serrin s overdetermined problem and constant mean curvature surfaces for all n 9 we find smooth entire epigraphs in rn namely smooth domains of the form x rn xn f x1 xn 1 which are not half spaces and in which a problem of the form u f u 0 in has a positive bounded solution with 0 dirichlet boundary data and constant neumann boundary data on this answers negatively for large dimensions a question by berestycki caffarelli and nirenberg in 1971 serrin proved that a bounded domain where such an overdetermined problem is solvable must be a ball in analogy to a famous result by alexandrov that states that an embedded compact surface with constant mean curvature cmc in euclidean space must be a sphere in lower dimensions we succeed in providing examples for domains whose boundary is close to large dilations of a given cmc surface where serrin s overdetermined problem is solvable

stability and eigenvalue estimates of linear weingarten hypersurfaces in a sphere abstract let m be an n dimensional compact hypersurface without boundary in a unit sphere s n 1 1 m is called a linear weingarten hypersurface if c r d h e 0 where c d and e are constants with c 2 d 2 0 r and h denote the scalar curvature and the mean curvature of m respectively by the gauss equation we can rewrite the condition c r d h e 0 as n 1 e h 2 a h b where h 2 is the 2nd mean curvature a b and e are constants such that a 2 e 2 0 when e 0 it reduces to the constant mean curvature case in this paper we obtain some stability results about linear weingarten hypersurfaces which generalize the stability results about the hypersurfaces with constant mean curvature or with constant scalar curvature we show that linear weingarten hypersurfaces satisfying n 1 h 2 a h b where a and b are constants can be characterized as critical points of the functional m a n h d v for volume preserving variations we prove that such a linear weingarten hypersurface is stable if and only if it is totally umbilical and non totally geodesic we also obtain optimal upper bounds for the first and second eigenvalues of the jacobi operator of linear weingarten hypersurfaces

a geometric approach to the classification of the equilibrium shapes of self gravitating fluids the classification of the equilibrium shapes that a self gravitating fluid can take in a riemannian manifold is a classical problem in mathematical physics in this paper it is proved that the equilibrium shapes are isoparametric submanifolds some geometric properties of them are also obtained e g classification and existence for some riemannian spaces and relationship with the isoperimetric problem and the group of isometries of the manifold our approach to the problem is geometrical and allows to study the equilibrium shapes on general riemannian spaces

levy processes capacity and hausdorff dimension we use the recently developed multiparameter theory of additive levy processes to establish novel connections between an arbitrary levy process x in r d and a new class of energy forms and their corresponding capacities we then apply these connections to solve two long standing problems in the folklore of the theory of levy processes first we compute the hausdorff dimension of the image x g of a nonrandom linear borel set g c r where x is an arbitrary levy process in r d our work completes the various earlier efforts of taylor proc cambridge phil soc 49 1953 31 39 mckean duke math j 22 1955 229 234 blumenthal and getoor illinois j math 4 1960 370 375 j math mech 10 1961 493 516 millar z wahrsch verw gebiete 17 1971 53 73 pruitt j math mech 19 1969 371 378 pruitt and taylor z wahrsch verw gebiete 12 1969 267 289 hawkes z wahrsch verw gebiete 19 1971 90 102 j london math soc 2 17 1978 567 576 probah theory related fields 112 1998 1 11 hendricks ann math stat 43 1972 690 694 ann probab 1 1973 849 853 kahane publ math orsay 83 02 1983 74 105 recent progress in fourier analysis 1985b 65 121 becker kern meerschaert and scheffler monatsh math 14 2003 91 101 and khoshnevisan xiao and zhong ann probab 31 2003a 1097 1141 where dimx g is computed under various conditions on g x or both we next solve the following problem kahane 1983 publ math orsay 83 02 74 105 when x is an isotropic stable process what is a necessary and sufficient analytic condition on any two disjoint borel sets f g c r such that with positive probability x f n x g is nonempty prior to this article this was understood only in the case that x is a brownian motion khoshnevisan 1999 trans amer math soc 351 2607 2622 here we present a solution to kahane s problem for an arbitrary levy process x provided the distribution of x t is mutually absolutely continuous with respect to the lebesgue measure on r d for all t 0 as a third application of these methods we compute the hausdorff dimension and capacity of the preimage x 1 f of a nonrandom borel set f r d under very mild conditions on the process x this completes the work of hawkes probab theory related fields 112 1998 1 11 that covers the special case where x is a subordinator

index growth of hypersurfaces with constant mean curvature in this paper we give the precise index growth for the embedded hypersurfaces of revolution with constant mean curvature cmc 1 in mathbb r n delaunay unduloids when n 3 using the asymptotics result of korevaar kusner and solomon we derive an explicit asymptotic index growth rate for finite topology cmc 1 surfaces with properly embedded ends similar results are obtained for hypersurfaces with cmc bigger than 1 in hyperbolic space

the structure of a brownian bubble we examine local geometric properties of level sets of the brownian sheet and in particular we identify the asymptotic distribution of the area of sets which correspond to excursions of the sheet high above a given level in the neighborhood of a particular random point it is equal to the area of certain individual connected components of the random set s t b t b s whereb is a standard brownian motion andb is essentially a bessel process of dimension 3 this limit distribution is studied and in particular explicit formulas are given for the probability that a point belongs to a specific connected component and for the expected area of a component given the height of the excursion ofb t b s in this component these formulas are evaluated numerically and compared with the results from direct simulations ofb andb

large isoperimetric surfaces in initial data sets we study the isoperimetric structure of asymptotically flat riemannian 3 manifolds m g that are mathcal c 0 asymptotic to schwarzschild of mass m gt 0 refining an argument due to h bray we obtain an effective volume comparison theorem in schwarzschild we use it to show that isoperimetric regions exist in m g for all sufficiently large volumes and that they are close to centered coordinate spheres this implies that the volume preserving stable constant mean curvature spheres constructed by g huisken and s t yau as well as r ye as perturbations of large centered coordinate spheres minimize area among all competing surfaces that enclose the same volume this confirms a conjecture of h bray our results are consistent with the uniqueness results for volume preserving stable constant mean curvature surfaces in initial data sets obtained by g huisken and s t yau and strengthened by j qing and g tian the additional hypotheses that the surfaces be spherical and far out in the asymptotic region in their results are not necessary in our work

the spaces of index one minimal surfaces and stable constant mean curvature surfaces embedded in flat three manifolds it is proved that the spaces of index one minimal surfaces and stable constant mean curvature surfaces with genus greater than one in non fixed flat three manifolds are compact in a strong sense given a sequence of any of the above surfaces we can extract a convergent subsequence of both the surfaces and the ambient manifolds in the ck topology these limits preserve the topological type of the surfaces and the affine diffeomorphism class of the ambient manifolds some applications to the isoperimetric problem are given

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

area stationary surfaces in the heisenberg group h1 we consider area stationary surfaces perhaps with a volume constraint in the heisenberg group h1 endowed with its carnot caratheodory distance by analyzing the first variation of area we characterize c2 area stationary surfaces as those with mean curvature zero or constant if a volume preserving condition is assumed and such that the characteristic curves meet orthogonally the singular curves moreover a minkowski type formula relating the area the mean curvature and the volume is obtained for volume preserving area stationary surfaces enclosing a given region as a consequence of the characterization of area stationary surfaces we refine the bernstein type theorem given in jih hsin cheng jenn fang hwang andrea malchiodi paul yang minimal surfaces in pseudohermitian geometry ann sc norm super pisa cl sci 5 4 1 2005 129 177 mr 2165405 and nicola garofalo scott pauls the bernstein problem in the heisenberg group arxiv math dg 0209065 to describe all c2 entire area stationary graphs over the xy plane in a calibration argument shows that these graphs are globally area minimizing finally by using the description of the singular set in jih hsin cheng jenn fang hwang andrea malchiodi paul yang minimal surfaces in pseudohermitian geometry ann sc norm super pisa cl sci 5 4 1 2005 129 177 mr 2165405 the characterization of area stationary surfaces and the ruling property of constant mean curvature surfaces we prove our main results where we classify area stationary surfaces in with or without a volume constraint and non empty singular set in particular we deduce the following counterpart to alexandrov uniqueness theorem in euclidean space any compact connected c2 surface in area stationary under a volume constraint must be congruent to a rotationally symmetric sphere obtained as the union of all the geodesics of the same curvature joining two points as a consequence we solve the isoperimetric problem in assuming c2 smoothness of the solutions 13 completely reducible hypersurfaces in a pencil pages 672 688 j v pereira s yuzvinsky preview pdf 324 k related articles 14 modules over motivic cohomology pages 689 727 oliver rondigs paul arne ostvaer preview pdf 303 k related articles 15 the fundamental group of period domains over finite fields pages 728 741 sascha orlik preview pdf 154 k related articles articles 1 15

area stationary surfaces inside the sub riemannian three sphere we consider the sub riemannian metric gh on mathbb s 3 given by the restriction of the riemannian metric of curvature 1 to the plane distribution orthogonal to the hopf vector field we compute the geodesics associated to the carnot caratheodory distance and we show that depending on their curvature they are closed or dense subsets of a clifford torus we study area stationary surfaces with or without a volume constraint in mathbb s 3 g h by following the ideas and techniques by ritore and rosales area stationary surfaces in the heisenberg group mathbb h 1 arxiv math dg 0512547 we introduce a variational notion of mean curvature characterize stationary surfaces and prove classification results for complete volume preserving area stationary surfaces with non empty singular set we also use the behaviour of the carnot caratheodory geodesics and the ruling property of constant mean curvature surfaces to show that the only c2 compact connected embedded surfaces in mathbb s 3 g h with empty singular set and constant mean curvature h such that h sqrt 1 h 2 is an irrational number are clifford tori finally we describe which are the complete rotationally invariant surfaces with constant mean curvature in mathbb s 3 g h

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

the critical ensemble an efficient simulation of a macroscopic system at the critical point abstract it is well known that conventional simulation algorithms are inefficient for the statistical description of macroscopic systems exactly at the critical point due to the divergence of the corresponding relaxation time critical slowing down on the other hand the dynamics in the order parameter space is simplified significantly in this case due to the onset of self similarity in the associated fluctuation patterns as a consequence the effective action at the critical point obtains a very simple form in the present work we show that this simplified action can be used in order to simulate efficiently the statistical properties of a macroscopic system exactly at the critical point using the proposed algorithm we generate an ensemble of configurations resembling the characteristic fractal geometry of the critical system related to the self similar order parameter fluctuations as an example we simulate the one component real scalar field theory at the transition point t t c as a representative system belonging to the 3 d ising universality class

measuring the range of an additive levy process the primary goal of this paper is to study the range of the random field x t nj 1xj tj where x1 xn vspace 1pt are independent levy processes in rd to cite a typical result of this paper let us suppose that i denotes the levy exponent of xi for each i 1 n then under certain mild conditions we show that a necessary and sufficient condition for x rn to have positive d dimensional lebesgue measure is the integrability of the function rd nj 1r 1 j 1 this extends a celebrated result of kesten and of bretagnolle in the one parameter setting furthermore we show that the existence of square integrable local times is yet another equivalent condition for the mentioned integrability criterion this extends a theorem of hawkes to the present random fields setting and completes the analysis of local times for additive levy processes initiated in a companion by paper khoshnevisan xiao and zhong

rotation hypersurfaces in spaces of constant curvature rotation hypersurfaces in spaces of constant curvature are defined and their principal curvatures are computed a local characterization of such hypersurfaces with dimensions greater than two is given in terms of principal curvatures some special cases of rotation hypersurfaces with constant mean curvature in hyperbolic space are studied in particular it is shown that the well known conjugation between the helicoid and the catenoid in euclidean three space extends naturally to hyperbolic three space h3 in the latter case catenoids are of three different types and the explicit correspondence is given it is also shown that there exists a family of simply connected complete embedded nontotally geodesic stable minimal surfaces in h3

curvature estimates and sheeting theorems for weakly stable cmc hypersurfaces abstract weakly stable constant mean curvature cmc hypersurfaces are stable critical points of the area functional with respect to volume preserving deformations we establish a pointwise curvature estimate in the non singular dimensions and a sheeting theorem in all dimensions for weakly stable cmc hypersurfaces giving an effective version of the compactness theorem for weakly stable cmc hypersurfaces established in the recent work of the first and third named authors 6 our results generalize the curvature estimate and the sheeting theorem proven respectively by schoen simon yau and schoen simon for strongly stable hypersurfaces

majoration de la seconde valeur propre d un operateur de schrodinger sur une variete compacte et applications abstract in the first part of this paper we obtain an optimal upper bound for the second eigenvalue of schrodinger operators through using a conformal invariant i e conformal volume in the second part we obtain some applications of this estimate concerning the morse index of constant mean curvature or minimal hypersurfaces the last part deals with some isospectrality problems

applications of compactness results for harmonic maps to stable constant mean curvature surfaces for all lipschitz functions u such that m u da 0 in the above formula and da are the gradient and the area element on m and is the square of the norm of the second fundamental form of we shall call i the index form of the immersion note that i u is the second variation formula of the area for a normal variation leaving constant the volume functional with variational vector field u where is a unit vector field normal to the immersion complete details can be found in bdce in this paper we study stable periodic surfaces i e stable immersions m r3 where is a discrete subgroup of translations in r3 these surfaces are interesting since they are good models for naturally occurring periodic phenomena such as the formation of interfaces in block copolymers see tahh and the references cited there up to now our knowledge of stable immersions in complete flat three manifolds can be resumed in the following result see rr for a sketch of the proof

level sets of additive levy processes we provide a probabilistic interpretation of a class of natural capacities on euclidean space in terms of the level sets of a suitably chosen multiparameter additive levy process x we also present several probabilistic applications of the aforementioned potential theoretic connections they include areas such as intersections of levy processes and level sets as well as hausdorff dimension computations

λ hypersurfaces on shrinking gradient ricci solitons abstract we consider an orientable smooth non totally geodesic hypersurface properly immersed on the cylinder shrinking soliton s 2 k 1 k r l g f where g is the product metric and the weight function f is defined by f p x x 2 4 if has weighted weak index less than or equal to l 1 we show that splits off a linear space also we classify the f stable orientable complete hypersurfaces properly immersed on s 2 k 1 k r g f with f p t t 2 4 obtaining as result s 2 k 1 k t or s 2 k 1 k 1 r

on the stability of hypersurfaces in space forms abstract we define the notion of r s stability concerning closed hypersurfaces with higher order mean curvatures linearly related in a riemannian space form by supposing that such a hypersurface m n is contained either in an open hemisphere of the euclidean sphere or in the hyperbolic space we are able to show that m n is r s stable if and only if m n is a geodesic sphere moreover we obtain a suitable characterization of the r s stability through the analysis of the first eigenvalue of the jacobi operator associated to the corresponding variational problem

geography of the level sets of the brownian sheet we describe geometric properties of w wherew is a standard real valued brownian sheet in the neighborhood of the first hitp of the level set w along a straight line or smooth monotone curvel in such a neighborhood we use a decomposition of the formw s t b s b t x s t whereb s andb t are particular diffusion processes andx s t is comparatively small to show thatp is not on the boundary of any connected component of w rather components of this set form clusters nearp an integral test for thorn shaped neighborhoods ofl with tip atp that do not meet w is given we then analyse the position and size of clusters and individual connected components of w near such a thorn giving upper bounds on their height width and the space between clusters this provides a local picture of the level set our calculations are based on estimates of the length of excursions ofb andb and an accounting of the error termx

uniqueness of stable capillary hypersurfaces in a ball in this paper we prove that any immersed stable capillary hypersurfaces in a ball in space forms are totally umbilical our result also provides a proof of a conjecture proposed by sternberg and zumbrun j reine angew math 503 63 85 1998 we also prove a heintze karcher ros type inequality for hypersurfaces with free boundary in a ball which together with the new minkowski formula yields a new proof of alexandrov s theorem for embedded cmc hypersurfaces in a ball with free boundary

fractal geometry of critical systems we investigate the geometry of a critical system undergoing a second order thermal phase transition using a local description for the dynamics characterizing the system at the critical point t t c we reveal the formation of clusters with fractal geometry where the term cluster is used to describe regions with a nonvanishing value of the order parameter we show that treating the cluster as an open subsystem of the entire system new instanton like configurations dominate the statistical mechanics of the cluster we study the dependence of the resulting fractal dimension on the embedding dimension and the scaling properties isothermal critical exponent of the system taking into account the finite size effects we are able to calculate the size of the critical cluster in terms of the total size of the system the critical temperature and the effective coupling of the long wavelength interaction at the critical point we also show that the size of the cluster has to be identified with the correlation length at criticality finally within the framework of the mean field approximation we extend our local considerations to obtain a global description of the 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

min max theory for constant mean curvature hypersurfaces in this paper we develop a min max theory for the construction of constant mean curvature cmc hypersurfaces of prescribed mean curvature in an arbitrary closed manifold as a corollary we prove the existence of a nontrivial smooth closed almost embedded cmc hypersurface of any given mean curvature c moreover if c is nonzero then our min max solution always has multiplicity one

on the entropies of hypersurfaces with bounded mean curvature we are interested in the impact of entropies on the geometry of a hypersurface of a riemannian manifold in particular we will be able to compare the volume entropy of a hypersurface with that of the ambient manifold provided some geometric assumptions are satisfied this comparison will follow from the existence of an embedded tube around such a hypersurface among the consequences of our study of entropies we point out some new answers to a question of do carmo on stable euclidean hypersurfaces of constant mean curvature

extrinsic upper bounds for λ1 si m est compact alors il existe c r tel que 1 m c n m hu 2 volm pour toutes les sous varietes m compactes connexes a n dimensions isometriquement immergees de m 1 m est la premiere valeur propre du laplacien de m

stable hypersurfaces with constant scalar curvature it is well known that hypersurfaces m n with constant mean curvature in a riemannian manifold overline m n 1 c of constant sectional curvature c are solutions to the variational problem of extremizing the area function for volumepreserving variations

the isoperimetric problem for lens spaces we solve the isoperimetric problem in the lens spaces with large fundamental group namely we prove that the isoperimetric surfaces are either geodesic spheres or quotients of clifford tori we also show that the isoperimetric problem in the lens spaces l 3 1 and l 3 2 follows from the proof of the willmore conjecture by marques and neves

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

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

the second variation of nonorientable minimal submanifolds suppose m is a complete nonorientable minimal submanifold of a riemannian manifold n we derive a second variation formula for the area of m with respect to certain perturbations giving a sufficient condition for the instability of m some simple applications are given we show that the totally geodesic irip2 is the only stable surface in irip3 and we show the non existence of stable nonorientable cones in r we reproduce and marginally extend some known results in the truly non compact setting

a general halfspace theorem for constant mean curvature surfaces in this paper we prove a general halfspace theorem for constant mean curvature surfaces under certain hypotheses we prove that in an ambient space m 3 any constant mean curvature h 0 surface on one side of a constant mean curvature h 0 surface sigma 0 is an equidistant surface to sigma 0 the main hypotheses of the theorem are that sigma 0 is parabolic and the mean curvature of the equidistant surfaces to sigma 0 evolves in a certain way