We prove that the equilibrium mass support complement satisfies a quadratic identity. We briefly investigate this problem and then narrow our focus to the study of the topology of classical quadratic fields.
Notation Index
The logarithmic potential theory is a plane theory and as such is not applicable to problems in these areas. Along the way we prove new results in the classical setting of logarithmic and quadrature potential domains.
Overview and Main Results
- Logarithmic Potential Theory on Riemann Surfaces
- Quadrature Domains and Laplacian Growth on Surfaces
- Algebraic Quadrature Domains
- Coulomb Gas Ensembles and CFT
- Overview
- Logarithmic Potentials
- Energy Problem
In [30], Xia provides a definition of quadrature domains on Riemann surfaces and studies an analogue of the exponential transformation. One of the central concerns of logarithmic potential theory is the precise understanding of µeq.
Quadrature Domains
Subharmonic Quadrature Domains and Partial Balayage
The main theorems of this chapter show, under mild hypotheses, that the equilibrium measure has an a priori structure. Using the first structure theorem, we prove a formula for the equilibrium measure structure when Q is subharmonic in the neighborhood of S, but no other regularity is assumed.
Interaction Kernel
Existence, Uniqueness, and Properties of the Interaction Kernel
We have thus shown that the interaction kernel is unique up to an additive constant. In this example we calculate the interaction kernel for the torus with background measure equal to the volume measure onT2 induced by the flat metric, g.
Physical Interpretation
In the absence of an external field, a charge distribution µinR3 can never be in static equilibrium. In what follows we will be interested in positive charge distributions with minimal energy of configuration for a given total charge. If the external field is properly chosen, there will exist a corresponding charge distribution µeq of positive density and total charge one such.
Physically, any motion would conserve the total energy and thus reduce the potential energy, which contradicts the fact that the charge distribution given by µeq has a minimum potential energy. Due to our assumption that the background measurement does not interact with other charges or with itself, the potentialUµgenerated by the charge distribution isµisR.
The Equilibrium Measure
The fact that the definition of the null capacity set is independent of the metric follows from property (iii) of Gg. The importance of the concept of admissibility is that if Q is admissible, then KνQ is lower semi-continuous. Later in sections 3.5 and 3.6, we analyze the dependence of the equilibrium measure on the background measure and the external field.
For the proof of the following statement it will be convenient to repeat Lemmas 3.4.10 and 3.4.11 using a. It is easy to check that T is lower semi-continuous and using Proposition 3.2.9 we have.
Structure of the Equilibrium Measure
Structure Theorems
We used equation 3.45 in the first equation, the fact that the mass of ν(M)µeq−ν+∆gQvol˜ is zero in the second equation, equation 3.44 in the third equation (to show that |R. As in We define κ as the balayage of 1Sc[ν−∆gQvol˜ g] to ∂S We now show that the partial derivatives (in the sense of distributions) of Ψ are bounded from below.
Putting this together, it follows that the Laplacian of 3.73 is bounded from below in the sense of distributions. Since z∗∈S was arbitrary, it follows, with respect to any diagram, that the partial derivatives up to order two of (Uµνeq+Q) are equal to zero almost everywhere onS.
The Structure of the Equilibrium Measure on C
It is not difficult to see that µ =µeq- the equilibrium measure in the classical sense corresponds to the equilibrium measure on ˆC which corresponds to the background measure 2π1δ∞. We can then apply the structure theorems to obtain results about the structure of the equilibrium measure at the level. We begin by showing that a well-known result about the structure of the equilibrium measure in the plane follows directly from the second structure theorem.
Then we use the first structure theorem to prove a new result about the structure of the equilibrium measure in a very general setting. Using the first structure theorem, we prove a formula for the structure of the equilibrium measure when Q is subharmonic near S, but no other regularity is assumed.
Obstacle Problem
Etingof and Varchenko [28] outlined the beginnings of a theory of Laplacian growth on Riemannian surfaces equipped with a Riemannian metric. If a Riemannian surface,M, is equipped with a Riemannian metric,g, then we can define Laplacian growth onM analogously. We introduce a weighted version of Laplacian growth on Riemann surfaces which depends only on the complex structure and the weight.
We discuss an interesting relationship between logarithmic potential theory on Riemann surfaces and Q-weighted Laplace growth (where Q is the external field). In the classical case, this unifies the known methods of producing inner and outer Laplace growth in 2D.
Quadrature Domains on Surfaces
It is also easy to show that the disks are also one-point subharmonic quadrature domains. We prove that the bounded one-point harmonic and subharmonic quadrature domains on the Poincar disk are exactly the hyperbolic disks. Similarly, it is sufficient to show that hyperbolic disks centered at zero are subharmonic quadrature domains.
We begin by showing that hyperbolic discs centered at 0 with subharmonic quadrature domains areacare with the data (cδ0, µh). Thus, we have shown that hyperbolic disks that do not contain cycles are subharmonic quadrature domains.
Laplacian Growth on Surfaces
At the same time, we prove a new upper bound on the number of special points of a quadrature domain as a function of its degree. At the same time, we provide a new upper limit for the number of special points. More interestingly, the singular points are isolated points of the algebraic curve that defines the limit of Ω.
In particular, there was interest in providing upper and lower bounds on the number of special points. Let b denote the number of vertices on ∂Ω, c denote the connectivity of Ω, and d the number of double points of ∂Ω.
Algebraic Quadrature Domains
Conversely, if D is an AQD for R, then there exists a compact Riemann surface N satisfying the above hypotheses and a meromorphic function ψon N satisfying. By the definition of anAQD, there exists a univalent mapκ : D → E, where E is a finite-bounded Riemann surface which extends to a continuous injection fromD→ E. Then there exists a compact Riemann surface N of genus gc and a meromorphic function T in N such that
Since Dis is an AQD, there exists a finitely bounded Riemann surface, E, and a continuous and injective map φ:D → E, which is univalent on D. Since Dis is an AQD, there exists a univalent map κ : D → E, where E is a finitely bounded Riemann surface extending to a continuous injection of D → E.
Topology of Quadrature Domains
Let b denote the number of cusps on ∂Ω, c denote the connectivity of Ω, and let denote the number of doubles on ∂Ω. In case ¯S has critical values at ∂Ω, ˆA is defined in the same way, but with a small modification (see [24). LetΛ≡int(Ω), let ˜c denote the connectivity ofΛ, and let ˜s denote the number of special points inΛ.
The internal energy component in the integrand of 6.1 is related to the vacuum expectation of vertex operators. The proof follows the structure of Johansson's [18] proof of the convergence of the marginal measures of β-ensembles onR.
Fekete Points
Johansson [18] proved this result on Rand Hedenmalm and Makarov [15] later proved the analogue in the complex plane. We then show that the Bosonic free field on the cylinder can be realized as a limit of Fluctuations of a Coulomb gas ensemble on the cylinder. In a previous chapter we studied the equilibrium measure—the measure in P(M) that minimizes the energy functional.
It is interesting, but perhaps not surprising, that the Fekete measures converge to the equilibrium measure. It then follows from 6.30 and the definition of the equilibrium measure that I[µeq]=I[µ] and µ=µeq which proves 6.16.
The Boltzmann Gibbs Distribution
In the following, we will eliminate the dependence of β, ν and Q in the notation for Z and Π, unless it is necessary. For ease of notation, we will eliminate the metric dependence and denote the geodesic sphere centered at x with radius δ simply as B(x, δ). Johansson's limit measure theorem on Riemann surfaces Let Q be admissible and Qν,g continuous in the neighborhood of S.
Then it clearly suffices to show that for any sequence of measures {µzn}n∈N, wherezn∈An,η, µzn* µeqasn→ ∞(because µn,ηmax∈Pn(An,η)) holds. It suffices to show that for every subsequence {µni}i∈N there exists a further subsequence that weakly converges to µeq.
The Free Bosonic Field
- Conformal Structure
- Complex Structure
- Moduli Space
- Curvature
- The Classical View
- The Quantum View
It is easy to see that the set of all conformal diffeomorphisms F : (M,[g]c) → (M,[g]c) forms a group that we call Aut(M,[g]c). The group Aut[M,[g]c) is called the automorphism group and its elements are called conformal automorphisms. It is easy to see that this is true iffF∗[g]c = [h]c and equivalently F is a conformal diffeomorphism. It is an easy exercise to check that ωg is independent of the choice of the holomorphic coordinate and that soωg extends to a (1-1)-form on M.
We now concentrate on understanding the case of the free bosonic action and the free bosonic action coupled with curvature. For more general actions there will be no “split” of action and so the family of measurements {µn}∞n=1 will generally not be consistent and thus may not give rise to a measure of Hg.
Bosonic Conformal Fields
Correlation Functions
In conformal field theory, we are mainly interested in fields that have conformal invariant correlation functions. In other words, if the free bosonic field coupled with the curvature is conformal, then so is the free bosonic field. So a necessary condition for the free bosonic field coupled to the curvature to be conformal is that it be automorphic.
In the previous section, it suffices to show that ifgis is automorphic, then both ˜E[Φ(x)] and the free bosonic field are conformally invariant. It is easy to see that ι is invariant under a holomorphic coordinate change and thus extends to a globally defined (1-1) form.
Next Steps: Linear Statistics on Coulomb Gas Ensembles and CFT
The study of these linear statistics in the special case where M = C,ˆ ν = 2πδ∞, andβ = 2, are of great interest and arise from the study of the distribution of eigenvalues of ensembles of normal random matrices. In the statement of the theorem, *denotes weak convergence, N denotes the normal distribution, and fS denotes the function defined on S by f|S and defined on Scas the harmonic extension of f|∂S to Sc. If M is a general compact Riemann surface and the external field is chosen such that the equilibrium mass support is all M, an analog of Theorem 6.6.1 would realize bosonic fields on compact Riemann surfaces as boundaries weak linear statistics for the Coulomb Ensemble gas.
Gustafsson, Singular and special points on quadratic domains from an algebraic geometric perspective, J. Neumann, From the Fundamental Theorem of Algebra to Astrophysics: A "Harmonic" Path, Notices of the AMS 55, Issue 6 (2008).
