majorization doubly stochastic matrices and comparison of eigenvalues abstract 1 basic properties of majorization 2 isotone maps and algebraic operations 3 double sub and superstochasticity 4 doubly stochastic matrices 5 doubly stochastic matrices with minimum permanent 6 comparison of eigenvalues 7 doubly stochastic maps

some ways of constructing furuta type inequalities abstract let h be an operator mean associated with an operator monotone function h and let a b be positive operators we investigate the following furuta type inequality for some fixed continuous function h a b 0 a r r x h r b r 1 where x h is the positive solution of h x b 1 the map h r r plays a central role in constructing a furuta type inequality similar to the role of the map p r r 1 p r in the following part of the furuta inequality a b 0 p r 1 b r 2 a p b r 2 1 r p r b p r 1 r p r we obtain this map h r r from the following ando hiai type inequality a b 0 a h b 1 a r h b r 1 r 1

the golden thompson trace inequality is complemented we prove a class of trace inequalities which complements the golden thompson inequality for example tr epa epb 2 p tr ea b holds for all p 0 when a and b are hermitian matrices and denotes the geometric mean we also prove related trace inequalities involving the logarithmic function namely p 1tr x log yp 2xpyp 2 tr x log x log y p 1tr x log xp 2ypxp 2 for all p 0 when x and y are nonnegative matrices these inequalities supply lower and upper bounds on the relative entropy

large isoperimetric regions in asymptotically hyperbolic manifolds we show the existence of isoperimetric regions of sufficiently large volumes in general asymptotically hyperbolic three manifolds furthermore we show that large coordinate spheres are uniquely isoperimetric in manifolds that are schwarzschild anti desitter at infinity these results have important repercussions for our understanding of spacelike hypersurfaces in lorentzian space times which are asymptotic to null infinity in fact as an application of our results we verify the asymptotically hyperbolic penrose inequality in the special case of the existence of connected isoperimetric regions of all volumes

a minkowski inequality for hypersurfaces in the anti de sitter schwarzschild manifold we prove a sharp inequality for hypersurfaces in the n dimensional anti de sitter schwarzschild manifold for general n 3 this inequality generalizes the classical minkowski inequality for surfaces in the three dimensional euclidean space and has a natural interpretation in terms of the penrose inequality for collapsing null shells of dust the proof relies on a new monotonicity formula for inverse mean curvature flow and uses a geometric inequality established by the first author in 3 c 2015 wiley periodicals inc

multiple solutions of periodic boundary value problems for first order differential equations by applying the well known leggett williams multiple fixed point theorem and fixed point theorem of cone expansion and compression this paper investigates the existence of multiple positive solutions of periodic boundary value problems for first order differential equations meanwhile two examples are worked out to demonstrate the main results

isoperimetric and weingarten surfaces in the schwarzschild manifold we show that any star shaped convex hypersurface with constant weingarten curvature in the desitter schwarzschild manifold is a sphere of symmetry moreover we study an isoperimetric problem for bounded domains in the doubled schwarzschild manifold we prove the existence of an isoperimetric surface for any value of the enclosed volume and we completely describe the isoperimetric surfaces for very large enclosed volume this complements work in h bray s thesis where isoperimetric surfaces homologous to the horizon are studied

some sphere theorems in linear potential theory in this paper we analyze the capacitary potential due to a charged body in order to deduce sharp analytic and geometric inequalities whose equality cases are saturated by domains with spherical sy

unique isoperimetric foliations of asymptotically flat manifolds in all dimensions the question of isoperimetry what is the largest amount of volume that can be enclosed by a given amount of area can be traced back to antiquity 1 the first mathematically rigorous results are as recent as the nineteenth century the question of isoperimetry and its close relative the analysis of minimal surfaces are two of the model problems of the geometric calculus of variations the list of geometries where an explicit answer to the question of isoperimetry is available is short we provide an overview of available results in appendix h in this paper and 22 we extend this list by a class of riemannian manifolds m g for which we describe all large isoperimetric regions completely we refer the reader to sect 2 for the precise definitions of all terms in the statement of our first main theorem

large outlying stable constant mean curvature spheres in initial data sets we give examples of asymptotically flat three manifolds m g which admit arbitrarily large constant mean curvature spheres that are far away from the center of the manifold this resolves a question raised by huisken and yau invent math 124 281 311 1996 on the other hand we show that such surfaces cannot exist when m g has nonnegative scalar curvature this result depends on an intricate relationship between the scalar curvature of the initial data set and the isoperimetric ratio of large stable constant mean curvature surfaces

a generalization of araki s log majorization abstract we generalize araki s log majorization to the log convexity theorem for the eigenvalues of a p 1 2 b p a p 1 2 as a function of p 0 where a b are positive semidefinite matrices and are positive linear maps between matrix algebras similar generalizations of the log majorization of ando hiai type for the weighted geometric mean a b are also given including a lemma on general operator means a e of a and a projection e

twin solutions to singular semipositone problems abstract in this paper by a specially constructed cone and the fixed point index theory we investigate the existence of multiple positive solutions for the following singular semipositone problem y f t y 0 t 0 1 y 0 y 1 0 the nonlinear term f t y may be singular at t 0 t 1 and y 0 also may be negative for some values of t and y and is a positive parameter

embedded area constrained willmore tori of small area in riemannian three manifolds i minimization we construct embedded willmore tori with small area constraint in riemannian three manifolds under some curvature condition used to prevent mobius degeneration the construction relies on a lyapunov schmidt reduction to this aim we establish new geometric expansions of exponentiated small symmetric clifford tori and analyze the sharp asymptotic behaviour of degenerating tori under the action of the mobius group in this first work we prove two existence results by minimizing or maximizing a suitable reduced functional in particular we obtain embedded area constrained willmore tori or equivalently toroidal critical points of the hawking mass under area constraint in compact 3 manifolds with constant scalar curvature and in the double schwarzschild space in a forthcoming paper new existence theorems will be achieved via morse theory

log majorization and complementary golden thompson type inequalities abstract we obtain a log majorization result for power means of positive semidefinite matrices this implies matrix norm inequalities for unitarily invariant norms which are considered as complementary to the golden thompson one other log majorization results are also obtained we give logarithmic trace inequalities and determinant inequalities as applications of our log majorizations

on large volume preserving stable cmc surfaces in initial data sets let m g be a complete 3 dimensional asymptotically flat manifold with everywhere positive scalar curvature we prove that given a compact subset k subset m all volume preserving stable constant mean curvature surfaces of sufficiently large area will avoid k this complements the results of g huisken and s t yau and of j qing and g tian on the uniqueness of large volume preserving stable constant mean curvature spheres in initial data sets that are asymptotically close to schwarzschild with mass m gt 0 the analysis in g huisken and s t yau and in j qing and g tian takes place in the asymptotic regime of m here we adapt ideas from the minimal surface proof of the positive mass theorem by r schoen and s t yau and develop geometric properties of volume preserving stable constant mean curvature surfaces to handle surfaces that run through the part of m that is far from euclidean

embedded weingarten tori in s3 abstract in this paper we show that an embedded weingarten surface in s 3 of genus 1 must be rotationally symmetric provided that certain structure conditions are satisfied the argument involves an adaptation of our proof of lawson s conjecture for embedded minimal tori

jensen and minkowski inequalities for operator means and anti norms abstract jensen inequalities for positive linear maps of choi and hansen pedersen type are established for a large class of operator matrix means such as some p means and some kubo ando means these results are also extensions of the minkowski determinantal inequality to this end we develop the study of anti norms a notion parallel to the symmetric norms in matrix analysis including functionals like schatten q norms for a parameter q 1 and the minkowski functional det 1 n a

existence and characterization of regions minimizing perimeter under a volume constraint inside euclidean cones we study the problem of existence of regions separating a given amount of volume with the least possible perimeter inside a euclidean cone our main result shows that nonexistence for a given volume implies that the isoperimetric profile of the cone coincides with the one of the half space this allows us to give some criteria ensuring existence of isoperimetric regions for instance local convexity of the cone at some boundary point we also characterize which are the stable regions in a convex cone i e second order minima of perimeter under a volume constraint from this it follows that the isoperimetric regions in a convex cone are the euclidean balls centered at the vertex intersected with the cone