isospectral conformal metrics on 3 manifolds theorem let g uj go be a sequence of conformal metrics satisfying the following conditions i vol m gj ao for some positive constant ao ii f r2 gj lp gj 12 dvj a 0 i e for each q defined on m we have

second eigenvalue of schrodinger operators and mean curvature let m be a compact immersed submanifold of the euclidean space the hyperbolic space or the standard sphere for any continuous potential q on m we give a sharp upper bound for the second eigenvalue of the operator q in terms of the total mean curvature of m and the mean value of q moreover we analyze the case where this bound is achieved as a consequence of this result we obtain an alternative proof for the alikakos fusco conjecture concerning the stability of the interface in the allen cahn reaction diffusion model

isospectral sets of conformally equivalent metrics il existe des varietes m a metriques conformement equivalentes g et g telles que g soit isospectrale a g

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

the classification of complete stable area stationary surfaces in the heisenberg group h1 abstract we prove that any c 2 complete orientable connected stable area stationary surface in the sub riemannian heisenberg group h 1 is either a euclidean plane or congruent to the hyperbolic paraboloid t x y

on stable constant mean curvature surfaces with free boundary ros and vergasta geom dedicata 56 19 33 1995 proved among others interesting results a theorem which states that an immersed orientable compact stable constant mean curvature surface sigma with free boundary in a closed ball b subset mathbb r 3 must be a planar equator a spherical cap or a surface of genus 1 with at most two boundary components in this article by using a modified hersch type balancing argument we complete their work by proving that sigma cannot have genus 1

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

compactness of conformal metrics with integral bounds on curvature in this thesis we show that a sequence of conformal metrics on a compact n dimensional riemannian manifold n 4 which has an upper bound on volume and an upper bound on the lp norm of the curvature tensor for fixed p n 2 has a subsequence which converges in c if n 3 we have the same result if we assume in addition that the scalar curvature has an l2 bound as corollaries we have the compactness of a sequence of conformal metrics on a compact three manifold which are isospectral with respect to either the standard or conformal laplacian and the result of lelong ferrand that any compact manifold with non compact conformal group is conformally equivalent to the standard sphere

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

from constant mean curvature hypersurfaces to the gradient theory of phase transitions given a nondegenerate minimal hypersurface in a riemannian manifold we prove that for all e small enough there exists ue a critical point of the allen cahn energy ee u e2 u 2 1 u2 2 whose nodal set converges to as e tends to 0 moreover if is a volume nondegenerate constant mean curvature hypersurface then the same conclusion holds with the function ue being a critical point of ee under some volume constraint

eigenvalues of the laplacian in two dimensions the eigenvalue problem for the laplace operator in two dimensions is classical in mathematics and physics nevertheless computational methods for estimating the eigenvalues are still of much current interest particularly in applications to acoustic and electromagnetic waveguides although our primary interest is with the computational methods there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and in addition are of interest in themselves these results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds some of the more powerful techniques available are illustrated by applying them to a model problem

double aggregation operators this paper introduces the so called double aggregation operators which allow to combine two input lists of information coming from different sources into a single output several basic properties of these operators such as symmetry neutral and annihilator elements idempotency etc are studied and illustrated with some examples in addition the classes of additive and comonotone additive double aggregation operators are characterized

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

residues of the eta function for an operator of dirac type abstract we compute the asymptotics of tr l 2 pe tp 2 where p is a first order operator of dirac type this is equivalent to evaluating the residues of the eta function

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

methods for computing lower bounds to eigenvalues of self adjoint operators new approaches for computing tight lower bounds to the eigenvalues of a class of semibounded self adjoint operators are presented that require comparatively little a priori spectral information and permit the effective use of among others finite element trial functions a variant of the method of intermediate problems making use of operator decompositions having the form t t is reviewed and then developed into a new framework based on recent inertia results in the weinstein aronszajn theory this framework provides greater flexibility in analysis and permits the formulation of a final computational task involving sparse well structured matrices although our derivation is based on an intermediate problem formulation our results may be specialized to obtain either the temple lehmann method or weinberger s matrix method

bounds to eigenvalues of the laplacian on l shaped domain by variational methods in this study the bounds for eigenvalues of the laplacian operator on an l shaped domain are determined by adopting some special functions in goerisch method for lower bounds and in traditional rayleigh ritz method for upper bounds very accurate bounds to eigenvalues for the problem are obtained numerical results show that these functions can also be successfully used to solve the problem on the region with other reentrant angle

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

trace identities for commutators with applications to the distribution of eigenvalues we prove trace identities for commutators of operators which are used to derive sum rules and sharp universal bounds for the eigenvalues of periodic schrodinger operators and schrodinger operators on immersed manifolds in particular we prove bounds on the eigenvalue lambda n 1 in terms of the lower spectrum bounds on ratios of means of eigenvalues and universal monotonicity properties of eigenvalue moments which imply sharp versions of lieb thirring inequalities in the case of a schrodinger operator on an immersed manifold of dimension d we derive a version of reilly s inequality bounding the eigenvalue lambda n 1 of the laplace beltrami operator by a universal constant times vertical bar vertical bar h vertical bar n 2 infinity 2 d

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

strongly stable surfaces in sub riemannian 3 space forms abstract a surface of constant mean curvature cmc equal to h in a sub riemannian 3 manifold is strongly stable if it minimizes the functional area 2 h volume up to second order in this paper we obtain some criteria ensuring strong stability of surfaces in sasakian 3 manifolds we also produce new examples of c 1 complete cmc surfaces with empty singular set in the sub riemannian 3 space forms by studying those ones containing a vertical line as a consequence we are able to find complete strongly stable non vertical surfaces with empty singular set in the sub riemannian hyperbolic 3 space m 1 in relation to the bernstein problem in m 1 we discover strongly stable c entire minimal graphs in m 1 different from vertical planes these examples are in clear contrast with the situation in the first heisenberg group where complete strongly stable surfaces with empty singular set are vertical planes finally we analyze the strong stability of cmc surfaces of class c 2 and non empty singular set in the sub riemannian 3 space forms when these surfaces have isolated singular points we deduce their strong stability even for variations moving the singular set

closed hypersurfaces driven by mean curvature and inner pressure we introduce a hyperbolic equation that describes the motion of closed hypersurfaces in a riemannian manifold with surface tension and inner pressure as driving forces in the case of spherical surfaces this equation can be considered as an idealized mathematical model for a moving soap bubble the equation is derived as an euler lagrange equation from a suitable action integral it is a quasi linear degenerate hyperbolic pde of second order that describes the motion of the surfaces extrinsically our main results are the solution of the cauchy problem by means of the nash moser inverse function theorem a continuation criterion and stability estimates 2012 wiley periodicals inc

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

universal inequalities for the eigenvalues of schrodinger operators on submanifolds we establish inequalities for the eigenvalues of schr o dinger operators on compact submanifolds possibly with nonempty boundary of euclidean spaces of spheres and of real complex and quaternionic projective spaces which are related to inequalities for the laplacian on euclidean domains due to payne polya and weinberger and to yang but which depend in an explicit way on the mean curvature in later sections we prove similar results for schr o dinger operators on homogeneous riemannian spaces and more generally on any riemannian manifold that admits an eigenmap into a sphere as well as for the kohn laplacian on subdomains of the heisenberg group among the consequences of this analysis are an extension of reilly s inequality bounding any eigenvalue of the laplacian in terms of the mean curvature and spectral criteria for the immersibility of manifolds in homogeneous spaces

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

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

a survey of weak connectives and the preservation of their properties by aggregations the aim of this paper is to choose diverse definitions of generalized logical connectives and to present them in a coherent order from the weakest to the richest such a rich list of notions allows us to consider the problem of admissible aggregations in the presented classes of unary and binary operations this gives a contribution to the discussion of tolerance analysis in soft computing decision making approximate reasoning and fuzzy control first we present a short survey of the development of mv logic s connectives next we discuss postulates used for generalized logical connectives finally we describe families of weak connectives and indicate the preservation of their properties by some aggregation functions

eigenvalue inclusions via domain decomposition we describe a method for the calculation of guaranteed bounds for the k lowest eigenvalues of secondorder problems with neumann boundary conditions using p2 approximations for the eigenfunctions a

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

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

a unique extremal metric for the least eigenvalue of the laplacian on the klein bottle we prove the following conjecture recently formulated by jakobson nadirashvili and polterovich cite jnp on the klein bottle mathbb k the metric of revolution g 0 9 1 8 cos 2v 2 over 1 8 cos 2v left du 2 dv 2 over 1 8 cos 2v right 0 le u frac pi 2 0 le v pi is the emph unique extremal metric of the first eigenvalue of the laplacian viewed as a functional on the space of all riemannian metrics of given area the proof leads us to study a hamiltonian dynamical system which turns out to be completely integrable by quadratures

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

bifurcation and local rigidity of constant second mean curvature hypersurfaces in riemannian warped products abstract in a riemannian warped product i f m n where i r is an open interval f is a positive real function defined on i and m n is a compact riemannian manifold without boundary we use equivariant bifurcation theory in order to establish sufficient conditions in terms of f and the spectrum of the laplacian on m n that allow us to guarantee the existence of bifurcation instants or the local rigidity of a certain family of open sets whose boundaries are h 2 hypersurfaces namely whose boundaries are hypersurfaces with constant second mean curvature h 2 for each of our results we have provided a considerable number of examples that verify all the assumptions under consideration

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

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

aggregation functions construction methods conjunctive disjunctive and mixed classes in this second part of our state of the art overview on aggregation theory based again on our recent monograph on aggregation functions we focus on several construction methods for aggregation functions and on special classes of aggregation functions covering the well known conjunctive disjunctive and mixed aggregation functions some fields of applications are included

a classification theorem for helfrich surfaces in this paper we study the functional mathcal w lambda 1 lambda 2 which is the sum of the willmore energy lambda 1 weighted surface area and lambda 2 weighted volume for surfaces immersed in mathbb r 3 this coincides with the helfrich functional with zero spontaneous curvature our main result is a complete classification of all smooth immersed critical points of the functional with lambda 1 ge 0 and small l 2 norm of tracefree curvature with no assumption on the growth of the curvature in l 2 at infinity this not only improves the gap lemma due to kuwert and schatzle for willmore surfaces immersed in mathbb r 3 but also implies the non existence of critical points of the functional satisfying the energy condition for which the surface area and enclosed volume are positively weighted

high order eigenvalues for the helmholtz equation in complicated non tensor domains through richardson extrapolation of second order finite differences we apply second order finite differences to calculate the lowest eigenvalues of the helmholtz equation for complicated non tensor domains in the plane using different grids which sample exactly the border of the domain we show that the results obtained applying richardson and pade richardson extrapolations to a set of finite difference eigenvalues corresponding to different grids allow us to obtain extremely precise values when possible we have assessed the precision of our extrapolations comparing them with the highly precise results obtained using the method of particular solutions our empirical findings suggest an asymptotic nature of the fd series in all the cases studied we are able to report numerical results which are more precise than those available in the literature

a note on willmore minimizing klein bottles in euclidean space abstract we show that 3 1 k r 4 0 n 4 is the unique minimizer among immersed klein bottles in its conformal class where s 4 r 4 is a stereographic projection and 3 1 is the bipolar surface of lawson s 3 1 surface 11 we conjecture that 3 1 is the unique minimizer among immersed klein bottles into r n n 4 whose existence the authors and p breuning proved in 2

computing eigenmodes ofelliptic operators using radial basis functions radial basis function rbf approximations have been successfully used to solve boundary value problems numerically we show that rbfs can also be used to compute eigenmodes of elliptic operators particular attention is given to the laplacian operator in two dimensions including techniques to avoid degradation of the solution near the boundaries for regions with corner singularities special functions must be added to the basis to maintain good convergence numerical results compare favorably to basic finite element methods

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

growth of heat trace and heat content asymptotic coefficients abstract we show in the smooth category that the heat trace asymptotics and the heat content asymptotics can be made to grow arbitrarily rapidly in the real analytic context however this is not true and we establish universal bounds on their growth

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

isoperimetric regions in cones we consider cones c 0 m n and prove that if the ricci curvature of c is nonnegative then geodesic balls about the vertex minimize perimeter for given volume if strict inequality holds then they are the only stable regions

a limit method for solving period problems on minimal surfaces we introduce a new technique to solve period problems on minimal surfaces called limit method if a family of surfaces has weierstras data converging to the data of a known example and this presents a transversal solution of periods then the original family contains a sub family with closed periods

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

the first steklov eigenvalue conformal geometry and minimal surfaces we consider the relationship of the geometry of compact riemannian manifolds with boundary to the first nonzero eigenvalue 1 of the dirichlet to neumann map steklov eigenvalue for surfaces with genus and k boundary components we obtain the upper bound 1l 2 k for 0 and k 1 this result was obtained by weinstock in 1954 and is sharp we attempt to find the best constant in this inequality for annular surfaces 0 and k 2 for rotationally symmetric metrics we show that the best constant is achieved by the induced metric on the portion of the catenoid centered at the origin which meets a sphere orthogonally and hence is a solution of the free boundary problem for the area functional in the ball for a general class of not necessarily rotationally symmetric metrics on the annulus which we call supercritical we prove that 1 l is dominated by that of the critical catenoid with equality if and only if the annulus is conformally equivalent to the critical catenoid by a conformal transformation which is an isometry on the boundary motivated by the annulus case we show that a proper submanifold of the ball is immersed by steklov eigenfunctions if and only if it is a free boundary solution we then prove general upper bounds for conformal metrics on manifolds of any dimension which can be properly conformally immersed into the unit ball in terms of certain conformal volume quantities we show that these bounds are only achieved when the manifold is minimally immersed by first steklov eigenfunctions we also use these ideas to show that any free boundary solution in two dimensions has area at least and we observe that this implies the sharp isoperimetric inequality for free boundary solutions in the two dimensional case

verified eigenvalue evaluation for the laplacian over polygonal domains of arbitrary shape the finite element method fem is applied to bound leading eigenvalues of the laplace operator over polygonal domains compared with classical numerical methods most of which can only give concrete eigenvalue bounds over special domains of symmetry our proposed algorithm can provide concrete eigenvalue bounds for domains of arbitrary shape even when the eigenfunction has a singularity the problem of eigenvalue estimation is solved in two steps first we construct a computable a priori error estimation for the fem solution of poisson s problem which holds even for nonconvex domains with reentrant corners second new computable lower bounds are developed for the eigenvalues because the interval arithmetic is implemented throughout the computation the desired eigenvalue bounds are expected to be mathematically correct we illustrate several computation examples such as the cases of an l shaped domain and a crack domain to demonstrate the efficiency and flexibility of the proposed method

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 geometric complexity of special lagrangian t2 cones we prove a number of results on the geometric complexity of special lagrangian slg t2 cones in 3 every slg t2 cone has a fundamental integer invariant its spectral curve genus we prove that the spectral curve genus of an slg t2 cone gives a lower bound for its geometric complexity i e the area the stability index and the legendrian index of any slg t2 cone are all bounded below by explicit linearly growing functions of the spectral curve genus we prove that the cone on the clifford torus which has spectral curve genus zero in s5 is the unique slg t2 cone with the smallest possible legendrian index and hence that it is the unique stable slg t2 cone this leads to a classification of all rigid index 1 slg cone types in dimension three for cones with spectral curve genus two we give refined lower bounds for the area the legendrian index and the stability index one consequence of these bounds is that there exist s1 invariant slg torus cones of arbitrarily large area legendrian and stability indices we explain some consequences of our results for the programme due to joyce to understand the most common three dimensional isolated singularities of generic families of slg submanifolds in almost calabi yau manifolds

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

effective versions of the positive mass theorem the study of stable minimal surfaces in riemannian 3 manifolds m g with non negative scalar curvature has a rich history in this paper we prove rigidity of such surfaces when m g is asymptotically flat and has horizon boundary as a consequence we obtain an effective version of the positive mass theorem in terms of isoperimetric or more generally closed volume preserving stable cmc surfaces that is appealing from both a physical and a purely geometric point of view we also include a proof of the following conjecture of schoen an asymptotically flat riemannian 3 manifold with non negative scalar curvature that contains an unbounded area minimizing surface is isometric to flat mathbb r 3

minimal immersions of surfaces by the first eigenfunctions and conformal area let 0 m s be a minimal immersion of a compact surface into a unit sphere then the linear functions of 0 are eigenfunctions for the laplacian of m corresponding to the eigenvalue 2 2 the main purpose of this paper is to study those minimal surfaces for which 2 is exactly the first non zero eigenvalue of its laplacian this kind of immersions have a peculiar behaviour among all compact minimal surfaces of the sphere and they appear naturally when one considers different geometric problems as li and yau have shown in 6 the methods that we use in this paper are based for the most part on 6 it is known that the only metric on a 2 dimensional sphere admitting a minimal immersion into s by the first eigenfunctions is the standard one this follows for example from the fact that the multiplicity of the first eigenvalue for such a metric is at most three see the cheng work 3 our first result shows that it is possible to extend this property for an arbitrary compact surface in the following way

maximization of the second positive neumann eigenvalue for planar domains we prove that the second positive neumann eigenvalue of a bounded simply connected planar domain of a given area does not exceed the first positive neumann eigenvalue on a disk of a twice smaller area this estimate is sharp and attained by a sequence of domains degenerating to a union of two identical disks in particular this result implies the polya conjecture for the second neumann eigenvalue the proof is based on a combination of analytic and topological arguments as a by product of our method we obtain an upper bound on the second eigenvalue for conformally round metrics on odd dimensional spheres

curvature estimates for cmc surfaces in three dimensional manifolds in this paper we give curvature estimates for strongly stable cmc surfaces in complete three dimensional manifolds

critical potentials of the eigenvalues and eigenvalue gaps of schrodinger operators abstract let m be a compact riemannian manifold with or without boundary and let be its laplace beltrami operator for any bounded scalar potential q we denote by i q the i th eigenvalue of the schrodinger type operator q acting on functions with dirichlet or neumann boundary conditions in case m we investigate critical potentials of the eigenvalues i and the eigenvalue gaps g i j j i considered as functionals on the set of bounded potentials having a given mean value on m we give necessary and sufficient conditions for a potential q to be critical or to be a local minimizer or a local maximizer of these functionals for instance we prove that a potential q l m is critical for the functional 2 if and only if q is smooth 2 q 3 q and there exist second eigenfunctions f 1 f k of q such that j f j 2 1 in particular 2 as well as any i admits no critical potentials under dirichlet boundary conditions moreover the functional 2 never admits locally minimizing potentials

binary survival aggregation functions we introduce a transformation that acts on binary aggregation functions and that generalizes the transformation that maps copulas a well studied class of binary aggregation functions with a profound probabilistic interpretation to their associated survival copulas the new transformation called double flipping is the composition of two elementary flipping transformations introduced earlier each operating on one of the arguments of the aggregation function we lay bare the relationships between these elementary flipping operations and double flipping we study invariants under these transformations furthermore we characterize the transformations that preserve flippability most importantly we characterize different subclasses of flippable aggregation functions in particular aggregation functions that have an absorbing element or that have a neutral element in this investigation the key role played by quasi copulas and their dual operations is highlighted these findings support the introduction of the term survival aggregation function

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

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

universal inequalities for the eigenvalues of laplace and schrodinger operators on submanifolds we establish inequalities for the eigenvalues of schrodinger oper ators on compact submanifolds possibly with nonempty boundary of euclidean spaces of spheres and of real complex and quaternionic projective spaces which are related to inequalities for the laplacian on euclidean domains due to payne polya and weinberger and to yang but which depend in an explicit way on the mean curvature in later sections we prove similar results for schrodinger operators on homogeneous riemannian spaces and more generally on any rie mannian manifold that admits an eigenmap into a sphere as well as for the kohn laplacian on subdomains of the heisenberg group among the consequences of this analysis are an extension of reilly s inequality bounding any eigenvalue of the laplacian in terms of the mean curvature and spectral criteria for the immersibility of manifolds in homogeneous spaces