tauberian theorems and stability of one parameter semigroups the main result is the following stability theorem let t t t tzo be a bounded co semigroup on a reflexive space x denote by a the generator of t and by a a the spectrum of a if a a n ir is countable and no eigenvalue of a lies on the imaginary axis then limt boo t t x o for all x e x

geodesics in the space of kahler cone metrics i in this paper we study the dirichlet problem of the geodesic equation in the space of k ahler cone metrics cal h cal b that is equivalent to a homogeneous complex monge amp ere equation whose boundary values consist of k ahler metrics with cone singularities our approach concerns the generalization of the function space defined by donaldson to the case of k ahler manifolds with boundary moreover we introduce a subspace cal h c of cal h cal b which we define by prescribing appropriate geometric conditions our main result is the existence uniqueness and regularity of c 1 1 cal b geodesics whose boundary values lie in cal h c moreover we prove that such geodesic is the limit of a sequence of c 2 alpha cal b approximate geodesics under the c 1 1 cal b norm as a geometric application we prove the metric space structure of cal h c

on power bounded operators abstract the main result asserts that for any contraction t on an arbitrary banach space x t n t n 1 0 as n if and only if the spectrum of t has no points on the unit circle except perhaps z 1 this theorem is extended for t t n where is a function of spectral synthesis on the unit circle as an application we generalize the so called zero two law of ornstein and sucheston and zaharopol to positive contraction on a very large class of banach lattices

geodesic convexity of small neighborhood in the space of kahler potentials abstract we show that given k 4 0 j min 1 4 k 4 4 any point in the space of non degenerate smooth kahler potentials has a small neighborhood with respect to c k norm s t any two points in this neighborhood can be connected by a geodesic of at least c k j regularity

strong stability of bounded evolution families and semigroups abstract we prove several characterizations of strong stability of uniformly bounded evolution families u t s t s 0 of bounded operators on a banach space x i e we characterize the property lim t u t s x 0 for all s 0 and all x x these results are connected to the asymptotic stability of the well posed linear nonautonomous cauchy problem u t a t u t t s 0 u s x x x in the autonomous case i e when u t s t t s for some c 0 semigroup t t t 0 we present in addition a range condition on the generator a of t t t 0 which is sufficient for strong stability this condition is more general than the condition in the ablv theorem involving countability of the imaginary part of the spectrum of a

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

minimal 2 spheres in 3 spheres we prove that any manifold diffeomorphic to s3 and endowed with a generic metric contains at least two embedded minimal 2 spheres the existence of at least one minimal 2 sphere was obtained by simon and smith in 1983 our approach combines ideas from min max theory and mean curvature flow we also establish the existence of smooth mean convex foliations in 3 manifolds we apply our methods to solve a problem posed by s t yau in 1987 on whether the planar 2 spheres are the only minimal spheres in ellipsoids centered about the origin in r4 finally considering the example of degenerating ellipsoids we show that the assumptions in the multiplicity 1 conjecture and the equidistribution of widths conjecture are in a certain sense sharp

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

isoperimetry for semilinear torsion problems in riemannian two manifolds abstract given m 2 g a smooth complete riemannian 2 manifold this note discusses three types of isoperimetric properties arising from u z o g o g z u d a g the non negative solution to the semilinear 0 1 torsion problem g u u in o subject to u o 0 for a relatively compact domain o m 2 with smooth boundary o

almost rigidity and the extinction time of positively curved ricci flows we prove that ricci flows with almost maximal extinction time must be nearly round provided that they have positive isotropic curvature when crossed with mathbb r 2 as an application we show that positively curved metrics on s 3 and rp 3 with almost maximal width must be nearly round

the extremal function associated to intrinsic norms through the study of the degenerate complex monge ampere equation we establish the optimal regularity of the extremal function associated to intrinsic norms of chern levine nirenberg and bedford taylor we prove a conjecture of chern levine nirenberg on the extended intrinsic norms on complex manifolds and verify bedford taylor s representation formula for these norms in general

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

complex legendre duality we introduce complex generalizations of the classical legendre transform operating on kahler metrics on a compact complex manifold these legendre transforms give explicit local isometric symmetries for the mabuchi metric on the space of kahler metrics around any real analytic kahler metric answering a question originating in semmes work

theorems of katznelson tzafriri type for contractions for t a contraction on a hilbert space and a function in the disc algebra we show that if 0 on spt then limn t tn 0 the same method gives a new proof of recent results of katznelson tzafriri several applications are given

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

estimates for the extinction time for the ricci flow on certain 3 manifolds and a question of perelman we show that the ricci flow becomes extinct in finite time on any riemannian 3 manifold without aspherical summands in this note we prove some bounds for the extinction time for the ricci flow on certain 3 manifolds our interest in this comes from a question that grisha perelman asked the first author at a dinner in new york city on april 25th of 2003 his question was what happens to the ricci flow on the 3 sphere when one starts with an arbitrary metric in particular does the flow become extinct in finite time he then went on to say that one of the difficulties in answering this is that he knew of no good way of constructing minimal surfaces for such a metric in general however there is a natural way of constructing such surfaces and that comes from the min max argument where the minimal of all maximal slices of sweep outs is a minimal surface see for instance 2 the idea is then to look at how the area of this min max surface changes under the flow geometrically the area measures a kind of width of the 3 manifold and as we will see for certain 3 manifolds those like the 3 sphere whose prime decomposition contains no aspherical factors the area becomes zero in finite time corresponding to that the solution becomes extinct in finite time moreover we will discuss a possible lower bound for how fast the area becomes zero very recently perelman posted a paper see 9 answering his original question about finite extinction time however even after the appearance of his paper then we still think that our slightly different approach may be of interest in part because it is in some ways geometrically more natural in part because it also indicates that lower bounds should hold and in part because it avoids using the curve shortening flow that he simultaneously with the ricci flow needed to invoke and thus our approach is in some respects technically easier let m 3 be a smooth closed orientable 3 manifold and let g t be a one parameter family of metrics on m evolving by the ricci flow so

dissipative operators on banach spaces a bounded linear operator t on a banach space is said to be dissipative if ett 1 for all t 0 we show that if t is a dissipative operator on a banach space then a limt ettt sup t ir b if t ir is contained in i 2 i 2 then limt ettsint sup sin t ir some related problems are also discussed

some results of katznelson tzafriri type abstract for semigroups s r z and z k we show that if t is a representation of s by contractions on a hilbert space then inf t s t t f t sup f sp u t s where f l 1 s and sp u t s is the unitary spectrum of t with respect to s for t a representation of a suitable semigroup s by contractions on a banach space we give sharp conditions on sp u t s which guarantee that the equality above holds these conditions concern the thinness of sp u t s in the harmonic analysis sense these results are related to theorems of katznelson tzafriri type which give conditions guaranteeing that inf t s t t f t vanishes