View source for Quadratic differential

''on a Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q0760401.png" />''

A rule which associates to each local parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q0760402.png" /> (cf. [[Local uniformizing parameter|Local uniformizing parameter]]) mapping a parametric neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q0760403.png" /> into the extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q0760404.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q0760405.png" />), a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q0760406.png" /> such that for any local parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q0760407.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q0760408.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q0760409.png" /> non-empty, the following holds in this intersection:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604011.png" /> is the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604012.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604013.png" /> under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604014.png" />. A quadratic differential is often denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604015.png" />, to which is attributed the invariance with respect to the choice of the local parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604016.png" />, as indicated by (1). In other words, a quadratic differential is a non-linear differential of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604017.png" /> on a Riemann surface.

The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604018.png" /> entering into the definition of a quadratic differential are ordinarily assumed to be measurable or even analytic. In the latter case the quadratic differential is called analytic. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604019.png" /> is called a zero (or pole) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604020.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604021.png" /> if for each local parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604022.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604023.png" /> has a zero (or pole) of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604024.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604025.png" />. The zeros and poles of a quadratic differential are called critical points of it. The zeros and simple poles are called finite critical points and their totality is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604026.png" />. The set of all poles of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604027.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604028.png" />. If a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604029.png" /> has a tangent at each of its points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604030.png" /> with respect to the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604031.png" />, with tangent vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604032.png" />, and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604034.png" /> is said to be positive, and one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604035.png" />, on the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604036.png" />. If (2) holds with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604037.png" /> sign replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604038.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604039.png" /> is called negative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604040.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604041.png" />. Each maximal regular curve on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604042.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604043.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604044.png" />) is called a trajectory (or orthogonal trajectory) of the quadratic differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604045.png" />.

A quadratic differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604046.png" /> defined on a finite Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604047.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604048.png" /> if the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604049.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604050.png" /> is either empty or consists of a finite number of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604051.png" /> and arcs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604052.png" /> on each of which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604053.png" /> is regular and positive or negative. If, furthermore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604054.png" /> is empty or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604055.png" /> is regular and positive on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604056.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604057.png" /> is called a positive quadratic differential on the Riemann surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604058.png" />. The metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604059.png" />, called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604061.png" />-metric, is single-valued on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604062.png" /> and invariant with respect to the choice of the local parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604063.png" />.

In some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604064.png" /> of any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604065.png" />, the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604066.png" /></td> </tr></table>

is regular, single-valued and univalent for each choice of the sign of the integrand; furthermore, each maximal arc of a trajectory (or orthogonal trajectory) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604067.png" /> is converted under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604068.png" /> into a horizontal (or vertical) line interval. Therefore, through each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604069.png" /> passes a trajectory which is either an open arc or a Jordan curve on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604070.png" />. The topological and conformal structures of the family of trajectories in a small neighbourhood of each critical point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604071.png" /> are completely classified in their dependence on the order of the critical point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604072.png" /> and (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604073.png" /> is a pole of the second order and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604074.png" />) on

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076040/q07604075.png" /></td> </tr></table>

(see [[Local structure of trajectories|Local structure of trajectories]]). A description of the [[Global structure of trajectories|global structure of trajectories]] is known for finite Riemann surfaces and has many important applications (see also [[#References|[1]]]).

O. Teichmüller has investigated the role of quadratic differentials in the theory of extremal conformal and quasi-conformal mapping and in the solution of moduli problems of Riemann surfaces (see [[#References|[1]]]–[[#References|[3]]]). He formulated a principle according to which certain quadratic differentials can be associated with extremal problems in the geometric theory of functions, where to each type of extremal problem correspond specific singularities of the quadratic differential (poles), and the geometric properties of the solution are related in a suitable fashion to the structure of the trajectories of the quadratic differential. Inequalities for the coefficients of univalent functions (cf. [[Univalent function|Univalent function]]) have been proved in terms of quadratic differentials. A more general inequality for the coefficients of univalent functions in families of domains distributed on a finite Riemann surface is called the general coefficient theorem and is a concrete realization of the Teichmüller principle for a wide class of problems (see [[#References|[1]]], [[#References|[4]]]). The Teichmüller principle also enables one to establish a special coefficient theorem and to solve a large number of concrete extremal problems (see [[#References|[1]]], [[#References|[5]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.A. Jenkins, "Univalent functions and conformal mapping" , Springer (1958) {{MR|0096806}} {{ZBL|0083.29606}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M. Schiffer, D.C. Spencer, "Functionals of finite Riemann surfaces" , Princeton Univ. Press (1954) {{MR|0065652}} {{ZBL|0059.06901}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L.V. Ahlfors, L. Bers, , ''Spaces of Riemann surfaces and conformal mapping'' , Moscow (1961) (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> P.M. Tamrazov, "On the general coefficient theorem" ''Math. USSR Sb.'' , '''1''' : 1 (1967) pp. 49–59 ''Mat. Sb.'' , '''72''' : 1 (1967) pp. 59–71 {{MR|0217284}} {{ZBL|0164.37902}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> J.A. Jenkins, "Some area theorems and a special coefficient theorem" ''Illinois J. Math.'' , '''8''' : 1 (1964) pp. 80–99 {{MR|0160888}} {{ZBL|0131.07601}} </TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Strebel, "Quadratic differentials" , Springer (1984) (Translated from German) {{MR|0766263}} {{MR|0753331}} {{MR|0743423}} {{ZBL|0547.30038}} {{ZBL|0547.30001}} </TD></TR></table>