classification of rotational special weingarten surfaces of minimal type in mathbb s 2 times mathbb r and mathbb h 2 times mathbb r in this paper we classify the complete rotational special weingarten surfaces in mathbb s 2 times mathbb r and mathbb h 2 times mathbb r i e rotational surfaces in mathbb s 2 times mathbb r and mathbb h 2 times mathbb r whose mean curvature h and extrinsic curvature ke satisfy h f h2 ke for some function f in mathcal c 1 0 infty such that f 0 0 and 4x f x 2 1 for any x 0 furthermore we show the existence of non complete examples of such surfaces

the codazzi equation for surfaces abstract in this paper we study the classical codazzi equation in space forms from an abstract point of view and use it as an analytic tool to derive global results for surfaces in different ambient spaces in particular we study the existence of holomorphic quadratic differentials the uniqueness of immersed spheres in geometric problems height estimates and the geometry and uniqueness of complete or properly embedded weingarten surfaces

convexity at infinity in cartan hadamard manifolds and applications to the asymptotic dirichlet and plateau problems we study the asymptotic dirichlet and plateau problems on cartan hadamard manifolds satisfying the so called strict convexity abbr sc condition the main part of the paper consists in studying the sc condition on a manifold whose sectional curvatures are bounded from above and below by certain functions depending on the distance to a fixed point in particular we are able to verify the sc condition on manifolds whose curvature lower bound can go to infty and upper bound to 0 simultaneously at certain rates or on some manifolds whose sectional curvatures go to infty faster than any prescribed rate these improve previous results of anderson borbely and ripoll and telichevsky we then solve the asymptotic plateau problem for locally rectifiable currents with mathbb z 2 multiplicity in acartan hadamard manifold satisfying the sc condition given any compact topologically embedded k 1 dimensional submanifold of partial infty m 2 le k le n 1 as the boundary data we also solve the asymptotic plateau problem for locally rectifiable currents with mathbb z multiplicity on any rotationally symmetric manifold satisfying the sc condition given a smoothly embedded submanifold as the boundary data these generalize previous results of anderson bangert and lang moreover we obtain new results on the asymptotic dirichlet problem for a large class of pdes in particular we are able to prove the solvability of this problem on manifolds with super exponential decay to infty of the curvature

area bounds for free boundary minimal surfaces in a geodesic ball in the sphere abstract we extend to higher dimensions earlier sharp bounds for the area of two dimensional free boundary minimal surfaces contained in a geodesic ball of the round sphere this follows work of brendle and fraser schoen in the euclidean case

the inverse mean curvature flow and the riemannian penrose inequality let m be an asymptotically flat 3 manifold of nonnegative scalar curvature the riemannian penrose inequality states that the area of an outermost minimal surface n in m is bounded by the adm mass m according to the formula n 16 m 2 we develop a theory of weak solutions of the inverse mean curvature flow and employ it to prove this inequality for each connected component of n using geroch s monotonicity formula for the adm mass our method also proves positivity of bartnik s gravitational capacity by computing a positive lower bound for the mass purely in terms of local geometry

a rigidity result for overdetermined elliptic problems in the plane let f 0 r be a locally lipschitz function and r 2 a c 1 domain whose boundary is unbounded and connected if there exists a positive bounded solution to the overdetermined elliptic problem u f u 0 in u 0 on u 1 on we prove that is a half plane in particular we obtain a partial answer to a question raised by h berestycki l caffarelli and l nirenberg in 1997

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

analytic and geometric background of recurrence and non explosion of the brownian motion on riemannian manifolds we provide an overview of such properties of the brownian motion on complete non compact riemannian manifolds as recurrence and non explosion it is shown that both properties have various analytic characterizations in terms of the heat kernel green function liouville properties etc on the other hand we consider a number of geometric conditions such as the volume growth isoperimetric inequalities curvature bounds etc which are related to recurrence and non explosion

nonintegrable dynamics of the triplet triplet spatiotemporal interaction in this paper we examine the coupling of two wave triplets sharing two common modes the analysis is performed in the solitonic sector of the parameter space where uncoupled solutions departing from linearly unstable homogeneous initial conditions evolve into a collection of regularly interspersed spatiotemporally localized spikes the uncoupled system is integrable but coupling destroys integrability as coupling grows energy transfer to smaller spatial scales does appear and becomes faster not only in linearly unstable but also in linearly stable cases chaos in low dimensional subsystems appears to be responsible for the transfer we perform a series of numerical tests to verify this idea

principal lines on surfaces immersed with constant mean curvature configurations of lines of principal curvature on constant mean curva ture immersed surfaces are studied analytical models for these configurations near general isolated umbilical points and particular types of ends are found from the existence of transversal invariant measures for the foliations by principal lines established here follows that the union of recurrent lines of principal curvature is an open set examples illustrating all possible cases are given 1 introduction this paper is devoted to the study of the possible configurations of lines of principal curvature and umbilical points of surfaces immersed in r3 with constant mean curvature except in the case where the image of the immersion is contained in a plane or a sphere the umbilical points are isolated 10 11 a very precise analytical model is found for the family of principal lines near the isolated umbilical points this model is related to a result of h hopf 10 11 which permits the computation of the index of the umbilical point the behaviour of the principal lines at infinity near particular types of ends called here elementary ends of the immersion is also described through precise analytical models in connection with the global behaviour of the principal lines it is shown that their foliations have a natural smooth transversal invariant measure this implies that the compact principal lines cycles appear in cylindrical open sets and that the closures of recurrent principal lines have nonempty interior examples illustrating all possible situations are given however the recurrent lines have been found only for immer sions of tori into spaces which are not more than locally isometric to r3 the existence of such immersions into r3 seems to be a very difficult question related to a conjecture of h hopf 24 p 684 problem 63 2 from the point of view of principal lines an immersion with constant mean curvature presents a global configuration which is quite opposite to that of the principally structurally stable immersions studied in 4 5 for a large class of this kind of immersions it is shown that the principal cycles are isolated and do not present recurrent lines 4 5 examples of immersions of the sphere and torus into

mean convex hulls and least area disks spanning extreme curves we show that for any extreme curve in a 3 manifold m there exist a canonical mean convex hull containing all least area disks spanning the curve similar result is true for asymptotic case in open image in new window such that for any asymptotic curve open image in new window there is a canonical mean convex hull containing all minimal planes spanning applying this to quasi fuchsian manifolds we show that for any quasi fuchsian manifold there exist a canonical mean convex core capturing all essential minimal surfaces on the other hand we also show that for a generic c3 smooth curve in the boundary of c3 smooth mean convex domain in 3 there exist a unique least area disk spanning the curve

isoperimetric inequalities for complete proper minimal submanifolds in hyperbolic space the classical isoperimetric inequality for a domain sigma subset mathbb r k with smooth boundary partial sigma is displaystyle k k omega k mathrm vol sigma k 1 leq mathrm vol partial sigma k where equality holds if and only if sigma is a ball in mathbb r k

differential operators on monomial curves abstract let k be an algebraically closed field of characteristic 0 let n 0 be a numerical semigroup and let a k be the corresponding semigroup algebra we give an explicit description of the ring d d a of k linear differential operators on a the associated graded ring grd a and the module of derivations derk a we also classify all graded d modules which are finitely generated torsion free a modules of rank 1 considered as modules over the sub ring a d a

characterization of f extremal disks abstract we show uniqueness for overdetermined elliptic problems defined on topological disks with c 2 boundary i e positive solutions u to u f u 0 in m 2 g so that u 0 and u c t e along the unit outward normal along under the assumption of the existence of a candidate family to do so we adapt the galvez mira generalized hopf type theorem 19 to the realm of overdetermined elliptic problem when m 2 g is the standard sphere s 2 and f is a c 1 function so that f x 0 and f x x f x for any x r we construct such candidate family considering rotationally symmetric solutions this proves the berestycki caffarelli nirenberg conjecture in s 2 for this choice of f more precisely this shows that if u is a positive solution to u f u 0 on a topological disk s 2 with c 2 boundary so that u 0 and u c t e along then must be a geodesic disk and u is rotationally symmetric in particular this gives a positive answer to the schiffer conjecture d cf 33 35 for the first dirichlet eigenvalue and classifies simply connected harmonic domains cf 28 also called serrin problem in s 2

radial symmetry for elliptic boundary value problems on exterior domains by the alexandroff serrin method 2 14 of moving hyperplanes we obtain radial symmetry for the domain and the solutions of on an exterior domain subject to the overdetermined boundary conditions on at and in in particular the following conjecture from potential theory due to p gruber cf 11 8 is proved let or be a bounded smooth domain with a constant source distribution on and let be the induced single layer potential if is constant in then is a ball

interior curvature estimates and the asymptotic plateau problem in hyperbolic space we show that for a very general class of curvature functions defined in the positive cone the problem of finding a complete strictly locally convex hypersurface in h satisfying f 0 1 with a prescribed asymptotic boundary at infinity has at least one smooth solution with uniformly bounded hyperbolic principal curvatures moreover if is euclidean starshaped the solution is unique and also euclidean starshaped while if is mean convex the solution is unique we also show via a strong duality theorem that analogous results hold in de sitter space a novel feature of our approach is a global interior curvature estimate

on the geometry of static spacetimes we review geometrical properties of a static spacetime m g including geodesic completeness causality standard splittings compact m closed geodesics and geodesic connectedness we pay special attention to the critical quadratic behaviour at infinity of the coefficients 1 g k k being k a timelike irrotational killing vector field which essentially control completeness causality and geodesic connectedness recent references are specially discussed

d module structure of local cohomology modules of toric algebras let s be a toric algebra over a field k of characteristic 0 and let i be a monomial ideal of s we show that the local cohomology modules h i i s are of finite length over the ring of differential operators d s k generalizing the classical case of a polynomial algebra s as an application we compute the characteristic cycles of some local cohomology modules

the presence of an invariant manifold in the dissipative nonlinear four wave coupling abstract we examine the existence of an invariant manifold in the phase space structure presented by a dissipative four wave coupling we show that once in such manifold the system preserves two quantities as time evolves a quantity related to the energy of the system and another related to the exchange of energy between the waves we observe that the presence of this manifold allows the system to manifest stable solutions for a specific interval of the dissipation parameters out of this range the system brings up only uncoupled states no nonlinear saturated states the dissipation parameters affect only the transversal dynamics to the manifold changing how quickly the invariant manifold is reached in the invariant manifold the system exhibits a large number of coexisting periodic attractors limit cycles coupled states with an interwoven structure of basins of attraction the time series of conserved quantities and the properties of the lyapunov spectra are used to imply the existence of a lower dimensional invariant manifold since the dynamics in such invariant manifold preserves an energy like function we identify it as being conservative

finite dimensional representations of invariant differential operators let k be an algebraically closed field of characteristic 0 y k r k s and let g be an algebraic torus acting diagonally on the ring of algebraic differential operators v y we give necessary and sufficient conditions for d v g to have enough simple finite dimensional representations in the sense that the intersection of the kernels of all the simple finite dimensional representations is zero as an application we show that if k gl v is a representation of a reductive group k and if zero is not a weight of a maximal torus of k on v then d v k has enough finite dimensional representations we also construct examples of fcr algebras with any integer gk dimension 3

eigenvalue estimates for minimal surfaces in hyperbolic space this paper gives an upper bound for the first eigenvalue of the universal cover of a complete stable minimal surface in hyperbolic space and a sharper one for least area disks

moving centre monotonicity formulae for minimal submanifolds and related equations abstract monotonicity formulae play a crucial role for many geometric pdes especially for their regularity theories for minimal submanifolds in a euclidean ball the classical monotonicity formula implies that if such a submanifold passes through the centre of the ball then its area is at least that of the equatorial disk recently brendle and hung proved a sharp area bound for minimal submanifolds when the prescribed point is not the centre of the ball which resolved a conjecture of alexander hoffman and osserman their proof involves asymptotic analysis of an ingeniously chosen vector field and the divergence theorem in this article we prove a sharp moving centre monotonicity formula for minimal submanifolds which implies the aforementioned area bound we also describe similar moving centre monotonicity formulae for stationary p harmonic maps mean curvature flow and the harmonic map heat flow

stability of hypersurfaces with constant mean curvature soit m n compacte orientable et soit x m n r n 1 une immersion a courbure moyenne constante non nulle alors x est stable si et seulement si x m r n 1 est une sphere s n r n 1

geometry and topology of some overdetermined elliptic problems we study necessary conditions on the geometry and the topology of domains in r2 that support a positive solution to a classical overdetermined elliptic problem u f u 0in u 0on u constanton the ideas and tools we use come from constant mean curvature surface theory in particular we obtain a partial answer to a question posed by h berestycki l caffarelli and l nirenberg in 1997 we investigate also some boundedness properties of the solution u some of our results generalize to higher dimensions

on the geometric and topological rigidity of hyperbolic 3 manifolds theorem 0 1 let n be a closed hyperbolic 3 manifold containing an embedded hyperbolic tube of radius log 3 2 549306 about a closed geodesic then i iff m n is a homotopy equivalence where m is an irreducible 3 manifold then f is homotopic to a homeomorphism ii if f g m s n are homotopic homeomorphisms then f is isotopic to g iii the space of hyperbolic metrics on n is path connected

onset of spatiotemporal chaos in a nonlinear system we describe the onset of spatiotemporal chaos in a spatially extended nonlinear dynamical system as a result of the loss of transversal stability of an invariant manifold representing a spatially homogeneous and temporally chaotic state the onset of spatiotemporal chaos is characterized by the switching between spatially homogeneous and nonhomogeneous states with statistical properties of on off intermittency

hypersurfaces of constant mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity in this paper we study the problem of finding smooth complete hypersurfaces of constant mean curvature in hyperbolic space with a prescribed asymptotic boundary at infinity our main results are proved by deriving a priori global gradient estimates based on the classical maximum principle for elliptic equations

on the dirichlet problem for minimal graphs in hyperbolic space soit r n n 2 un domaine borne on etudie le probleme de dirichlet suivant f f i f j f ij 1 df 2 n f 0 dans f 0 dans f 0 sur on considere le comportement a la frontiere de la solution

on partially and globally overdetermined problems of elliptic type we consider some elliptic pdes with dirichlet and neumann data prescribed on some portion of the boundary of the domain and we obtain rigidity results that give a classication of the solution and of the domain in particular we nd mild conditions under which a partially overdetermined problem is in fact globally overdetermined this enables to use several classical results in order to classify all the domains that admit a solution of suitable general partially overdetermined problems these results may be seen as solutions of suitable inverse problems that is to say given that an overdetermined system possesses a solution we nd the shape of the admissible domains models of these type arise in several areas of mathematical physics and shape optimization

higher rank hyperbolicity the large scale geometry of hyperbolic metric spaces exhibits many distinctive features such as the stability of quasi geodesics the morse lemma the visibility property and the homeomorphism between visual boundaries induced by a quasi isometry we prove a number of closely analogous results for spaces of rank n ge 2 in an asymptotic sense under some weak assumptions reminiscent of nonpositive curvature for this purpose we replace quasi geodesic lines with quasi minimizing locally finite n cycles of r n volume growth prime examples include n cycles associated with n quasiflats solving an asymptotic plateau problem and producing unique tangent cones at infinity for such cycles we show in particular that every quasi isometry between two proper text cat 0 spaces of asymptotic rank n extends to a class of n 1 cycles in the tits boundaries

rings of differential operators on invariant rings of tori let k be an algebraically closed field of characteristic zero and g a torus acting diagonally on k for a subset 8 of s 1 2 s set up u e k u o if j e 8 then g acts on d up the ring of regular functions on up and we study the ring d d up g of all differential operators on the invariantvring more generally suppose that is a set of subsets of s such that each invariant ring d u g b e has the same quotient field we prove thatn g d d up g is noetherian and finitely generated as a k algebra now g acts on each d d up and there is a natural map 0 n d d ub n d d up g d ya g obtained by restriction of the differential operators we find necessary and sufficient conditions for 0 to be suuective and describe the kernel of the algebras n g d up g and n g a d up g carry a natural filtration given by the order of the differential operators we show that the associated graded rings are finitely generated commutative algebras and are gorensetin rings we also determine the centers of n ga d up5 g and n ga d d u g introduction throughout this paper k will be an algebraically closed field of characteristic zero if k is a commutative k algebra we denote by do k the set of k linear maps k k and if 1 a k linear map f k k belongs to dp k provided the map f r defined by f r s f rs rf s for s e k belongs to dp l k for all r e k the set d k upaodp k forms a subring of endk k called the ring of differential operators on k we consider k as a left d k module where f r f r for all f e d k r e k the ring of differential operators on the invariant ring of a finite group acting on a polynomial ring has been studied in ka l1 and l2 we carry out a similar study for the invariant ring of a torus we assume that the torus g acts on ks as a group of diagonal matrices if jb is a subset of s 1 2 s then g acts on ua u e ksluj o if j e jb also g acts on p u via gf u f g 1u for g e g f e g u 3 u e u 3 more generally suppose that for some set of subsets a of s we have y ue ua where each invariant ring p u c has the same quotient field f then each d p u 6 is a received by the editors june 18 1986 and in revised form december 11 1986 1980 mclthemcltiss subject clclssificcltion 1985 revision primary 16a33 secondary 13n05 14l30 p 1987 american mathematical s clety 0002 9947 87 1 00 25 per page

a weierstrass representation for linear weingarten spacelike surfaces of maximal type in the lorentz minkowski space in this work we extend the weierstrass representation for maximal spacelike surfaces in the 3 dimensional lorentz minkowski space to spacelike surfaces whose mean curvature is proportional to its gaussian curvature linear weingarten surfaces of maximal type we use this representation in order to study the gaussian curvature and the gauss map of such surfaces when the immersion is complete proving that the surface is a plane or the supremum of its gaussian curvature is a negative constant and its gauss map is a diffeomorphism onto the hyperbolic plane finally we classify the rotation linear weingarten surfaces of maximal type

two state on off intermittency caused by unstable dimension variability in periodically forced drift waves certain high dimensional dynamical systems present two or more attractors characterized by different energy branches for some parameter values the dynamics oscillates between these two branches in a seemingly random fashion a phenomenon called two state on off intermittency in this work we show that the dynamical mechanism underlying this intermittency involves the severe breakdown of hyperbolicity of the attractors through a mechanism known as unstable dimension variability we characterize the parametric evolution of this variability using statistical properties of the finite time lyapunov exponents as a model system that exhibits this behavior we consider periodically forced and damped drift waves in this spatiotemporal example there is a low dimensional chaotic attractor that is created by an interior crisis already presenting unstable dimension variability

asymptotic behavior of area minimizing currents in hyperbolic space on etudie le comportement asymptotique des courants localement rectifiables minimisant l aire dans un espace hyperbolique on considere l aspect theorie de la mesure geometrique et l aspect equation aux derivees partielles

renormalized area and properly embedded minimal surfaces in hyperbolic 3 manifolds we study the renormalized area functional mathcal a in the ads cft correspondence specifically for properly embedded minimal surfaces in convex cocompact hyperbolic 3 manifolds and somewhat more broadly poincare einstein spaces our main results include an explicit formula for the renormalized area of such a minimal surface y as an integral of local geometric quantities as well as formulae for the first and second variations of mathcal a which are given by integrals of global quantities over the asymptotic boundary loop of y all of these formulae are also obtained for a broader class of nonminimal surfaces the proper setting for the study of this functional when the ambient space is hyperbolic requires an understanding of the moduli space of all properly embedded minimal surfaces with smoothly embedded asymptotic boundary we show that this moduli space is a smooth banach manifold and develop a mathbb z valued degree theory for the natural map taking a minimal surface to its boundary curve we characterize the nondegenerate critical points of mathcal a for minimal surfaces in mathbb h 3 and finally discuss the relationship of mathcal a to the willmore functional

density and spectrum of minimal submanifolds in space forms let varphi m m rightarrow n n be a minimal proper immersion in an ambient space suitably close to a space form mathbb n n k of curvature k le 0 in this paper we are interested in the relation between the density function theta r of m and the spectrum of its laplace beltrami operator in particular we prove that if theta r has subexponential growth when k 0 or sub polynomial growth k 0 along a sequence then the spectrum of m m is the same as that of the space form mathbb n m k notably the result applies to anderson s smooth solutions of plateau s problem at infinity on the hyperbolic space independently of their boundary regularity we also give a simple condition on the second fundamental form that ensures m to have finite density in particular we show that minimal submanifolds with finite total curvature in the hyperbolic space also have finite density

quantum error correcting codes and 4 dimensional arithmetic hyperbolic manifolds using 4 dimensional arithmetic hyperbolic manifolds we construct some new homological quantum error correcting codes they are low density parity check codes with linear rate and distance ne their rate is evaluated via euler characteristic arguments and their distance using z2 systolic geometry this construction answers a question of zemor on cayley graphs surface codes and the limits of homological coding for quantum error correction in proceedings of second international workshop on coding and cryptology iwcc lecture notes in computer science vol 5557 2009 pp 259 273 who asked whether homological codes with such parameters could exist at all

flattening results for elliptic pdes in unbounded domains with applications to overdetermined problems semilinear elliptic pdes are dealt with either in the half space or in overdetermined settings we obtain several new results and also give new proofs of celebrated theorems by exploiting some geometric analysis of level sets a geometric inequality and a pointwise gradient estimate

quasi minimizing surfaces in hyperbolic space let h 3 g denote 3 dimensional hyperbolic space with constant curvature 1 we are interested in surfaces f in h 3 with the following quasi minimality property if it f h 3 0 1 which does not depend on the homotopy if this condition is satisfied with q 1 then f is called homotopically minimizing a surface f c h 3 with this property is not necessarily area minimizing in contrast to the situation in higher dimensions cf 13 our main result is

two parameter bifurcation study of the regularized long wave equation we perform a two parameter bifurcation study of the driven damped regularized long wave equation by varying the amplitude and phase of the driver increasing the amplitude of the driver brings the system to the regime of spatiotemporal chaos stc a chaotic state with a large number of degrees of freedom several global bifurcations are found including codimension two bifurcations and homoclinic bifurcations involving three tori and the manifolds of steady waves leading to the formation of chaotic saddles in the phase space we identify four distinct routes to stc they depend on the phase of the driver and involve boundary and interior crises intermittency the ruelle takens scenario the feigenbaum cascade an embedded saddle node homoclinic and other bifurcations this study elucidates some of the recently reported dynamical phenomena

an asymptotic theorem for minimal surfaces and existence results for minimal graphs in mathbb h 2 times mathbb r in this paper we prove a general and sharp asymptotic theorem for minimal surfaces in mathbb h 2 times mathbb r as a consequence we prove that there is no properly immersed minimal surface whose asymptotic boundary is a jordan curve homologous to zero in partial infty mathbb h 2 times mathbb r such that is contained in a slab between two horizontal circles of partial infty mathbb h 2 times mathbb r with width equal to we construct vertical minimal graphs in mathbb h 2 times mathbb r over certain unbounded admissible domains taking certain prescribed finite boundary data and certain prescribed asymptotic boundary data our admissible unbounded domains in mathbb h 2 times 0 are non necessarily convex and non necessarily bounded by convex arcs each component of its boundary is properly embedded with zero one or two points on its asymptotic boundary satisfying a further geometric condition

d modules bernstein sato polynomials and f invariants of direct summands abstract we study the structure of d modules over a ring r which is a direct summand of a polynomial or a power series ring s with coefficients over a field we relate properties of d modules over r to d modules over s we show that the localization r f and the local cohomology module h i i r have finite length as d modules over r furthermore we show the existence of the bernstein sato polynomial for elements in r in positive characteristic we use this relation between d modules over r and s to show that the set of f jumping numbers of an ideal i r is contained in the set of f jumping numbers of its extension in s as a consequence the f jumping numbers of i in r form a discrete set of rational numbers we also relate the bernstein sato polynomial in r with the f thresholds and the f jumping numbers in r

complete linear weingarten surfaces of bryant type a plateau problem at infinity in this paper we study a large class of weingarten surfaces which includes the constant mean curvature one surfaces and flat surfaces in the hyperbolic 3 space we show that these surfaces can be parametrized by holomorphic data like minimal surfaces in the euclidean 3 space and we use it to study their completeness we also establish some existence and uniqueness theorems by studing the plateau problem at infinity when is a given curve on the ideal boundary the asymptotic boundary of a complete surface in our family and how many embedded solutions are there

d modules on smooth toric varieties abstract let x be a smooth toric variety cox introduced the homogeneous coordinate ring s of x and its irrelevant ideal b let a denote the ring of differential operators on spec s we show that the category of d modules on x is equivalent to a subcategory of graded a modules modulo b torsion additionally we prove that the characteristic variety of a d module is a geometric quotient of an open subset of the characteristic variety of the associated a module and that holonomic d modules correspond to holonomic a modules

extremal domains on hadamard manifolds abstract we investigate the geometry and topology of extremal domains in a hadamard manifold i e domains that support a positive solution to an overdetermined elliptic problem oep first we study narrow properties of such domains and characterize the boundary at infinity we give an upper bound for the hausdorff dimension of its boundary at infinity and how the domain behaves at infinity this shows interesting relations with the singular yamabe problem later we focus on extremal domains in the hyperbolic space symmetry and boundedness properties will be shown in a certain sense we extend levitt rosenberg s theorem to oeps which suggests a strong relation with constant mean curvature hypersurfaces in particular we are able to prove the berestycki caffarelli nirenberg conjecture under certain assumptions either on the boundary at infinity of the extremal domain or on the oep itself also a height estimate for solutions on extremal domains in a hyperbolic space will be given

the geometry of finite topology bryant surfaces we prove that in the three dimensional hyperbolic space h 3 the properly embedded constant mean curvature one surfaces with finite topology are of finite total curvature and each ends are regular this imply in particular that the hrosphere is the only properly embedded simply connected constant mean curvature one surface in h 3

erratum on the dirichlet problem for minimal graphs in hyperbolic space r n 0 that can be identified with the asymptotic infinity of hn 1 let nn 1 hn 1 be a closed embedded n 1 dimensional submanifold then there exists a complete absolutely area minimizing integral n current asymptotic to nn 1 at infinity see for example 1 2 it was also noted in 1 2 that in the case n 6 currents obtained are smoothly embedded complete hypersurfaces in the case n 7 as in the euclidean case there can be a singular set of hausdorff dimension at most n 7

the topology and geometry of embedded surfaces of constant mean curvature si m est une surface proprement plongee dans r 3 avec une courbure moyenne constante non nulle alors m n est pas homeomorphe a une surface fermee avec un seul point elimine soit m une surface proprement plongee dans r 3 a courbure moyenne constante non nulle si m est homeomorphe a une surface fermee avec 2 points elimines alors m est une distance bornee a partir d une ligne droite soit m une surface proprement plongee de courbure moyenne non nulle qui est homeomorphe a une surface fermee avec 3 points elimines alors m reste une distance bornee a partir d un plan

gelfand kirillov dimension and associated graded modules abstract let s a 0sa be a filtered algebra over a field k and m ma a filtered s module with associated graded module gr m m a m a 1 over gr s s a s a 1 we consider conditions which ensure that gr m and m have the same gelfand kirillov dimension and show that this equality holds in particular if gr s is an affine commutative k algebra

quantifying singularities with differential operators abstract the f signature of a local ring of prime characteristic is a numerical invariant that detects many interesting properties for example this invariant detects non singularity and strong f regularity however it is very difficult to compute motivated by different aspects of the f signature we define a numerical invariant for rings of characteristic zero or p 0 that exhibits many of the useful properties of the f signature we also compute many examples of this invariant including cases where the f signature is not known we also obtain a number of results on symbolic powers and bernstein sato polynomials

minimal and weakly trapped submanifolds in standard static spacetimes abstract in this article we obtain some rigidity results for spacelike submanifolds of arbitrary codimension in standard static spacetimes in particular under appropriate assumptions we prove that they must be contained in a spacelike slice which enables us to characterize minimal closed submanifolds of codimension two we also prove new rigidity results for complete non compact weakly trapped submanifolds of arbitrary codimension in these ambient spacetimes

regularity at infinity for area minimizing hypersurfaces in hyperbolic space equipped with the hyperbolic metric y z dxz dy2 a standard compactification of ih involves adding the boundary r 0 w so that i i is simply the one point compactification of the euclidean closed half space r 0 suppose 0 7 any interior singularities of m must remain in a bounded region of hyperbolic space near points of f in the euclidean topology m w f may thus be described as the graph of a function this function is the solution of an interesting partial differential equation that becomes degenerate along the part of the boundary corresponding to f the second author has recently established l a boundary higher regularity result for this equation which implies in particular that m w f is r if f is cgk for k 2 3 finally in case f bounds a star

stability of hypersurfaces of constant mean curvature in riemannian manifolds hypersurfaces m n with constant mean curvature in a riemannian manifold overline m n 1 display many similarities with minimal hypersurfaces of overline m n 1 they are both solutions to the variational problem of minimizing the area function for certain variations in the first case however the admissible variations are only those that leave a certain volume function fixed for precise definitions see sect 2 this isoperimetric character of the variational problem associated to hypersurfaces of constant mean curvature introduces additional complications in the treatment of stability of such hypersurfaces

complete minimal varieties in hyperbolic space it is a basic problem in the subject of minimal varieties to determine the class of complete minimal and area minimizing varieties m k in a given riemannian manifold n besides being of their own intrinsic interest the cases r of euclidean space and mn sn of the sphere are fundamental in understanding the local structure of minimal varieties due to the existence of tangent cones to general minimal varieties there has been much study of the complete areaminimizing varieties in these cases in particular relating to the bernstein conjecture and minimizing cones e g l2 f1 nevertheless many deep problems remain unsolved this paper is centered around the question of existence of complete areaminimizing varieties in hyperbolic space h here by way of contrast to the situation in ir n or s we find a comparatively complete solution to the problem of existence in fact the main result of this paper exhibits a wealth of complete area minimizing varieties in h in all dimensions and codimensions more precisely we have