear systems have vector relative degree

fractional laplacian in conformal geometry abstract in this note we study the connection between the fractional laplacian operator that appeared in the recent work of caffarelli and silvestre and a class of conformally covariant operators in conformal geometry

l sub 2 gain analysis of nonlinear systems and nonlinear state feedback h sub infinity control previously obtained results on l2 gain analysis of smooth nonlinear systems are unified and extended using an approach based on hamilton jacobi equations and inequalities and their relation to invariant manifolds of an associated hamiltonian vector field on the basis of these results a nonlinear analog is obtained of the simplest part of a state space approach to linear h sub infinity control namely the state feedback h sub infinity optimal control problem furthermore the relation with h sub infinity control of the linearized system is dealt with

lagrangian mean curvature flow for entire lipschitz graphs ii we prove longtime existence and estimates for smooth solutions to a fully nonlinear lagrangian parabolic equation with locally c 1 1 initial data u 0 satisfying either 1 1 eta i n le d 2u 0 le 1 eta i n for some positive dimensional constant eta 2 u 0 is weakly convex everywhere or 3 u 0 verifies a large supercritical lagrangian phase condition

expanding solitons with non negative curvature operator coming out of cones we consider ricci flow of complete riemannian manifolds which have bounded non negative curvature operator non zero asymptotic volume ratio and no boundary we prove scale invariant estimates for these solutions using these estimates we show that there is a limit solution obtained by scaling down this solution at a fixed point in space this limit solution is an expanding soliton coming out of the asymptotic cone at infinity

a non compactness result on the fractional yamabe problem in large dimensions abstract let x n 1 g be an n 1 dimensional asymptotically hyperbolic manifold with conformal infinity m n h the fractional yamabe problem addresses to solve p g h u c u n 2 n 2 u 0 on m where c r and p g h is the fractional conformal laplacian whose principal symbol is the laplace beltrami operator on m in this paper we construct a metric on the half space x r n 1 which is conformally equivalent to the unit ball for which the solution set of the fractional yamabe equation is non compact provided that n 24 for 0 and n 25 for 1 where 0 1 is a certain transition exponent the value of turns out to be approximately 0 940197

invariant surface area functionals and singular yamabe problem in 3 dimensional cr geometry abstract we express two cr invariant surface area elements in terms of quantities in pseudohermitian geometry we deduce the euler lagrange equations of the associated energy functionals many solutions are given and discussed in relation to the singular cr yamabe problem we show that one of the energy functionals appears as the coefficient up to a constant multiple of the log term in the associated volume renormalization

guaranteed performance in semi global trajectory tracking for matched disturbance nonlinear systems in this article we present a semi global trajectory tracking approach that guarantees a priori computable l 2 and l performance bounds for matched disturbance control affine systems the proposed controller is derived by combining a standard inverse control technique with an extended nonlinear robust state feedback the latter is based on a control lyapunov function used for stabilising one operating point inside the considered state space a difference gradient formulation of this lyapunov function is then applied to prove stabilisation along any trajectory in the considered state space results for l 2 and l bounded disturbances will be presented and further extended to the case of actuator uncertainties and disturbance offsets the theoretical contributions are verified applying them to a numerical example

nonlinear inversion based output tracking an inversion procedure is introduced for nonlinear systems which constructs a bounded input trajectory in the preimage of a desired output trajectory in the case of minimum phase systems the trajectory produced agrees with that generated by hirschorn s inverse dynamic system however the preimage trajectory is noncausal rather than unstable in the nonminimum phase case in addition the analysis leads to a simple geometric connection between the unstable manifold of the system zero dynamics and noncausality in the nonminimum phase case with the addition of stabilizing feedback to the preimage trajectory asymptotically exact output tracking is achieved tracking is demonstrated with a numerical example and compared to the well known byrnes isidori regulator rather than solving a partial differential equation to construct a regulator the inverse is calculated using a picard like interaction when preactuation is not possible noncausal inverse trajectories can be truncated resulting in the tracking error transients found in other control schemes

ricci flow under local almost non negative curvature conditions abstract we find a local solution to the ricci flow equation under a negative lower bound for many known curvature conditions under a non collapsing assumption the flow exists for a uniform amount of time during which the curvature stays bounded below by a controllable negative number the curvature conditions we consider include 2 non negative and weakly pic 1 cases of which the results are new we complete the discussion of the almost preservation problem by bamler cabezas rivas wilking and the 2 non negative case generalizes a result in 3d by simon topping to higher dimensions as an application we use the local ricci flow to smooth a metric space which is the limit of a sequence of manifolds with the almost non negative curvature conditions and show that this limit space is bi holder homeomorphic to a smooth manifold

guaranteed robustness bounds for matched disturbance nonlinear systems this paper considers input affine nonlinear systems with matched disturbances and shows how to compute an a priori upper bound of the h attenuation level achieved by the optimal l 2 controller and the suboptimal h central controller the case where the disturbance contains a constant term is also discussed these bounds are shown to depend only on the function mapping the control input to the performance variable this result is used to derive a robust control design for a special but practically important class of non input affine nonlinear systems consisting of the series connection of a nonlinear state and input dependent map and of a nonlinear input affine dynamical system approximate inversion of the nonlinear static map leads to a robust control problem which fits into the framework the effectiveness of the theoretical results is shown by its use for the robust control design of a diesel engine test bench

quasi interpretations a way to control resources this paper presents in a reasoned way our works on resource analysis by quasi interpretations the controlled resources are typically the runtime the runspace or the size of a result in a program execution quasi interpretations allow the analysis of system complexity a quasi interpretation is a numerical assignment which provides an upper bound on computed functions and which is compatible with the program operational semantics the quasi interpretation method offers several advantages i it provides hints in order to optimize an execution ii it gives resource certificates and iii finding quasi interpretations is decidable for a broad class which is relevant for feasible computations by combining the quasi interpretation method with termination tools here term orderings we obtained several characterizations of complexity classes starting from ptime and pspace

on the regularity of solutions to the yamabe equation and the existence of smooth hyperboloidal initial data for einstein s field equations the regularity of the solutions to the yamabe problem is considered in the case of conformally compact manifolds and negative scalar curvature the existence of smooth hyperboloidal initial data for einstein s field equations is demonstrated

incomplete yamabe flows and removable singularities abstract we study the yamabe flow on a riemannian manifold of dimension m 3 minus a closed submanifold of dimension n and prove that there exists an instantaneously complete solution if and only if n m 2 2 in the remaining cases 0 n m 2 2 including the borderline case we show that the removability of the n dimensional singularity is necessarily preserved along the yamabe flow in particular the flow must remain geodesically incomplete as long as it exists this is contrasted with the two dimensional case where instantaneously complete solutions always exist

plurisubharmonic functions and the structure of complete kahler manifolds with nonnegative curvature in this paper we study global properties of continuous plurisubharmonic functions on complete noncompact kahler manifolds with nonnegative bisectional curvature and their applications to the structure of such manifolds we prove that continuous plurisubharmonic functions with reasonable growth rate on such manifolds can be approximated by smooth plurisubharmonic functions through the heat flow deformation optimal liouville type theorem for the plurisubharmonic functions as well as a splitting theorem in terms of harmonic functions and holomorphic functions are established the results are then applied to prove several structure theorems on complete noncompact kahler manifolds with nonnegative bisectional or sectional curvature

complete conformal metrics with negative scalar curvature in compact riemannian manifolds soit m g une variete de riemann compacte de dimension n 3 et soit une sous variete lisse fermee de dimension d alors il y a une metrique conforme complete sur m m a courbure scalaire negative constante si et seulement si d n 2 2

uniform practical nonlinear output regulation in this paper we present a solution to the problem of asymptotic and practical semiglobal regulation by output feedback for nonlinear systems a key feature of the proposed approach is that practical regulation is achieved uniformly with respect to the dimension of the internal model and to the gain of the stabilizer near the zero error manifold this property renders the approach interesting for a number of real cases by bridging the gap between output regulation theory and advanced engineering applications simulation results regarding meaningful control problems are also presented

eigenvalues of triangle frac r 2 on manifolds with nonnegative curvature operator in this paper we show that the eigenvalues of triangle frac r 2 are nondecreasing under the ricci flow for manifolds with nonnegative curvature operator then we show that the only steady ricci breather with nonnegative curvature operator is the trivial one which is ricci flat

minimal two spheres and the topology of manifolds with positive curvature on totally isotropic two planes there has been much interest among differential geometers in finding relationships between curvature and topology of riemannian manifolds for the most part efforts have been directed towards understanding the implications of positive or negative sectional curvature ricci curvature or scalar curvature but there are other hypotheses on the curvature which also deserve investigation in this article we will consider the topological implications of a new curvature assumption positive curvature on totally isotropic two planes and we will prove via the sacks uhlenbeck theory of minimal two spheres that a compact simply connected riemannian manifold of dimension at least four with positive curvature on totally isotropic two planes is homeomorphic to a sphere as corollaries we will obtain a proof via minimal two spheres for riemannian manifolds of dimension at least four of the berger klingenberg rauch toponogov sphere theorem strengthened to pointwise quarter pinching as well as a proof that every simply connected compact riemannian manifold with positive curvature operators is homeomorphic to a sphere let m be an n dimensional riemannian manifold with tangent space tpm at the point p e m recall that the curvature operator at p is the self adjoint

a lie algebraic approach to ricci flow invariant curvature conditions and harnack inequalities we consider a subset s of the complex lie algebra so n c and the cone c s of curvature operators which are nonnegative on s we show that c s defines a ricci flow invariant curvature condition if s is invariant under adso n c the analogue for kahler curvature operators holds as well although the proof is very simple and short it recovers all previously known invariant nonnegativity conditions as an application we reprove that a compact kahler manifold with positive orthogonal bisectional curvature evolves to a manifold with positive bisectional curvature and is thus biholomorphic to cp moreover the methods can also be applied to prove harnack inequalities we consider a lie algebra g endowed with a scalar product which is invariant under the adjoint representation of the lie algebra the reader should think of g either as the space so n of skew adjoint endomorphism of r with the scalar product a b 1 2 tr ab or of the lie subalgebra u n so 2n corresponding to the unitary group u n so 2n endowed with the induced scalar product we consider the space of selfadjoint endomorphisms of s g every selfadjoint endomorphism r s g is determined by the corresponding bilinear form x y 7 rx y the extension of this form to a complex bilinear form r g r c g r c c will be denoted with the same letter r notice that for any x g r c the number r x x is real where x 7 x is complex conjugation also recall that the space of algebraic curvature operators s b so n is a linear subspace of s so n similarly the space of algebraic kahler curvature operators s k u n is a linear subspace of s u n the subspaces are also invariant under the ricci flow ode on s g r r r where rx y 12 tr adx r ady r for x y g we have the following basic result theorem 1 let s be a subset of the complex lie algebra g r c and let gc denote a lie group with lie algebra g r c if s is invariant under the adjoint representation of gc then for h r the set c s h r s g r v v h for all v s is invariant under the ode r r r in many cases s is scaling invariant and then h 0 is the only meaningful choice for h 0 the set c s c s 0 is a cone and the curvature condition c s can be thought of as a nonnegativity condition we recall that for a o n invariant subset c s so n we say that a manifold satisfies c if the curvature operator

inverse mean curvature flow with forced term abstract in this paper we study the evolution of convex hypersurfaces by inverse mean curvature minus an external force field c we prove that the flow will preserve the convexity for any c if c 1 h on the initial surface we prove that the flow will expand the hypersurface for all time and after rescaling the hypersurface will converge to a sphere if c 1 h on the initial surface we show that the maximum existence time of the flow is finite and the hypersurface will contract to a point when approaching the final time

on a simple maximum principle technique applied to equations on the circle abstract for a class of general quasilinear equations on s 1 we show that by a very simple maximum principle technique as long as the solution stays finite all of its derivatives also remain finite some specific examples are given under suitable assumptions we also derive exponential decay of the derivatives of the solution

lie algebraic curvature conditions preserved by the hermitian curvature flow the purpose of this paper is to prove that the hermitian curvature flow hcf on an hermitian manifold m g j preserves many natural curvature positivity conditions following ideas of wilking j reine angew math 679 223 247 2013 for an text ad gl t 1 0 m invariant subset s subset text end t 1 0 m and a nice function f text end t 1 0 m rightarrow mathbb r we construct a convex set of curvature operators c s f which is invariant under the hcf varying s and f we prove that the hcf preserves griffiths positivity dual nakano positivity positivity of holomorphic orthogonal bisectional curvature lower bounds on the second scalar curvature as an application we prove that stationary periodic solutions to the hcf can exist only on manifolds m with the trivial canonical bundle on the universal cover widetilde m

large solutions for yamabe and similar problems on domains in riemannian manifolds we present a unifled approach to study large positive solutions i e u x 1 as x of the equation u hu i k u if in an arbitrary domain we assume u is convex and grows su ciently fast as u 1 equations of this type arise in geometry yamabe problem two dimensional curvature equation probability superdifiusion we prove that both existence and uniqueness are local properties of points of the boundary i e they depend only on properties of in arbitrary small neighborhood of each boundary point we also flnd several new necessary and su cient conditions for existence and uniqueness of large solutions including an existence theorem on domains with fractal boundaries

a fully nonlinear flow for two convex hypersurfaces in riemannian manifolds we consider a one parameter family of closed embedded hypersurfaces moving with normal velocity g kappa big sum i j frac 1 lambda i lambda j 2 kappa big 1 where lambda 1 le cdots le lambda n denote the curvature eigenvalues and kappa is a nonnegative constant this defines a fully nonlinear parabolic equation provided that lambda 1 lambda 2 2 kappa in contrast to mean curvature flow this flow preserves the condition lambda 1 lambda 2 2 kappa in a general ambient manifold our main goal in this paper is to extend the surgery algorithm of huisken sinestrari to this fully nonlinear flow this is the first construction of this kind for a fully nonlinear flow as a corollary we show that a compact riemannian manifold satisfying overline r 1313 overline r 2323 ge 2 kappa 2 with non empty boundary satisfying lambda 1 lambda 2 2 kappa is diffeomorphic to a 1 handlebody the main technical advance is the pointwise curvature derivative estimate the proof of this estimate requires a new argument as the existing techniques for mean curvature flow due to huisken sinestrari haslhofer kleiner and brian white cannot be generalized to the fully nonlinear setting to establish this estimate we employ an induction on scales argument this relies on a combination of several ingredients including the almost convexity estimate the inscribed radius estimate as well as a regularity result for radial graphs we expect that this technique will be useful in other situations as well

an upper bound on the derivational complexity of knuth bendix orderings the derivational complexity of a terminating rewrite system is a measure for the maximal length of rewrite sequences we study the influence of certain standard termination criteria on the derivational complexity in this paper we prove a uniform multiple recursive upper bound for knuth bendix orderings this continues work by hofbauer and lautemann 13 where it has been shown that primitive recursive bounds are impossible

an optimal gap theorem by solving the cauchy problem for the hodge laplace heat equation for d closed positive 1 1 forms we prove an optimal gap theorem for kahler manifolds with nonnegative bisectional curvature which asserts that the manifold is flat if the average of the scalar curvature over balls of radius r centered at any fixed point o is a function of o r 2 furthermore via a relative monotonicity estimate we obtain a stronger statement namely a positive mass type result asserting that if m g is not flat then liminf r to infty frac r 2 v o r int b o r mathcal s y d mu y 0 for any o m

should model based inverse inputs be used as feedforward under plant uncertainty bounds on the size of the plant uncertainties are found such that the use of the inversion based feedforward input improves the output tracking performance when compared to the use of feedback alone the output tracking error is normalized by the size of the desired output and used as a measure of the output tracking performance the worst case performance is compared for two cases 1 with the use of feedback alone and 2 with the addition of the feedforward input it is shown that inversion based feedforward controllers can lead to performance improvements at frequencies w where the uncertainty spl delta jw in the nominal plant is smaller than the size of the nominal plant g sub 0 jw divided by its condition number k sub g0 jw i e spl par spl delta jw spl par sub 2 spl par g sub 0 jw spl par sub 2 k sub g0 jw a modified feedforward input is proposed that only uses the model information in frequency regions where plant uncertainty is sufficiently small the use of this modified inverse with any feedback results in improvement of the output tracking performance when compared to the use of the feedback alone

mean curvature flows of lagrangian submanifolds with convex potentials this article studies the mean curvature flow of lagrangian submanifolds in particular we prove the following global existence and convergence theorem if the potential function of a lagrangian graph in t 2n is convex then the flow exists for all time and converges smoothly to a flat lagrangian submanifold we also discuss various conditions on the potential function that guarantee global existence and convergence

new li yau hamilton inequalities for the ricci flow via the space time approach we prove li yau hamilton inequalties that extend hamilton s matrix inequality for solutions of the ricci flow with nonnegative curvature operators to obtain our extensions we apply the space time formalism of s c chu and one of the authors to solutions of the ricci flow modified by a cosmological constant then we adjoin to the ricci flow the evolution of a 1 form and a 2 form flowing by a system of heat type equations by a rescaling argument the inequalities we obtain in this manner yield new inequalities which are reminiscent of the linear trace inequality of hamilton and one of the authors

gluing eguchi hanson metrics and a question of page in 1978 gibbons pope and page proposed a physical picture for the ricci flat kahler metrics on the k3 surface based on a gluing construction in this construction one starts from a flat torus with 16 orbifold points and resolves the orbifold singularities by gluing in 16 small eguchi hanson manifolds that all have the same orientation this construction was carried out rigorously by topiwala lebrun singer and donaldson in 1981 page asked whether the above construction can be modified by reversing the orientations of some of the eguchi hanson manifolds this is a subtle question if successful this construction would produce einstein metrics that are neither kahler nor self dual in this paper we focus on a configuration of maximal symmetry involving eight small eguchi hanson manifolds of each orientation that are arranged according to a chessboard pattern by analyzing the interactions between eguchi hanson manifolds with opposite orientation we identify a nonvanishing obstruction to the gluing problem thereby destroying any hope of producing a metric of zero ricci curvature in this way using this obstruction we are able to understand the dynamics of such metrics under ricci flow as long as the eguchi hanson manifolds remain small in particular for the configuration described above we obtain an ancient solution to the ricci flow with the property that the maximum of the riemann curvature tensor blows up at a rate of t 1 2 while the maximum of the ricci curvature converges to 0 2016 wiley periodicals inc

mean curvature flow of pinched submanifolds to spheres the evolution of hypersurfaces by their mean curvature has been studied by many authors since the appearance of gerhard huisken s seminal paper hu1 more recently mean curvature flow of higher codimension submanifolds has also received attention in this paper we prove a result analogous to that of hu1 for submanifolds of any codimension let f0 n rn k be a smooth immersion of a compact manifold the mean curvature flow with initial condition f0 is a smooth family of immersions f 0 t rn k satisfying

output stabilization via nonlinear luenberger observers the present paper addresses the problem of the existence of an output feedback law that asymptotically steers to zero prescribed outputs while keeping all state variables bounded for any initial conditions in a given compact set the problem can be viewed as an extension of the classical problem of semiglobally stabilizing the trajectories of a controlled system to a compact set the problem also encompasses a version of the classical problem of output regulation under only a weak minimum phase assumption it is shown that there exists a controller solving the problem at hand the paper is deliberately focused on theoretical results regarding the existence of such a controller practical aspects involving the design and the implementation of the controller are left to a forthcoming work

sharp differential estimates of li yau hamilton type for positive p p forms on kahler manifolds in this paper we study the heat equation of hodge laplacian deformation of p p forms on a kahler manifold after identifying the condition and establishing that the positivity of a p p form solution is preserved under such an invariant condition we prove the sharp differential harnack in the sense of liyau hamilton estimates for the positive solutions of the hodge laplacian heat equation we also prove a nonlinear version coupled with the kahler ricci flow and some interpolating matrix differential harnack type estimates for both the kahler ricci flow and the ricci flow 2011 wiley periodicals inc

mean curvature flow of mean convex hypersurfaces in the last 15 years white and huisken sinestrari developed a far reaching structure theory for the mean curvature flow of mean convex hypersurfaces their papers provide a package of estimates and structural results that yield a precise description of singularities and of high curvature regions in a mean convex flow in the present paper we give a new treatment of the theory of mean convex and k convex flows this includes 1 an estimate for derivatives of curvatures 2 a convexity estimate 3 a cylindrical estimate 4 a global convergence theorem 5 a structure theorem for ancient solutions and 6 a partial regularity theorem our new proofs are both more elementary and substantially shorter than the original arguments our estimates are local and universal a key ingredient in our new approach is the new noncollapsing result of andrews 2 some parts are also inspired by the work of perelman 32 33 in a forthcoming paper 17 we will give a new construction of mean curvature flow with surgery based on the methods established in the present paper note added in may 2015 since the first version of this paper was posted on arxiv in april 2013 the estimates have been used to construct mean convex flow with surgery in 3 by brendle and huisken 5 in september 2013 and in another paper by the authors in april 2014 2016 wiley periodicals inc

ricci flows with unbounded curvature we show that any noncompact riemann surface admits a complete ricci flow g t t in 0 infty which has unbounded curvature for all t in 0 infty

complete riemannian manifolds with pointwise pinched curvature an analogous bonnet myers theorem is obtained for a complete and positively curved n dimensional n 3 riemannian manifold m n we prove that if n 4 and the curvature operator of m n is pointwise pinched or if n 3 and the ricci curvature of m 3 is pointwise pinched then m n is compact

derivation lengths and order types of knuth bendix orders abstract the derivation length function of a finite term rewriting system terminating via a knuth bendix order is shown to be bounded by the ackermann function applied to a single exponential function this result is essentially optimal as there are rewrite systems with such derivation lengths in a second part the order types of knuth bendix orders over finite signatures are classified within the ordinals up to

mean curvature flow with surgeries of two convex hypersurfaces we consider a closed smooth hypersurface immersed in euclidean space evolving by mean curvature flow it is well known that the solution exists up to a finite singular time at which the curvature becomes unbounded the purpose of this paper is to define a flow after singularities by a new approach based on a surgery procedure compared with the notions of weak solutions existing in the literature the flow with surgeries has the advantage that it keeps track of the changes of topology of the evolving surface and thus can be applied to classify all geometries that are possible for the initial manifold our construction is inspired by the procedure originally introduced by hamilton for the ricci flow and then employed by perelman in the proof of thurston s geometrization conjecture in this paper we consider initial hypersurfaces which have dimension at least three and are two convex that is such that the sum of the two smallest principal curvatures is nonnegative everywhere under these assumptions we construct a flow with surgeries which has uniformly bounded curvature until the evolving manifold is split in finitely many components with known topology as a corollary we obtain a classification up to diffeomorphism of the hypersurfaces under consideration

termination proofs for term rewriting systems by lexicographic path orderings imply multiply recursive derivation lengths abstract it is shown that a termination proof for a term rewriting system using a lexicographic path ordering yields a multiply recursive bound on the length of derivations measured in the depth of the starting term this result is essentially optimal since for every multiply recursive function a rewrite system which reduces under the lexicographic path ordering can be found such that its derivation length cannot be bounded by

a sharp bound for the inscribed radius under mean curvature flow we consider a family of embedded mean convex hypersurfaces which evolve by the mean curvature flow it follows from general results of white that the inscribed radius at each point on the hypersurface is at least frac c h where c is a positive constant that depends only on the initial data andrews recently gave a new proof of that fact using the maximum principle in this paper we show that the inscribed radius is at least frac 1 1 delta h at each point where the curvature is sufficiently large