walter s theorem for fusion systems we determine the saturated 2 fusion systems in which the centralizer of some fully centralized involution contains a component that is the 2 fusion system of a large group of lie type over a field of odd order

one sided complete stable minimal surfaces we prove that there are no complete one sided stable minimal surfaces in the euclidean 3 space we classify least area surfaces in the quotient of mathbb r 3 by one or two linearly independent translations and we give sharp upper bounds of the genus of compact two sided index one minimal surfaces in non negatively curved ambient spaces finally we estimate from below the index of complete minimal surfaces in flat spaces in terms of the topology of the surface

reduced fusion systems over 2 groups of small order we prove when s is a 2 group of order at most 29 that each reduced fusion system over s is the fusion system of a finite simple group and is tame it then follows that each saturated fusion system over a 2 group of order at most 29 is realizable what is most interesting about this result is the method of proof we show that among 2 groups with order in this range the ones which can be sylow 2 subgroups of finite simple groups are almost completely determined by criteria based on bender s classification of groups with strongly 2 embedded subgroups

reduced fusion systems over p groups with abelian subgroup of index p ii abstract for an odd prime p we look at simple fusion systems over a finite nonabelian p group s which has an abelian subgroup a of index p when s has more than one such subgroup we reduce this to a case already studied by ruiz and viruel when a is the unique abelian subgroup of index p in s and is not essential equivalently is not radical in the fusion system we give a complete list of all possibilities which can occur this includes several families of exotic fusion systems including some which have proper strongly closed subgroups

immersions minimales premiere valeur propre du laplacien et volume conforme generalisation de quelques resultats de li yau en toute dimension caracterisation de la metrique admettant des immersions minimales par ses premieres fonctions propres

the global attractor of g navier stokes equations with linear dampness on r2 in this paper the long time behaviors of g navier stokes equations with linear dampness on r 2 were investigated by using the energy equation method the existence of the global attractor for the equations was proved without the restriction of the forcing term belonging to some weighted sobolev space moreover the estimation of the hausdorff and fractal dimensions of such attractors were also obtained

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

subcentric linking systems for part of this research the author was supported by the danish national research foundation through the centre for symmetry and deformation dnrf92

dynamics of the g navier stokes equations the 2d g navier stokes equations has the following form u t u u u p f with the continuity equation gu 0 where g is a suitable smooth real valued function for the restricted function g roh showed the existence of the global attractors for the periodic boundary conditions one note that we get the 2d navier stokes equations for g 1 therefore in this paper we are interested in the behavior of the global attractors of the 2d g navier stokes equations as g 1 in the proper sense and will prove that the semiflows generated by the projection of the solutions of the g navier stokes equations into the solution space of the navier stokes equations is robust at the global attractor of the navier stokes equations with respect to g for that we will use the robustness theorem developed by sell and you

on picard groups of blocks of finite groups we show that the subgroup of the picard group of ap block ofa finite group given by bimodules with endopermutation sources modulo theautomorphism group of a source algebra is determined locally in terms of thefusion system on a defect group we show that the picard group of a block overa complete discrete valuation ringoof characteristic zero with an algebraicclosurekoffpas residue field is a colimit of finite picard groups of blocks overp adic subrings ofo we apply the results to blocks with an abelian defectgroup and frobenius inertial quotient and specialise this further to blockswith cyclic or klein four defect groups

existence and uniqueness of linking systems chermak s proof via obstruction theory we present a version of a proof by andy chermak of the existence and uniqueness of centric linking systems associated with arbitrary saturated fusion systems this proof differs from the one in ch2 in that it is based on the computation of derived functors of certain inverse limits this leads to a much shorter proof but one which is aimed mostly at researchers familiar with homological algebra

control of fixed points and existence and uniqueness of centric linking systems a chermak has recently proved that to each saturated fusion system over a finite p group there is a unique associated centric linking system b oliver extended chermak s proof by showing that all the higher cohomological obstruction groups relevant to unique existence of centric linking systems vanish both proofs indirectly assume the classification of finite simple groups we show how to remove this assumption thereby giving a classification free proof of the martino priddy conjecture concerning the p completed classifying spaces of finite groups our main tool is a 1971 result of the first author on control of fixed points by p local subgroups this result is directly applicable for odd primes and we show how a slight variation of it allows applications for p 2 in the presence of offenders

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

asymptotic behavior of stochastic g navier stokes equations on a sequence of expanding domains the limiting dynamics of stochastic 2d nonautonomous g navier stokes equations defined on a sequence of expanding domains are investigated where the limiting domain is unbounded by generalizing the energy equation method we show that the sequence of expanding cocycles is weakly equicontinuous and strongly equiasymptotically compact which lead to both existence and upper semicontinuity of the null expansion of the corresponding random attractor when the bounded domain approaches to the unbounded domain the limiting dynamics of stochastic 2d nonautonomous g navier stokes equations defined on a sequence of expanding domains are investigated where the limiting domain is unbounded by generalizing the energy equation method we show that the sequence of expanding cocycles is weakly equicontinuous and strongly equiasymptotically compact which lead to both existence and upper semicontinuity of the null expansion of the corresponding random attractor when the bounded domain approaches to the unbounded domain

on stable compact minimal submanifolds of riemannian product manifolds abstract in this paper we prove a classification theorem for stable compact minimal submanifolds of a riemannian product of an m 1 dimensional m 1 3 hypersurface m 1 in euclidean space and any riemannian manifold m 2 when the sectional curvature k m 1 of m 1 satisfies 1 m 1 1 k m 1 1 in particular when the ambient space is an m dimensional m 3 compact hypersurface m in euclidean space if the sectional curvature k m of m satisfies 1 m 1 k m 1 then we conclude that there exist no stable compact minimal submanifolds in m

the classical theory of minimal surfaces we present here a survey of recent spectacular successes in classical minimal surface theory we highlight this article with the theorem that the plane the helicoid the catenoid and the one parameter family frtgt2 0 1 of riemann minimal examples are the only complete properly embedded minimal planar domains in r 3 the proof of this result depends primarily on work of colding

minimal hypersurfaces with bounded index we prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index and volume if the ambient dimension is greater than three in a riemannian manifold m n g 3 le n le 7 can degenerate loosely speaking our results show that embedded minimal hypersurfaces with bounded index behave qualitatively like embedded stable minimal hypersurfaces up to controlled errors several compactness finiteness theorems follow from our local picture

fusion systems and amalgams we study reduced fusion systems from the point of view of their essential subgroups using the classification by goldschmidt and fan of amalgams of prime index to analyze certain pairs of such subgroups our results are applied here to study reduced fusion systems over 2 groups of order at most 64 and also reduced fusion systems over 2 groups having abelian subgroups of index two more applications are given in later papers

reduced tame and exotic fusion systems we define here two new classes of saturated fusion systems reduced fusion systems and tame fusion systems these are motivated by our attempts to better understand and search for exotic fusion systems fusion systems which are not the fusion systems of any finite group our main theorems say that every saturated fusion system reduces to a reduced fusion system which is tame only if the original one is realizable and that every reduced fusion system which is not tame is the reduction of some exotic nonrealizable fusion system

general curvature estimates for stable h surfaces in 3 manifolds applications we obtain an estimate for the norm of the second fundamental form of stable h surfaces in riemannian 3 manifolds with bounded sectional curvature our estimate depends on the distance to the boundary of the surface and on the bound on the sectional curvature but not on the manifold itself we give some applications in particular we obtain an interior gradient estimate for h sections in killing submersions

fusion systems over non resistant a2 groups abstract in this paper we study the saturated fusion systems f over potential non resistant a 2 groups of order greater than p 4 for each case in the classification f is constrained and realized by a finite group

the dimension of attractor of the 2d g navier stokes equations abstract the g navier stokes equations in spatial dimension 2 were introduced by roh as u t u u u p f with the continuity equation g u 0 where g is a suitable smooth real valued function roh proved the existence of global solutions and the global attractor for the spatial periodic and dirichlet boundary conditions roh also proved that the global attractor a g of the g navier stokes equations converges in the sense of upper continuity to the global attractor a 1 of the navier stokes equations as g 1 in the proper sense in this paper we will estimate the dimension of the global attractor a g for the spatial periodic and dirichlet boundary conditions then we will see that the upper bounds for the dimension of the global attractors a g converge to the corresponding upper bounds for the global attractor a 1 as g 1 in the proper sense

saturated fusion systems over mathcal a _2 groups in this paper we study the saturated fusion systems mathcal f over mathcal a 2 groups s of order greater than p 4 this study requires the analysis of the possible saturated fusion systems in terms of the mathcal f automorphisms of the possible mathcal f centric and mathcal f radical subgroups also for each case in the classification either s is resistant or mathcal f is constrained the classification identifies a large family of resistant groups complementing the results on resistant groups found by various authors such as green minh stancu and diaz ruiz viruel moreover the classification provides some interesting examples of saturated fusion systems which have an mathcal f essential abelian subgroup of index p

local uniformly upper semi continuity of random attractor for g navier stokes equation this paper is concerned with the locally uniform convergence from a family of pullback random attractors to a deterministic attractor we establish criteria by using joint pathwise convergence of the cocycles collective locally uniform compactness and eventually deterministic of the random attractors as an application of the abstract result it is shown that the family of random attractors for the stochastic g navier stokes equation is local uniformly upper semi continuous as the density of random noise tends to zero

automorphisms of p local compact groups abstract self equivalences of classifying spaces of p local compact groups are well understood by means of the algebraic structure that gives rise to them but explicit descriptions are lacking in this paper we use robinson s construction of an amalgam g realising a given fusion system to produce a split epimorphism from the outer automorphism group of g to the group of homotopy classes of self homotopy equivalences of the classifying space of the corresponding p local compact group