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an important class of problems in mathematics take the following prototypical form : given a riemmanian manifold , find a conformally equivalent metric satisfying certain prescribed properties . conformal mapping problems _ arise ubiquitiously , and their solutions are invaluable to many areas of applied mathematics , physics , and engineering .    inspired largely by these applications , considerable work has been done in developing notions of _ discrete conformal maps_. of particular interest have been discretizations that preserve certain structural properties and characteristics of their continuum counterparts . such discretizations have shown to contain a surprisingly profound and rich theory of their own , and their study is an exciting and flourishing area of mathematics today .    the main purpose of this paper is to note a correspondence between continuum conformal mapping problems that arise in the study of vortices in the abelian higgs field theory ( see @xcite ) and certain discrete conformal mapping problems studied in @xcite . the abelian higgs field theory first arose as a phenomenological model of superconductivity . the model admits vortex solutions , which are topogically stable , finite energy solutions to the field equations . physically , the vortices are crucial in understanding the meissner effect of magnetic repulsion in superconductors . it was observed by witten @xcite that in some circumstances , the equations describing vortices are closely related to the liouville equation for the conformal factor of constant curvature surfaces , which he exploited to construct explicit vortex solutions . this was further generalized by recent work by @xcite , who showed that the vortex equations can be reformulated as a conformal mapping problem involving prescribed linear combinations of the gaussian curvature and the riemannian volume form .    on the other hand , @xcite studied discrete conformal maps of triangulations , in which metrics correspond to edge lengths and conformal transformations to scale factors associated with vertices . these notions were generalized to hyperbolic and spherical triangulations ( see @xcite ) . several types of associated mapping problems were studied and shown to be solvable by variational principles . we argue that these discrete conformal problems correspond naturally to continuum problems involving combinations of the curvature and volume form . we then explain how the discrete conformal theory can be used to construct discrete vortex solutions . acknowledgements : we graciously thank thilo rrig for help producing figure [ fig : vortices ] using the software varylab @xcite . this research was supported by the dfg collaborative research center trr 109 `` discretization in geometry and dynamics . '' the classical abelian higgs field theory is defined on a hermitian line bundle @xmath0 over a closed , oriented two dimensional riemannian manifold @xmath1 . the fields are a section @xmath2 of @xmath3 , and a hermitian @xmath4-connection @xmath5 on @xmath3 . the action functional of the higgs field theory is given by @xmath6 =   \int_m \left ( | f_a|^2 - 2 c \ ; | d_a \phi |^2 + \left(c | \phi |^2 - c_0 \right)^2 \right ) \ ; dv,\end{aligned}\ ] ] where @xmath7 and @xmath8 are real constants , @xmath9 is the curvature of the connection , and @xmath10 is the covariant derivative , and @xmath11 is the riemannian volume element .    in a trivializing coordinate chart , the connection can be represented by a differential one form @xmath12 . the curvature two form is @xmath13 , and the covariant derivative @xmath14 . the euler - lagrange equations are given by @xmath15 where @xmath16 is the hodge star . the vortex equations are a set of first order equations describing stationary points of the abelian higgs action ; when @xmath17 , solutions of the vortex equations are global minima of the action functional . the equations are found after rearranging the action by exploiting the complex structure on @xmath1 . recall that the hodge star induces a complex structure and gives a decomposition of complexified forms @xmath18 , and of the exterior derivative @xmath19 . there is correspondingly a decomposition @xmath20 and a splitting @xmath21 . in addition , denote by @xmath22 the contraction with volume form ; on a two dimensional manifold , @xmath23 for any @xmath24 , and @xmath25 is the pointwise inner product on forms induced by the riemannian metric .    the action ( [ eq : action ] ) can then be rewritten ( see @xcite ) as @xmath26 & = \int_m \left ( \left ( \lambda f_a + ( c_0 - c | \phi |^2 ) \right)^2   - c | \bar \partial_a \phi | ^ 2 \right)\ ; dv + \int_m c_0 f_a.\end{aligned}\ ] ] since the integral of the curvature is quantized and independent of the connection @xmath27 where @xmath28 is the chern number and an invariant of the bundle @xmath29 , the action can then be written @xmath30= \int_m \left ( \left ( \lambda f_a + ( c_0 - c | \phi |^2 ) \right)^2   - 2 c | \bar \partial_a \phi | ^ 2 \right ) \ ; dv + 4 \pi n c_0 . \label{eq : action2}\end{aligned}\ ] ]    the _ vortex equations _ demand that the each term of the integrand vanishes : @xmath31 the vortex equations imply the euler - lagrange ( [ eq : el ] ) equations . moreover , when @xmath32 , each term of the integrand ( [ eq : action2 ] ) is positive definite , from which it follows that solutions to the vortex equations minimize the action . the first vortex equation ( [ eq : vorteq1 ] ) says that @xmath33 is gauge - covariantly holomorphic . it is convenient to assume complex coordinates @xmath34 , in which the metric takes the conformal form @xmath35 the connection can be written @xmath36 , where by unitarity , @xmath37 .    in these coordinates , the equation ( [ eq : vorteq1 ] ) can be expanded as @xmath38 and be integrated @xmath39 this yields for the curvature @xmath40    since @xmath33 is gauge covariantly holomorphic , it has isolated zeroes at points @xmath41 , called the _ vortex centers _ , with multiplicities @xmath42 , called _ vortex numbers_. from ( [ eq : vorcurv ] ) , near each vortex center @xmath41 , the curvature has a singularity as @xmath43 of the form @xmath44 it follows that the total vortex number is given by @xmath45    turning to the second vortex equation ( [ eq : vorteq2 ] ) , using @xmath46 , one finds @xmath47 where @xmath48 is the laplace - beltrami operator . in terms of the function @xmath49 defined by @xmath50 , this can be written as @xmath51      by rescaling first the metric and then shifting @xmath52 ( rescaling @xmath33 ) , the constants @xmath7 and @xmath8 can without loss of generality be assumed to take the values @xmath53 , giving nine vortex equations . however , since @xmath33 is covariantly holomorphic and has only zeroes , ( [ eq : vortnumber ] ) implies that @xmath54 . then , since by ( [ eq : vorteq2 ] ) @xmath55 it follows that solutions only exist if the integrand on the right side can be positive . this discludes the four cases @xmath56 , and yields five remaining vortex equations :    1 . taubes vortex . 2 .   @xmath58 . bradlow vortex . 3 .   @xmath59 . ambjrn - olesen vortex . ( see @xcite ) 4 . @xmath60 . jackiw - pi vortex . popov vortex . the gaussian curvature @xmath62 under conformal scaling of the metric @xmath63 transforms as @xmath64 so the taubes vortex equation corresponds to the liouville equation in constant negative curvature . this was pursued further in @xcite , who study the conformally rescaled metric @xmath65 the baptista metric is degenerate , with cone type singularities at the vortex centers where @xmath66 vanishes , with excess angle @xmath67 . the curvature @xmath68 of the baptista metric away from the vortex centers can be computed using liouville s equation , and using the vortex equation ( [ eq : vortliouville ] for @xmath69 to obtain @xmath70 rearranging gives @xmath71 denoting by @xmath72 and @xmath73 the volume forms , the baptista equation can equivalently be written as @xmath74 a field configuration @xmath75 is a solution to the vortex equations if and only if the associated baptista metric satisfies ( [ eq : bapt ] ) away from the vortex centers . a _ triangulated surface _ consists of a compact oriented surface @xmath76 , a finite simplicial complex @xmath77 , and a homeomorphism @xmath78 . combinatorially , the complex @xmath77 consists of vertices @xmath79 , edges @xmath80 , and faces @xmath81 . we will enumerate vertices , and denote by @xmath82 the edge adjacent to vertices @xmath83 and @xmath84 , and by @xmath85 the face adjacent to vertices @xmath86 and @xmath87 . a _ geodesically triangulated surface _ is a triangulated surface such that image of each edge of @xmath77 under the homeomorphism @xmath88 is geodesic in @xmath76 . a _ euclidean triangulated surface _ is a geodesically triangulated surface with a smooth riemannian metric defined on @xmath89 such each point of @xmath89 is locally isometric to the euclidean plane . in a _ hyperbolic triangulated surface _ , each point of @xmath89 is locally isometric to the hyperbolic plane . in a _ spherical triangulated surface _ , each point of @xmath89 is locally isometric to the sphere . such triangulated surfaces can be obtained by gluing a finite collection of either euclidean , hyperbolic , or spherical triangles along edges . these surfaces have constant gaussian curvature except at vertices where , the metric may have a conical singularity . we denote by @xmath90 the total angle about vertex @xmath91 . a discrete metric on @xmath77 is the pair @xmath92 , where @xmath93 describes the geometry of the faces , and @xmath94 the length of edges . the map @xmath95 satisfies the triangle inequalities @xmath96 in addition , for the case of spherical geometry @xmath97 must satisfy @xmath98 a discrete metric satisfying these inequalities defines a unique triangulated surface that we will denote @xmath99 .      the distance function @xmath100 on a triangulated surface @xmath101 is defined as @xmath102 the distance function can be understood by imagining the hyperbolic plane and sphere as isometrically embedded in @xmath103 and @xmath104 respectively ; then @xmath105 is the ambient distance of two points seperated by intrinsic distance @xmath95 , see figure [ fig : ll ] . two discrete metrics @xmath106 and @xmath107 are _ discrete conformally equivalent _ if there exists a function ( logarthmic scale factor ) @xmath108 such that @xmath109note that we do not require the geometries @xmath110 and @xmath111 to be the same . for the euclidean geometry , this definition first appeared in @xcite , for the spherical and hyperbolic geometry in @xcite , and in all generality in @xcite .      in @xcite , several conformal mapping and uniformization problems were studied in detail . in this note , we focus in particular on the following special case :    [ cmp ] given a triangulated surface with discrete metric @xmath92 , find a conformally equivalent metric @xmath112 with prescribed conical angle @xmath113 and prescribed geometry @xmath114 \{euclidean , hyperbolic , spherical}.    let us now describe the analagous continuum conformal mapping problem . the conical defect angle has often been considered as a discretization of the guassian curvature . more precisely ( see @xcite ) , for a euclidean triangulation , the defect angle is a discretization of curvature two form @xmath115 . this is evident from the scaling behaviour : upon rescaling , both the curvature form and angular defect are preserved . for the more general case of hyperbolic and spherical triangulations , the conical defect angle should be understood as a discretization of a linear combination of the curvature two form and the volume form . this is motivated as follows :    consider the voronoi cell @xmath116 of a vertex @xmath117 . it consists of a finite number of geodesic triangles @xmath118 for @xmath119 , and has a piecewise geodesic boundary @xmath120 consisting of smooth segments @xmath121 as shown in figure [ fig : vor ] . ( p0 ) at ( -1,.1 ) ; ( p1 ) at ( -.6,-.5 ) ; ( p2 ) at ( 0,-.8 ) ; ( p3 ) at ( .6,-.3 ) ; ( p4 ) at ( .5,.5 ) ; ( p5 ) at ( -.4,.8 ) ; ( c0 ) at ( 0,0 ) ; ( e0 ) at ( 0.3 , -0.95 ) ;    ( p0)(p1)(p2)(p3)(p4)(p5)(p0 ) ; ( c0)(p0 ) ; ( c0)(p1 ) ; ( c0)(p2 ) ; ( c0)(p3 ) ; ( c0)(p4 ) ; ( c0)(p5 ) ; ( p2)(e0 ) ;    ; ; ; ;    the integral of the curvature form over @xmath116 is given by the gauss - bonnet theorem , including terms accounting for boundary as @xmath122 here @xmath123 is the turning angle between the adjacent segments @xmath124 and @xmath125 ( see figure [ fig : vor ] ) . rewriting in terms of internal angles gives : @xmath126 for spherical / hyperbolic triangles , the area @xmath127 of a triangle is related to the angles of the triangle by @xmath128 so that in any case @xmath129 where @xmath130 for @xmath131 spherical , euclidean , and hyperbolic respectively .    substituting this expression into the sum ( [ eq : sum2 ] ) gives : @xmath132 or upon rearranging @xmath133    thus , the conical angle @xmath134 corresponds most naturally to the two form @xmath135 . the analogous continuous mapping problem is :    [ ccmp ] given a riemannian manifold @xmath136 , find a conformally equivalent metric with prescribed linear combination of curvature form and area form @xmath137 where @xmath138 . the discrete conformal mapping problem [ cmp ] can be solved by variational principles that we briefly review now ( see @xcite for further details ) . fix a triangulated surface @xmath139 , along with a target geometry @xmath140 and target angles @xmath141 . the _ feasible region _ @xmath142 is the subset of logarthmic conformal factors @xmath143 such that conformally transformed @xmath144 defined via equation ( [ eq : confrmequiv ] ) defines a discrete metric in the @xmath111 geometry ; i.e. @xmath145 should satisfy the triangle inequalities ( [ eq : tri ] ) , and also inequality ( [ eq : sphc ] ) in the case that @xmath140 is spherical . it is convenient to introduce the logarithmic lengths @xmath146 in terms of @xmath147 , equation ( [ eq : confrmequiv ] ) takes the form @xmath148    the functional @xmath149 is defined as @xmath150 the function @xmath151 here is defined as @xmath152 where @xmath153 is the lobachevsky function @xmath154 the angles @xmath155 and @xmath156 are related to the triangle ( in @xmath140 geometry ) with edge lengths @xmath157 and its circumcircle as shown in figure ( [ fig : tricirc ] ) . \(a ) at ( 0.173648 , 0.984808 ) ; ( b ) at ( -0.939693 , -0.34202 ) ; ( c ) at ( 0.642788 , -0.766044 ) ; ( c1 ) at ( 0.939693 , -0.34202 ) ; ( a1 ) at ( 0.642788 , 0.766044 ) ; plot ( cos(),sin ( ) ) ; ( a)(b ) node [ above , left , midway ] ; ( b)(c ) node [ below , midway ] ; ( c)(a ) node [ right , midway ] ; ; ; ;    [ fig : tricirc ]    algebraically , the @xmath158 and @xmath159 are related by @xmath160 and the @xmath158 can be computed explicitly from the edge lengths using the formulas @xmath161    every solution of the conformal mapping problem [ cmp ] corresponds via ( [ eq : confrmequiv ] ) to a critical point of the functional @xmath162 . conversely , a critical point of @xmath162 within the feasible region @xmath142 gives a solution to the conformal mapping problem [ cmp ] . furthermore it was shown in @xcite for @xmath111 euclidean , and in @xcite for @xmath111 hyperbolic that the functional @xmath162 is convex in @xmath142 , and can be extended to a convex function defined on @xmath163 . this is crucial both theoretically and practically . the convexity implies that critical points of @xmath162 are global minima of the functional . moreover , when @xmath111 is hyperbolic , the functional is strictly convex and so has a unique minimum direction . when @xmath111 is euclidean , the minimum of the functional is unique up to an overall scale . practically , for numerical calculations , this implies that one can utilize powerful numerical tools developed for convex optimization .    in the case @xmath111 is spherical , @xmath164 is not convex , and solutions may not correspond to minima . in the context of the vortex equations , this corresponds to the fact that vortices with @xmath165 do not correspond to minima of the higgs action functional , and are not stable . nonetheless , with some modification ( see @xcite ) , the standard numerical methods can be succesfully employed . on the other hand , the existence of solutions to the conformal mapping problem [ cmp ] is more complicated . solutions to the discrete conformal mapping problem may fail to exist due to violated triangle equalities ( [ eq : tri ] ) . on the other hand , if we regard the underlying surface @xmath76 with constant curvature and conical singularities as the fundamental independent of its triangulation , then it is natural to consider re - triangulations . indeed , the theory of discrete conformal equivalence can be extended to combinatorially inequivalent triangulations ( see @xcite ) . this generalization leads to progress with the existence problem . in particular , for compact surfaces , a discrete uniformization theorem has been proven in @xcite . in this section , we demonstrate how the discrete conformal mapping problem can be used to construct vortex solutions . comparing the vortex equation in the form ( [ eq : bapt ] ) to the conformal mapping problem [ ccmp ] , it is clear how to proceed : given a riemannian manifold , we choose a triangulation @xmath166 with @xmath110 chosen so that @xmath167 . the conical angles @xmath168 can then be calculated at each vertex . a solution to the vortex equation then corresponds to solving the conformal mapping problem [ cmp ] with geometry @xmath111 chosen so that @xmath169 , and identical conical angles @xmath170 except at vortex centers , where @xmath171 . bobenko , s. sechelmann , b. springborn . discrete conformal maps : boundary value problems , circle domains , fuchsian and schottky uniformization . _ advances in discrete differential geometry . _ springer berlin ( 2016 ) , 156
we establish a connection between recent developments in the study of vortices in the abelian higgs models , and in the theory of structure - preserving discrete conformal maps . 
 we explain how both are related via conformal mapping problems involving prescribed linear combinations of the curvature and volume form , and show how the discrete conformal theory can be used to construct discrete vortex solutions .