singular riemannian foliations on nonpositively curved manifolds we prove the nonexistence of a proper singular riemannian foliation admitting section in compact manifolds of nonpositive curvature then we give a global description of proper singular riemannian foliations admitting sections on hadamard manifolds in addition by using the theory of taut immersions we provide a short proof of this result in the special case of a polar action

uniform trajectory attractor for non autonomous reaction diffusion equations with caratheodory s nonlinearity abstract we consider the problem of uniform long time behavior of all globally defined weak solutions of a non autonomous reaction diffusion system with caratheodory s nonlinearity satisfying standard sign and polynomial growth assumptions the main contributions of this paper are i the existence of a uniform trajectory attractor for all globally defined weak solutions of non autonomous reaction diffusion equations with caratheodory s nonlinearity ii sufficient conditions for the existence of a uniform trajectory attractor in strongest topologies and iii new topological properties of weak solutions

uniform fatou s lemma abstract fatou s lemma is a classic fact in real analysis stating that the limit inferior of integrals of functions is greater than or equal to the integral of the inferior limit this paper introduces a stronger inequality that holds uniformly for integrals on measurable subsets of a measurable space the necessary and sufficient condition under which this inequality holds for a sequence of finite measures converging in total variation is provided this statement is called the uniform fatou lemma and it holds under the minor assumption that all the integrals in the inequality are well defined the uniform fatou lemma improves the classic fatou lemma in the following directions the uniform fatou lemma states a more precise inequality it provides the necessary and sufficient condition and it deals with variable measures various corollaries of the uniform fatou lemma are formulated the examples in this paper demonstrate that a the uniform fatou lemma may indeed provide a more accurate inequality than the classic fatou lemma b the uniform fatou lemma does not hold if convergence of measures in total variation is relaxed to setwise convergence

solutions to the overdetermined boundary problem for semilinear equations with position dependent nonlinearities abstract we show that a wide range of overdetermined boundary problems for semilinear equations with position dependent nonlinearities admits nontrivial solutions the result holds true both on r n and on compact riemannian manifolds as a byproduct of the proofs we also obtain some rigidity or partial symmetry results for solutions to overdetermined problems on riemannian manifolds of nonconstant curvature

the capelli identity the double commutant theorem and multiplicity free actions 0 the capelli identity cal 3 w p 39 is one of the most celebrated and useful formulas of classical invariant theory w d cl z the double commutant theorem w p 91 is likewise a basic result in the general theory of associative algebras both play key roles in weyl s book the classical groups the main purpose of this paper is to demonstrate a close connection between the two in the context of multiplicity free actions ekj of groups on vector spaces the focus of our discussion will be the structure of the differential operators which commute with a multiplicity free action applications include a new derivation of some formulas of shimura shl and rubenthaler and schiffmann rs for b functions associated to hermitian symmetric spaces and a construction of interesting sets of generators for the center of the universal enveloping algebra of gin we also give a detailed discussion of certain aspects of multiplicity free representations here is an overview of the contents of the paper in sect 1 we review the classical capelli identities we observe how their existence is predicted by a double commutant result 1 6 and 1 8 and we show how they can be used to compute b functions we remark that capelli understood that his operators generate the center of the algebra generated by the polarization operators see b1 p 77 also capelli s motivation in introducing his operators was to explain a formula of cayley essentially the computation of a b function capelli s point of view was remarkably modern and structuralist in certain ways more modern even than that of weyl sections 2 through 9 provide a conceptual context for understanding the capelli identities as a feature of multiplicity free actions they contain a general discussion of the structure of g the algebra of polynomial coefficient differential operators which commute with a given group g of linear transformations the results of the discussion are summarized in theorem 9 1 which says in particular that the polynomial coefficient differential operators commuting with a

the role of critical exponents in blow up theorems the sequel abstract in 27 fujita showed that for positive solutions the initial value problem in r n for u t u u p with p 1 exhibited the following behavior if p p c 1 2 n then the initial value problem does not have any nontrivial non negative solution existing on r n 0 a global solution whereas if p p c there exist global small data positive solutions as well as solutions which are non global we call such a result a blow up theorem of fujita type in 50 levine discussed the various theorems of this type that had appeared in the literature prior to 1990 in this paper we revisit the literature since 1990

submanifold geometry in symmetric spaces the classical local invariants of a submanifold in a space form are the first fundamental form the shape operators and the induced normal connection and they determine the submanifold up to ambient isometry one of the main topics in differential geometry is to study the relation between the local invariants and the global geometry and topology of submanifolds many remarkable results have been developed for submanifolds in space forms whose local invariants satisfy certain natural conditions the study of focal points of a submanifold in an arbitrary riemannian manifold arises from the morse theory of the energy functional on the space of paths in the riemannian manifold joining a fixed point to the submanifold the morse index theorem relates the geometry of a submanifold to the topology of this path space the focal structure is intimately related to the local invariants of the submanifold in the case of space forms one can go backwards and reconstruct the local invariants from the focal structure so it is not too surprising that most of the structure theory of submanifolds can be reformulated in terms of their focal structure what is perhaps surprising is a fact that became increasingly evident to the authors from their individual and joint research over the past decade while extending the theory of submanifolds to ambient spaces more general than space forms proves quite difficult if one tries to use the same approach as for the space forms at least for symmetric spaces it has proved possible to develop an elegant theory based on focal structure that reduces to the classical theory in the case of space forms this paper is an extended report on this theory and the authors believe that the methods developed herein provide important tools for a continuing study of the submanifold geometry in symmetric spaces first we set up some notations let n g be a riemannian manifold m an immersed submanifold of n and m the normal bundle of m the end point map m n of m is the restriction of the exponential map exp to m if v m x is a singular point of and the dimension of the kernel of d v is m then v is called a multiplicity m focal normal and exp v is called a multiplicity m focal point of m with respect to m in n the focal data m is defined to be the set of all pairs v m such that v is a multiplicity m focal normal of m the focal variety v m is the set of all pairs v m with v m m the main purpose of our paper is to study the global geometry and topology

coisotropic and polar actions on complex grassmannians the main result of the paper is the complete classification of the compact connected lie groups acting coisotropically on complex grassmannians this is used to determine the polar actions on the same manifolds

blow up and global existence of solutions to an inhomogeneous parabolic system abstract we study the critical phenomenon of a weakly coupled inhomogeneous parabolic system which is largely unexplored even though the homogeneous problem has been widely studied the conditions on the powers of the nonlinear terms are new compared with the well known homogeneous case the method appears to be different from the standard ones in the literature an application to the lame emden system is also given

parallel focal structure and singular riemannian foliations we give a necessary and sufficient condition for a submanifold with parallel focal structure to give rise to a global foliation of the ambient space by parallel and focal manifolds we show that this is a singular riemannian foliation with complete orthogonal transversals for this object we construct an action on the transversals that generalizes the weyl group action for polar actions

on the connectedness and asymptotic behaviour of solutions of reaction diffusion systems abstract in this paper we consider reaction diffusion systems in which the conditions imposed on the nonlinearity provide global existence of solutions of the cauchy problem but not uniqueness we prove first that for the set of all weak solutions the kneser property holds that is that the set of values attained by the solutions at every moment of time is compact and connected further we prove the existence and connectedness of a global attractor in both the autonomous and nonautonomous cases the obtained results are applied to several models of physical or chemical interest a model of fractional order chemical autocatalysis with decay the fitz hugh nagumo equation and the ginzburg landau equation

existence of solutions for an inhomogeneous fractional semilinear heat equation abstract we obtain necessary conditions and sufficient conditions on the existence of solutions to the cauchy problem for a fractional semilinear heat equation with an inhomogeneous term we identify the strongest spatial singularity of the inhomogeneous term for the solvability of the cauchy problem

blow up results and global existence of positive solutions for the inhomogeneous evolution p laplacian equations abstract this paper deals with the cauchy problem of inhomogeneous evolution p laplacian equations t u div u p 2 u u q w x with nonnegative initial data where p 1 q max 1 p 1 and w x 0 is a nonnegative continuous functions in r n we prove that q c p 1 n n p is its critical exponent provided that 2 n n 1 p n i e if q q c then every positive solution blows up in finite time whereas for q q c the equation possesses a global positive solution for some w x and some initial data meanwhile we also prove that its positive solutions blow up in finite time provided that n p

polar coordinates induced by actions of compact lie groups let g be a connected lie subgroup of the real orthogonal group 0 n for the action of g on r we construct linear subspaces a that intersect all orbits we determine for which g there exists such an a meeting all the g orbits orthogo nally groups that act transitively on spheres are obvious examples with few exceptions all possible g arise as the isotropy subgroups of riemannian symmetric spaces introduction let g be a compact lie group acting on a real vectorspace v and let k be a g invariant inner product on v having a linear cross section a c v of

a classification of multiplicity free actions abstract the action of a complex algebraic groupgon an affine varietyvis said to be multiplicity free if each irreducible representation ofgoccurs at most once in the ring c v of regular functions onv this paper concerns the classical setting wherevis a vector space andgacts linearly onv the irreducible linear multiplicity free actions have been classified by v kac the focus here is on nonirreducible linear multiplicity free actions we exhibit many such actions which do not decompose as direct products of irreducible actions our main results provide a classification for all linear multiplicity free actions on a vector space

taut submanifolds and foliations we give an equivalent description of taut submanifolds of complete riemannian manifolds as exactly those submanifolds whose normal exponential map has integrable fibers it turns out that every taut submanifold is also mathbb z 2 taut so that tautness is essentially the same as mathbb z 2 tautness in the case where the normal exponential map of a submanifold has integrable fibers we explicitely construct generalized bott samelson cycles for the critical points of the energy functionals on the path spaces which generically represent a basis for the mathbb z 2 cohomology we also consider singular riemannian foliations all of whose leaves are taut and discuss some of their main features using our characterization of taut submanifolds we are able to show that tautness of a singular riemannian foliation is actually a property of the quotient

polar actions on compact rank one symmetric spaces are taut we prove that the orbits of a polar action of a compact lie group on a compact rank one symmetric space are tautly embedded with respect to z2 coefficients

on the kneser property for the complex ginzburg landau equation and the lotka volterra system with diffusion as we have seen in the previous chapters when we consider the cauchy problem of a differential equation and uniqueness fails to hold or it is not known to hold then we have to consider a set of solutions corresponding to a given initial data

the critical exponents for the quasi linear parabolic equations with inhomogeneous terms abstract this paper deals with the critical exponents for the quasi linear parabolic equations in r n and with an inhomogeneous source or in exterior domains and with inhomogeneous boundary conditions for n 3 2 n and p max 1 1 we obtain that p c n 1 n 2 is the critical exponent of these equations furthermore we prove that if max 1 1 p p c then every positive solution of these equations blows up in finite time whereas these equations admit the global positive solutions for some f x and some initial data u 0 x if p p c meantime we also demonstrate that every positive solution of these equations blows up in finite time provided n 1 2 1 and p max 1 1

pullback attractors for a class of extremal solutions of the 3d navier stokes system the study of the asymptotic behavior of the weak solutions of the three dimensional 3d for short navier stokes system is a challenging problem which is still far to be solved in a satisfactory way in particular the existence of a global attractor in the strong topology is an open problem for which only some partial or conditional results are given see 3 4 6 15 17 19 20 27 38 concerning the existence of trajectory attractors some results are proved in 13 18 36 the main difficulty in this problem but not the only one is to prove the asymptotic compactness of solutions see 2 for a review on these questions

coisotropic and polar actions on compact irreducible hermitian symmetric spaces we obtain the full classification of coisotropic and polar isometric actions of compact lie groups on irreducible hermitian symmetric spaces

critical exponents of fujita type for inhomogeneous parabolic equations and systems abstract we consider the large time behavior of sign changing solutions of inhomogeneous parabolic equations and systems for example for u t u u p w x in r n 0 t we show the following if n 3 and r n w x dx 0 and 1 p n n 2 then all solutions blow up in finite time while if p n n 2 there are both global and nonglobal solutions we show by example that global solutions exist for all p 1 and w satisfying r n w x dx n 1 2 and r n w x dx 0 no solution can exist for all time extensions of the above result to various geometries and some other problems are indicated

invariance and stability of global attractors for multi valued impulsive dynamical systems abstract in this paper we consider infinite dimensional impulsive dynamical systems generated by evolution problems with impulses at the moments of intersection of trajectories with given subset of the phase space uniqueness of the corresponding cauchy problem is not assumed so we use multi valued semiflows to describe the long time dynamics of such systems considering global attractor as a compact minimal uniformly attracting set of the corresponding multi valued semiflow we propose a criterion of existence of such a set under additional mild assumptions we prove the invariance of the global attractor despite the fact that our approach does not require the invariance property of the set in the definition of the global attractor moreover we prove that the obtained global attractor is stable under perturbation of the right hand part of the evolution problem we also provide examples which show that in general the global attractor is not preserved under perturbation of impulsive parameters

hyperpolar homogeneous foliations on symmetric spaces of noncompact type a foliation f on a riemannian manifold m is homogeneous if its leaves coincide with the orbits of an isometric action on m a foliation f is polar if it admits a section that is a connected closed totally geodesic submanifold of m which intersects each leaf of f and intersects orthogonally at each point of intersection a foliation f is hyperpolar if it admits a flat section these notes are related to joint work with jose carlos diaz ramos and hiroshi tamaru about hyperpolar homogeneous foliations on riemannian symmetric spaces of noncompact type apart from the classification result which we proved in 1 we present here in more detail some relevant material about symmetric spaces of noncompact type and discuss the classification in more detail for the special case m slr 1 r sor 1 1 isometric actions on the real hyperbolic plane the special linear group g sl2 r a b c d a b c d r ad bc 1 acts on the upper half plane h z c z 0 by linear fractional transformations of the form a b c d z az b cz d by equipping h with the riemannian metric ds dx dy y2 dzdz z 2 z x iy

polar and coisotropic actions on kahler manifolds the main result of the paper is that a polar action on a compact irreducible homogeneous kahler manifold is coisotropic this is then used to give new examples of polar actions and to classify coisotropic and polar actions on quadrics

a classification of hyperpolar and cohomogeneity one actions an isometric action of a compact lie group on a riemannian manifold is called hyperpolar if there exists a closed connected submanifold that is flat in the induced metric and meets all orbits orthogonally in this article a classification of hyperpolar actions on the irreducible riemannian symmetric spaces of compact type is given since on these symmetric spaces actions of cohomogeneity one are hyperpolar i e normal geodesics are closed we obtain a classification of the homogeneous hypersurfaces in these spaces by computing the cohomogeneity for all hyperpolar actions this result implies a classification of the cohomogeneity one actions on compact strongly isotropy irreducible homogeneous spaces

cohomogeneity one actions on noncompact symmetric spaces of rank one we classify up to orbit equivalence all cohomogeneity one actions on the hyperbolic planes over the complex quaternionic and cayley numbers and on the complex hyperbolic spaces ch n 3 for the quaternionic hyperbolic spaces mh n n 3 we reduce the classification problem to a problem in quaternionic linear algebra and obtain partial results for real hyperbolic spaces this classification problem was essentially solved by elie cartan

the critical exponent of doubly singular parabolic equations abstract in this paper we study the cauchy problem of doubly singular parabolic equations u t div u u m t s x u p with non negative initial data here 1 m max 0 1 2 n satisfying 0 m 1 p 1 and s 0 we prove that if max 2 1 s n 1 m 2 then p c m m 1 s 2 1 s n 1 is the critical exponent i e if 1 p p c then every non trivial solution blows up in finite time but for p p c a positive global solution exists

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

curvature and symmetry of milnor spheres since milnor s discovery of exotic spheres mi one of the most intriguing problems in riemannian geometry has been whether there are exotic spheres with positive curvature it is well known that there are exotic spheres that do not even admit metrics with positive scalar curvature hi on the other hand there are many examples of exotic spheres with positive ricci curvature cf ch1 he po and na and this work recently culminated in wr where it is shown that every exotic sphere that bounds a parallelizable manifold has a metric of positive ricci curvature this includes all exotic spheres in dimension 7 so far however no example of an exotic sphere with positive sectional curvature has been found in fact until now only one example of an exotic sphere with nonnegative sectional curvature was known the so called gromoll meyer sphere gm in dimension 7 as one of our main results we prove

polar actions on complex hyperbolic spaces we classify polar actions on complex hyperbolic spaces up to orbit equivalence 1 introduction and main results a proper isometric lie group action on a riemannian manifold is called polar if there exists an immersed submanifold that meets every orbit orthogonally such a submanifold is then called a section of the action in the special case where the section is flat in its induced riemannian metric the action is called hyperpolar in this article we classify polar actions on complex hyperbolic spaces the motivation for our work can be traced back to the work of dadok 12 who classified polar representations on euclidean spaces and the paper of palais and terng 30 who proved fundamental properties of polar actions on riemannian manifolds several years later the problem of classifying hyperpolar actions on symmetric spaces of compact type was posed in 18 hyperpolar actions on irreducible symmetric spaces of compact type have been classified by the third named author in 20 the classification of polar actions on compact symmetric spaces of rank one was obtained by podesta and thorbergsson 31 this classification shows that there are finitely many examples of polar non hyperpolar actions on each compact symmetric space of rank one after having completed the classification of polar actions on irreducible hermitian sym metric spaces of compact type biliotti 11 formulated the following conjecture a polar action on an irreducible symmetric space of compact type and higher rank is hyperpolar the third author proved that the conjecture holds for symmetric spaces with simple isom etry group 21 and for the exceptional simple lie groups 22 in recent work lytchak 26 obtained a decomposition theorem for the more general case of singular polar foliations on nonnegatively curved not necessarily irreducible riemannian symmetric spaces in particular his result shows that polar actions on irreducible symmetric spaces of higher rank are hyperpolar if the cohomogeneity is at least three kollross and lytchak 24 then completed the proof that biliotti s conjecture holds and hence that the classification of polar actions on irreducible symmetric spaces of compact type follows from 20 and 31 note that the classification of polar actions on reducible symmetric spaces cannot be ob tained from the corresponding classification in irreducible ones however by the structural

a general theory of canonical forms si g est un groupe de lie compact et m une g variete riemannienne d orbites principales de codimension k alors une section ou une forme canonique pour m est une sous variete fermee lisse a k dimensions de m qui rencontre toutes les orbites de m orthogonalement

a new critical phenomenon for semilinear parabolic problems abstract we study the cauchy problem of the inhomogeneous semilinear parabolic equations u up ut w 0 onmn 0 with initial valueu0 0 wheremnis a riemannian manifold with possibly nonnegative ricci curvature there is an exponentp which is critical in the following sense when 1 p the problem has a global positive solution for somew w x 0 and someu0 0