totally real minimal tori in in this paper we show that all totally real minimal tori in open image in new window correspond to doubly periodic finite gap solutions of the tzitzeica equation open image in new window using the results on the tzitzeica equation in integrable systems theory we describe explicitly these tori by prym theta functions

a kind of helicoidal surfaces in 3 dimensional minkowski space abstract in this paper we constructed a helicoidal surface with a light like axis with prescribed mean curvature or gauss curvature given by smooth function in 3 dimensional minkowski space e 1 3 and solved an open problem left by beneki kaimakamis and papantoniou in j math anal appl 275 2002 586 614

surfaces immersed in lie algebras obtained from the sigma models we study some geometrical aspects of two dimensional orientable surfaces arising from the study of sigma models to this aim we employ an identification of with the lie algebra by means of which we construct a generalized weierstrass formula for immersion of such surfaces the structural elements of the surface like its moving frame the gauss weingarten and the gauss codazzi ricci equations are expressed in terms of the solution of the model defining it further the first and second fundamental forms the gaussian curvature the mean curvature vector the willmore functional and the topological charge of surfaces are expressed in terms of this solution we present detailed implementation of these results for surfaces immersed in and lie algebras

on spacelike hypersurfaces with constant mean curvature in the de sitter space soit n n n 1 c g une variete de lorentz a n 1 dimensions de courbure constante positive c soit m une hypersurface du genre espace complete a courbure moyenne constante h dans n on suppose que 1 h c 1 2 quand n 2 2 h 2 n 1 c 1 2 n quand n 3 alors m est totalement ombilicale

spectral theory for the periodic sine gordon equation a concrete viewpoint a summary of the spectral theory for quasiperiodic sine and sinh gordon equations is given analogies with whole line solitons and scattering theory motivates the discussion the relation between the ingredients in the inverse spectral solution of the periodic sine gordon equation and physical characteristics of sine gordon waves is emphasized the explicit topics covered are summarized in the table of contents in the introduction

the weierstrass enneper system for constant mean curvature surfaces and the completely integrable sigma model the integrability of a system which describes constant mean curvature surfaces by means of the adapted weierstrass enneper inducing formula is studied this is carried out by using a specific transformation which reduces the initial system to the completely integrable two dimensional euclidean nonlinear sigma model through the use of the apparatus of differential forms and cartan theory of systems in involution it is demonstrated that the general analytic solutions of both systems possess the same degree of freedom furthermore a new linear spectral problem equivalent to the initial weierstrass enneper system is derived via the method of differential constraints a new procedure for constructing solutions to this system is proposed and illustrated by several elementary examples including a multi soliton solution

a constant mean curvature surface and the dirac operator the shape of a surface with constant mean curvature cmc has been studied in mathematics and physics related to nonlinear integrable theory and harmonic map model theory in the study a fictitious linear dirac type operator appears as a tool of the calculus konopelchenko b g and taimanov i a 1996 j phys a math gen 29 1261 5 in this paper i confine the dirac field defined in to a thin surface embedded in and obtain a proper dirac operator for the thin surface then it completely agrees with the dirac type operator used in the calculus of the cmc surface theory in other words the mathematical dirac type operator is realized by a physical dirac particle

dressing cmc n noids the purpose of this paper is to construct cmc n noids with bubbletons we recall a specific class of dressing matrices which we will refer to as simple factor dressing matrices which in many known cases add bubbletons to a cmc surface our new surfaces are obtained by dressing known n noids with embedded delaunay ends by well chosen simple factor dressing matrices in such a way that the dressed surface is also a cmc n noid

representation theory of finite semigroups semigroup radicals and formal language theory in this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field generalizing results of rhodes for the field of complex numbers applications are given to obtain many new results as well as easier proofs of several results in the literature involving triangularizability of finite semigroups which semigroups have split basic semigroup algebras two sided semidirect product decompositions of finite monoids unambiguous products of rational languages products of rational languages with counter and cerny s conjecture for an important class of automata

constant mean curvature surfaces via an integrable dynamical system it is shown that the equation which describes constant mean curvature surfaces via the generalized weierstrass enneper induction has hamiltonian form its simplest finite dimensional reduction is the integrable hamiltonian system with two degrees of freedom this finite dimensional system admits action and classes of equivalence of its trajectories are in one to one correspondence with different helicoidal constant mean curvature surfaces thus the interpretation of well known delaunay and do carmo dajczer surfaces via an integrable finite dimensional hamiltonian system is established

a deformation of tori with constant mean curvature in to those in other space forms it is shown that tori with constant mean curvature in r 3 constructed by wente 7 can be deformed to tori with constant mean curvature in the hyperbolic 3 space or the 3 sphere

invariant conformal metrics on ⁿ in this paper we use the relationship between conformal metrics on the sphere and horospherically convex hypersurfaces in the hyperbolic space for giving sufficient conditions on a conformal metric to be radial under some constrain on the eigenvalues of its schouten tensor also we study conformal metrics on the sphere which are invariant by a k parameter subgroup of conformal diffeomorphisms of the sphere giving a bound on its maximum dimension moreover we classify conformal metrics on the sphere whose eigenvalues of the shouten tensor are all constant we call them isoparametric conformal metrics and we use a clas

n dimensional area of minimal rotational hypersurfaces in spheres abstract let m r n denote the set of compact minimal rotational hypersurfaces in the n 1 dimensional sphere s n 1 for any compact hypersurface m s n 1 let us denote by m its n dimensional area in this paper we show that for any integer n 2 the set of numbers a n m m m r n is a discrete set we also show that if m m r 3 then either m s 3 or m m 0 3 or m m 0 3 where m 0 n s n 1 n 1 n s 1 1 n finally for 2 n 100 we show numerically evidence that if m m r n then either m s n or m m 0 n or m 2 m 0 n we strongly believe that for any m m r n either m s n or m m 0 n or m 2 m 0 n for all n

flows of constant mean curvature tori in the 3 sphere the equivariant case we present a deformation for constant mean curvature tori in the 3 sphere we show that the moduli space of equivariant constant mean curvature tori in the 3 sphere is connected and we classify the minimal the embedded and the alexandrov embedded tori therein

flat surfaces in the hyperbolic 3 space in this paper we give a conformal representation of flat surfaces in the hyperbolic 3 space using the complex structure induced by its second fundamental form we also study some examples and the behaviour at infinity of complete flat ends

a new characterization of the clifford torus via scalar curvature pinching abstract let m n be a compact hypersurface with constant mean curvature h in s n 1 denote by s the squared norm of the second fundamental form of m we prove that there exists an explicit positive constant n depending only on n such that if h n and n h s n h n 23 then s n h and m is one of the following cases i s k k n s n k n k n 1 k n 1 ii s 1 1 1 2 s n 1 1 2 here n h n n 3 2 n 1 h 2 n n 2 2 n 1 n 2 h 4 4 n 1 h 2 and n h n 2 h 2 4 n 1 2 for n 6 12 24 we give examples to show that the assumption h n cannot be removed

positive continuous linear functionals on riesz spaces and applications to minimax theorems on deduit des egalites minimax pour des fonctions prenant des valeurs dans un espace topologique de riesz complet ordonne e a partir d egalites minimax faibles et en fin de compte par des theoremes minimax classiques pour des fonctions a valeurs reelles

on mean convex alexandrov embedded surfaces in the 3 sphere we investigate non compact mean convex alexandrov embedded surfaces in the round unit 3 sphere and show under which conditions it is possible to continuously deform these preserving mean convex alexandrov embeddedness

lawson s genus two surface and meromorphic connections we investigate the lawson genus 2 surface by methods from integrable system theory to this surface we derive a globally defined meromorphic dpw potential which is determined up to two unknown functions depending only on the spectral parameter

minimal immersions into space forms with two principal curvatures in 2 chern do carmo and kobayashi consider minimal immersions of an ndimensional manifold m into the unit sphere s n of the n n l dimensional euclidean space ir n n l with the property that the second fundamental form has constant length n 2 1 n if m is compact this is the smallest possible value for a non totally geodesic minimal immersion as above they prove that locally m is either a piece of a veronese surface n 2 n 2 or the product of two spaces of dimensions m and n m and of constant curvatures n m and n n m they also prove the corresponding global result under a compactness assumption for m the main feature of the situation is that the second fundamental form in any normal direction has two distinct eigenvalues and in general the respective eigenspaces will determine involutive distributions whose integral manifolds will give the local product structure this leads naturally to the study of minimal immersions into spaces of constant curvature such that the second fundamental form has in any normal direction at most two distinct eigenvalues this problem was studied by otsuki in 9 for the case of hypersurfaces he proves the following result see also remark 3 7

a hopf differential for constant mean curvature surfaces in s2 r and h2 r a basic tool in the theory of constant mean curvature cmc surfaces in space forms is the holomorphic quadratic differential discovered by h hopf in this paper we generalize this differential to immersed cmc surfaces in the product spaces s r and h r and prove a corresponding result about the geometry of immersed cmc spheres s in these target spaces

holomorphic representation of constant mean curvature surfaces in minkowski space consequences of non compactness in loop group methods abstract we give an infinite dimensional generalized weierstrass representation for spacelike constant mean curvature cmc surfaces in minkowski 3 space r 2 1 the formulation is analogous to that given by dorfmeister pedit and wu for cmc surfaces in euclidean space replacing the group su 2 with su 1 1 the non compactness of the latter group however means that the iwasawa decomposition of the loop group used to construct the surfaces is not global we prove that it is defined on an open dense subset after doubling the size of the real form su 1 1 and prove several results concerning the behavior of the surface as the boundary of this open set is encountered we then use the generalized weierstrass representation to create and classify new examples of spacelike cmc surfaces in r 2 1 in particular we classify surfaces of revolution and surfaces with screw motion symmetry as well as studying another class of surfaces for which the metric is rotationally invariant

geometry and integrability of topological antitopological fusion integrability of equations of topological antitopological fusion being proposed by cecotti and vafa describing the ground state metric on a given 2d topological field theory tft model is proved for massive tft models these equations are reduced to a universal form being independent on the given tft model by gauge transformations for massive perturbations of topological conformal field theory models the separatrix solutions of the equations bounded at infinity are found by the isomonodromy deformations method also it is shown that the ground state metric together with some part of the underlined tft structure can be parametrized by pluriharmonic maps of the coupling space to the symmetric space of real positive definite quadratic forms

inequality systems and minimax results without linear structure inequality systems and their connections with minimax results for real valued functions have been studied in different ways in several papers 1 2 4 5 7 9 133 in particular pomerol 131 takes into consideration the equivalence between the consistency of inequality systems and the resolvability of each inequality of the system it should be remarked that a minimax theorem can be directly derived from this equivalence the aim of this paper is to investigate the relationships between the con sistency of inequality systems and minimax results for functions defined on spaces without linear structure and taking values in a riesz space for real valued functions minimax results without linear structure go back to fan 3 throughout this paper

critical curves for a santalo problem in 3 space forms abstract l santalo 1956 studied in 15 the minimization of frenet first curvature dependent energies for spaces of curves constrained to lie in a surface of the euclidean 3 space in this paper we analyze an extension of that problem by considering energy functionals determined by lagrangians which depend not only on the curvature but also on the torsion acting on spaces of surface contained clamped curves of real 3 space forms

conformal hypersurfaces with the same gauss map in this paper we provide a complete classification of all hypersurfaces of euclidean space which admit conformal deformations other than the ones obtained through conformal diffeomorphisms of the ambient space preserving the gauss map elie cartan in one of his earlier papers in differential geometry classified all euclidean hypersurfaces of dimension at least five which admit conformal deformations other than the trivial ones obtained through compositions with conformal diffeomorphisms of the ambient space when all possible conformal deformations are trivial a hypersurface is called conformally rigid first cartan proved that a hypersurface is conformally rigid if at no point does there exist a principal curvature of multiplicity at least n 2 see ca cd2 or da using this he concluded that a conformally deformable hypersurface is either conformally flat or a 2 parameter envelope of spheres or planes of some very special type moreover the set of all conformal deformations is either a 1parameter family or there is just one other deformation in this paper we classify all euclidean hypersurfaces which admit nontrivial conformal deformations preserving the gauss map in the isometric case a complete classification for any codimension has been obtained by dajczer and gromoll dg2 while the conformal case but only for surfaces is due to vergasta ve2 theorem let f g mn rn 1 n 3 be conformal immersions of an ndimensional connected riemannian manifold with the same gauss map assume that on no open subset do they differ by a conformal diffeomorphism of rn l then f mn is part of one of the following examples while g mn is of the same type i a minimal real kaehler hypersurface ii a rotation hypersurface over a plane curve iii a rotation hypersurface over a minimal surface in r3 minimal real kaehler hypersurfaces have been completely classified by dajczer and gromoll in dg2 the assumption that f and g do not differ locally by a conformal deformation of the ambient space has been introduced in order to produce a global result without that assumption our proof shows that the only other possibility is to have an open subset where up to homothety received by the editors december 10 1993 1991 mathematics subject classification primary 53c42 1995 american mathematical society

constant mean curvature invariant surfaces and extremals of curvature energies abstract we determine the extremal curves of curvature energy functionals which generalize a variational problem studied by blaschke in r 3 the generalization is made by extending both the lagrangian energy itself and the ambient space to riemannian and lorentzian 3 space forms then we show that constant mean curvature cmc invariant surfaces in 3 space forms can be constructed locally as the evolution of the above extremals under their binormal flow with appropriate velocity moreover we see that these surfaces are intrinsically described by a warping function satisfying an ermakov milne pinney equation finally we use the previous findings to extend known results on isometric deformations of cmc surfaces and the lawson s correspondence of cmc cousins to our background spaces

some contributions to the theory of transformation monoids abstract the aim of this paper is to present some contributions to the theory of finite transformation monoids the dominating influence that permutation groups have on transformation monoids is used to describe and characterise transitive transformation monoids and primitive transitive transformation monoids we develop a theory that not only includes the analogs of several important theorems of the classical theory of permutation groups but also contains substantial information about the algebraic structure of the transformation monoids open questions naturally arising from the substantial paper of steinberg 2010 11 have been answered our results can also be considered as a further development in the hunt for a solution of the cern conjecture

rigidity theorems on smooth metric measure spaces with weighted poincare inequality we obtain some results on the rigidity of complete smooth metric measure spaces with weighted poincare inequality which generalize previous results in cheng and zhou 2009 dung 2012 dung and sung 2013 fu and li 2010 fu and yang 2012 lam 2010 li 2005 li and wang 2001 2006 wang 2012 as an application we prove that a complete n 1 2n2 stable minimal hypersurface in rn 1 with n 3 must have only one end

extremal curves for weighted elastic energy in surfaces of 3 space forms a variational problem closely related to the bending energy of curves contained is surfaces of real 3 space forms is considered we seek curves in a surface which are critical for the elastic energy when this is weighted by the total squared normal curvature energy under two different sets of constraints clamped curves and one free end curves of constant length we start by deriving the first variation formula and the corresponding euler lagrange equations and natural boundary conditions of these energies and characterize critical geodesics we show how surfaces locally foliated by critical geodesics can be found by using the fundamental theorem of submanifolds in order to find explicit solutions we classify complete rotation surfaces in a real space form for which every parallel is critical

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

groups and actions in transformation semigroups let s be a transformation semigroup of degree n to each element s in s we associate a permutation group g r s acting on the image of s and we find a natural generating set for this group it turns out that the mathcal r class of s is a disjoint union of certain sets each having size equal to the size of g r s as a consequence we show that two mathcal r classes containing elements with equal images have the same size even if they do not belong to the same mathcal d class by a certain duality process we associate to s another permutation group g l s on the image of s and prove analogous results for the mathcal l class of s finally we prove that the schutzenberger group of the mathcal h class of s is isomorphic to the intersection of g r s and g l s the results of this paper can also be applied in new algorithms for investigating transformation semigroups which will be described in a forthcoming paper

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

embedded minimal tori in s3 and the lawson conjecture we show that any embedded minimal torus in s 3 is congruent to the clifford torus this answers a question posed by h b lawson jr in 1970

periodic multiphase solutions of the kadomtsev petviashvili equation n phase solutions of the kadomtsev petviashvili kp equation 1970 that are periodic in space variables x and y were obtained and effectively investigated using the schottky uniformisation of which a short description is given many wave patterns are represented graphically as contour plots and as isometric projections for different parameter values of two three and four phase solutions of the kp equation

simple factor dressing and the lopez ros deformation of minimal surfaces in euclidean 3 space the aim of this paper is to investigate a new link between integrable systems and minimal surface theory the dressing operation uses the associated family of flat connections of a harmonic map to construct new harmonic maps since a minimal surface in 3 space is a willmore surface its conformal gauss map is harmonic and a dressing on the conformal gauss map can be defined we study the induced transformation on minimal surfaces in the simplest case the simple factor dressing and show that the well known lopez ros deformation of minimal surfaces is a special case of this transformation we express the simple factor dressing and the lopez ros deformation explicitly in terms of the minimal surface and its conjugate surface in particular we can control periods and end behaviour of the simple factor dressing this allows to construct new examples of doubly periodic minimal surfaces arising as simple factor dressings of scherk s first surface

the geometry of real sine gordon wavetrains the characterization ofreal n phase quasiperiodic solutions of the sine gordon equation has been an open problem in this paper we achieve this result employing techniques of classical algebraic geometry which have not previously been exploited in the soliton literature a significant by product of this approach is a naturalalgebraic representation of the full complex isospectral manifolds and an understanding of how the real isospectral manifolds are embedded by placing the problem in this general context these methods apply directly to all soliton equations whose multiphase solutions are related to hyperelliptic functions

generalized dpw method and an application to isometric immersions of space forms let g be a complex lie group and g denote the group of maps from the unit circle mathbb s 1 into g of a suitable class a differentiable map f from a manifold m into g is said to be of connection order a b if the fourier expansion in the loop parameter of the mathbb s 1 family of maurer cartan forms for f namely f lambda 1 rm d f lambda is of the form sum i a b alpha i lambda i most integrable systems in geometry are associated to such a map roughly speaking the dpw method used a birkhoff type splitting to reduce a harmonic map into a symmetric space which can be represented by a certain order 1 1 map into a pair of simpler maps of order 1 1 and 1 1 respectively conversely one could construct such a harmonic map from any pair of 1 1 and 1 1 maps this allowed a weierstrass type description of harmonic maps into symmetric spaces we extend this method to show that for a large class of loop groups a connection order a b map for a 0 b splits uniquely into a pair of a 1 and 1 b maps as an application we show that constant non zero sectional curvature submanifolds with flat normal bundle of a sphere or hyperbolic space split into pairs of flat submanifolds reducing the problem at least locally to the flat case to extend the dpw method sufficiently to handle this problem requires a more general iwasawa type splitting of the loop group which we prove always holds at least locally

statistical mechanics of elastica on a plane origin of the mkdv hierarchy in this paper the statistical mechanics of a non stretching elastica in two dimensional space using the path integral method is investigated in the calculation the modified korteweg de vries mkdv hierarchy naturally appeared in the equations including the temperature fluctuation we have classified the moduli of the closed elastica in a heat bath and summed the boltzmann weight with the thermal fluctuation over the moduli due to the bilinearity of the energy functional its exact partition function has been obtained by investigation of the system it is conjectured that an expectation value at a critical point of this system obeys the painleve equation of the first kind and its related equations are extended by the korteweg de vries hierarchy furthermore we also comment on the relation between the mkdv hierarchy and the becchi rouet stora transformation in this system

j simons type integral formula for hypersurfaces in a unit sphere abstract in this paper by investigating compact rotational hypersurfaces m n in a unit sphere s n 1 1 we get some integral formulas and then apply the integral formulas to characterize torus s 1 k n s n 1 n k n

embedded constant mean curvature tori in the three sphere we prove that any constant mean curvature embedded torus in the three dimensional sphere is axially symmetric and use this to give a complete classification of such surfaces for any given value of the mean curvature

sequences of willmore surfaces in this paper we develop the theory of willmore sequences for willmore surfaces in the 4 sphere we show that under appropriate conditions this sequence has to terminate in this case the willmore surface either is the twistor projection of a holomorphic curve into mathbb c mathbb p 3 or the inversion of a minimal surface with planar ends in mathbb r 4 these results give a unified explanation of previous work on the characterization of willmore spheres and willmore tori with non trivial normal bundles by various authors

complete minimal hypersurfaces in the hyperbolic space mathbb h 4 with vanishing gauss kronecker curvature we investigate 3 dimensional complete minimal hypersurfaces in the hyperbolic space h 4 with gauss kronecker curvature identically zero more precisely we give a classification of complete minimal hypersurfaces with gauss kronecker curvature identically zero a nowhere vanishing second fundamental form and a scalar curvature bounded from below

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

a minimax theorem for functions taking values in a riesz space on demontre des theoremes de minimax pour des fonctions prenant des valeurs dans un espace de riesz on donne une bonne extension de la version en espace de riesz du theoreme de hahn banach classique

rotational surfaces of constant astigmatism in space forms abstract a surface in a riemannian space is called surface of constant astigmatism if the difference between the principal radii of curvature at each point is a constant function in this paper we give a classification of all rotational surfaces of constant astigmatism in space forms we also prove that generating curves of such surfaces are critical points of a variational problem for a curvature energy using the description of these curves we locally construct all rotational surfaces of constant astigmatism as the associated binormal evolution surfaces from the generating curves

critical curves for the total normal curvature in surfaces of 3 dimensional space forms abstract a variational problem closely related to the bending energy of curves contained in surfaces of real 3 dimensional space forms is considered we seek curves in a surface which are critical for the total normal curvature energy and its generalizations we start by deriving the first variation formula and the corresponding euler lagrange equations of these energies and apply them to study critical special curves geodesics asymptotic lines lines of curvature on surfaces then we show that a rotation surface in a real space form for which every parallel is a critical curve must be a special type of a linear weingarten surface finally we give some classification and existence results for this family of rotation surfaces

deformations of surfaces associated with integrable gauss mainardi codazzi equations using the formulation of the immersion of a two dimensional surface into the three dimensional euclidean space proposed recently a mapping from each symmetry of integrable equations to surfaces in r3 can be established we show that among these surfaces the sphere plays a unique role indeed under the rigid su 2 rotations all integrable equations are mapped to a sphere furthermore we prove that all compact surfaces generated by the infinitely many generalized symmetries of the sine gordon equation are homeomorphic to a sphere we also find some new weingarten surfaces arising from the deformations of the modified kurteweg de vries and of the nonlinear schrodinger equations surfaces can also be associated with the motion of curves we study curve motions on a sphere and we identify a new integrable equation characterizing such a motion for a particular choice of the curve velocity

capillary surfaces in a convex cone we show that a compact embedded hypersurface s subset mathbb r n 1 with constant higher order mean curvature in a convex piecewise smooth cone c which is perpendicular to c is part of a hypersphere also we prove that an embedded disk type constant mean curvature surface s subset mathbb r 3 in a convex polyhedral cone c which makes constant contact angles with c is a spherical cap if c has at most five faces this condition on the number of faces can be dropped if c is a right cone over a regular n gon and the contact angles are the same on s

surfaces on lie groups on lie algebras and their integrability it is shown that the problem of the immersion of a 2 dimensional surface into a 3 dimensional euclidean space as well as then dimensional generalization of this problem is related to the problem of studying surfaces in lie groups and surfaces in lie algebras a particular case of the general formalism presented here implies that any surface can be characterized in terms of 2 2 matrices using an arbitrary parametrization it is also shown that this generality of parametrization is useful for studying integrable surfaces i e surfaces described by integrable equations in particular starting from a suitable lax pair i e a suitable integrable equation it is possible to construct explicitly large classes of integrable surfaces

nontrivial minimal surfaces in a class of finsler spheres of vanishing s curvature abstract the minimal surface theory in finsler geometry deserves to be well developed as that in riemannian geometry in this paper we derive the mean curvature of submanifolds in a general manifold by considering the busemann hausdorff measure and holmes thompson measure respectively we then study the rotationally invariant minimal surfaces in the finsler 3 sphere endowed with an metric f k k k k k 1 where is a smooth function s 3 k is the berger sphere e 4 k 1 and k is a killing one form of constant length along the hopf fibers of s 3 we define the energy of the minimal surfaces and by using the volume ratio function introduced by the author and y b shen we give the explicit local expressions of the rotationally invariant bh minimal and ht minimal surfaces in such sphere respectively as a special case we give a detailed study of the rotationally invariant ht minimal surfaces in the 3 sphere with square metric

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

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

a spectral curve approach to lawson symmetric cmc surfaces of genus 2 minimal and cmc surfaces in s 3 can be treated via their associated family of flat sl 2 mathbb c connections in this the paper we parametrize the moduli space of flat sl 2 mathbb c connections on the lawson minimal surface of genus 2 which are equivariant with respect to certain symmetries of lawson s geometric construction the parametrization uses hitchin s abelianization procedure to write such connections explicitly in terms of flat line bundles on a complex 1 dimensional torus this description is used to develop a spectral curve theory for the lawson surface this theory applies as well to other cmc and minimal surfaces with the same holomorphic symmetries as the lawson surface but different riemann surface structure additionally we study the space of isospectral deformations of compact minimal surface of genus g ge 2 and prove that it is generated by simple factor dressing