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Inmathematics, theSchwarz lemma, named afterHermann Amandus Schwarz, is a result in complex differential geometry that estimates the (squared) pointwise norm${\displaystyle |\mathrm{\partial }f{|}^2}$ of a holomorphic map${\displaystyle f:(X,g_X)\to (Y,g_Y)}$ between Hermitian manifolds under curvature assumptions on${\displaystyle g_X}$ and${\displaystyle g_Y}$.

Theclassical Schwarz lemma is a result incomplex analysis typically viewed to be aboutholomorphic functions from theopenunit disk to itself.

The Schwarz lemma has opened several branches of complex geometry, and become an essential tool in the use of geometric PDE methods in complex geometry.

Let be the openunit disk in thecomplex plane${\displaystyle \mathbb{C}}$ centered at theorigin, and let${\displaystyle f:\mathbf{D}\to \mathbb{C}}$ be aholomorphic map such that${\displaystyle f(0)=0}$ and${\displaystyle |f(z)|\le 1}$ on${\displaystyle \mathbf{D}}$.

Then${\displaystyle |f(z)|\le |z|}$ for all${\displaystyle z\in \mathbf{D}}$, and${\displaystyle |f^{\prime }(0)|\le 1}$.

Moreover, if${\displaystyle |f(z)|=|z|}$ for some non-zero${\displaystyle z}$ or${\displaystyle |f^{\prime }(0)|=1}$, then${\displaystyle f(z)=az}$ for some${\displaystyle a\in \mathbb{C}}$ with${\displaystyle |a|=1}$.^[1]

The proof, which first appears in a paper byCarathéodory,^[2] where it is attributed to Erhard Schmidt, is a straightforward application of themaximum modulus principle on the function

which is holomorphic on the whole of${\displaystyle \mathbb{D}}$, including at the origin (because${\displaystyle f}$ is differentiable at the origin and fixes zero). Now if${\displaystyle {\mathbb{D}}_r=\{z:|z|\le r\}}$ denotes the closed disk of radius${\displaystyle r}$ centered at the origin, then the maximum modulus principle implies that, for, given any${\displaystyle z\in {\mathbb{D}}_r}$, there exists${\displaystyle z_r}$ on the boundary of${\displaystyle {\mathbb{D}}_r}$ such that

${\displaystyle |g(z)|\le |g(z_r)|=\frac{|f(z_r)|}{|z_r|}\le \frac{1}{r}.}$

As${\displaystyle r\to 1}$ we get${\displaystyle |g(z)|\le 1}$.

Moreover, suppose that${\displaystyle |f(z)|=|z|}$ for some non-zero${\displaystyle z\in \mathbb{D}}$, or${\displaystyle |f^{\prime }(0)|=1}$. Then,${\displaystyle |g(z)|=1}$ at some point of${\displaystyle \mathbb{D}}$. So by the maximum modulus principle,${\displaystyle g(z)}$ is equal to a constant${\displaystyle a}$ such that${\displaystyle |a|=1}$. Therefore,${\displaystyle f(z)=az}$, as desired.

A variant of the Schwarz lemma, known as theSchwarz–Pick theorem (afterGeorg Pick), characterizes the analytic automorphisms of the unit disc, i.e.bijectiveholomorphic mappings of the unit disc to itself:

Let${\displaystyle f:\mathbf{D}\to \mathbf{D}}$ be holomorphic. Then, for all${\displaystyle z_1,z_2\in \mathbf{D}}$,

${\displaystyle \left|\frac{f(z_1)-f(z_2)}{1-\overline{f(z_1)}f(z_2)}\right|\le \left|\frac{z_1-z_2}{1-\overline{z_1}{z}_2}\right|}$

and, for all${\displaystyle z\in \mathbf{D}}$,

${\displaystyle \frac{\left|f^{\prime }(z)\right|}{1-{\left|f(z)\right|}^2}\le \frac{1}{1-{\left|z\right|}^2}.}$

The expression

${\displaystyle d(z_1,z_2)={\tanh }^{-1}\left|\frac{z_1-z_2}{1-\overline{z_1}{z}_2}\right|}$

is the distance of the points${\displaystyle z_1}$,${\displaystyle z_2}$ in thePoincaré metric, i.e. the metric in thePoincaré disk model forhyperbolic geometry in dimension two. The Schwarz–Pick theorem then essentially states that a holomorphic map of the unit disk into itselfdecreases the distance of points in the Poincaré metric. If equality holds throughout in one of the two inequalities above (which is equivalent to saying that the holomorphic map preserves the distance in the Poincaré metric), then${\displaystyle f}$ must be an analyticautomorphism of the unit disc, given by aMöbius transformation mapping the unit disc to itself.

An analogous statement on theupper half-plane${\displaystyle \mathbf{H}}$ can be made as follows:

Let${\displaystyle f:\mathbf{H}\to \mathbf{H}}$ be holomorphic. Then, for all${\displaystyle z_1,z_2\in \mathbf{H}}$,

${\displaystyle \left|\frac{f(z_1)-f(z_2)}{\overline{f(z_1)}-f(z_2)}\right|\le \frac{\left|z_1-z_2\right|}{\left|\overline{z_1}-z_2\right|}.}$

This is an easy consequence of the Schwarz–Pick theorem mentioned above: One just needs to remember that theCayley transform${\displaystyle W(z)=(z-i)/(z+i)}$ maps the upper half-plane${\displaystyle \mathbf{H}}$ conformally onto the unit disc${\displaystyle \mathbf{D}}$. Then, the map${\displaystyle W\circ f\circ W^{-1}}$ is a holomorphic map from${\displaystyle \mathbf{D}}$ onto${\displaystyle \mathbf{D}}$. Using the Schwarz–Pick theorem on this map, and finally simplifying the results by using the formula for${\displaystyle W}$, we get the desired result. Also, for all${\displaystyle z\in \mathbf{H}}$,

${\displaystyle \frac{\left|f^{\prime }(z)\right|}{\text{Im}(f(z))}\le \frac{1}{\text{Im}(z)}.}$

If equality holds for either the one or the other expressions, then${\displaystyle f}$ must be aMöbius transformation with real coefficients. That is, if equality holds, then

${\displaystyle f(z)=\frac{az+b}{cz+d}}$

with${\displaystyle a,b,c,d\in \mathbb{R}}$ and${\displaystyle ad-bc>0}$.

The proof of the Schwarz–Pick theorem follows from Schwarz's lemma and the fact that aMöbius transformation of the form

maps theunit circle to itself. Fix${\displaystyle z_1}$ and define the Möbius transformations

${\displaystyle M(z)=\frac{z_1-z}{1-\overline{z_1}z},\phi (z)=\frac{f(z_1)-z}{1-\overline{f(z_1)}z}.}$

Since${\displaystyle M(z_1)=0}$ and the Möbius transformation is invertible, the composition${\displaystyle \phi (f(M^{-1}(z)))}$ maps${\displaystyle 0}$ to${\displaystyle 0}$ and the unit disk is mapped into itself. Thus we can apply Schwarz's lemma, which is to say

${\displaystyle \left|\phi \left(f(M^{-1}(z))\right)\right|=\left|\frac{f(z_1)-f(M^{-1}(z))}{1-\overline{f(z_1)}f(M^{-1}(z))}\right|\le |z|.}$

Now calling${\displaystyle z_2=M^{-1}(z)}$ (which will still be in the unit disk) yields the desired conclusion

${\displaystyle \left|\frac{f(z_1)-f(z_2)}{1-\overline{f(z_1)}f(z_2)}\right|\le \left|\frac{z_1-z_2}{1-\overline{z_1}{z}_2}\right|.}$

To prove the second part of the theorem, we rearrange the left-hand side into thedifference quotient and let${\displaystyle z_2}$ tend to${\displaystyle z_1}$.

TheSchwarz–Ahlfors–Pick theorem provides an analogous theorem for hyperbolic manifolds.

De Branges' theorem, formerly known as the Bieberbach Conjecture, is an important extension of the lemma, giving restrictions on the higher derivatives of${\displaystyle f}$ at${\displaystyle 0}$ in case${\displaystyle f}$ isinjective; that is,univalent.

TheKoebe 1/4 theorem provides a related estimate in the case that${\displaystyle f}$ is univalent.

• Nevanlinna–Pick interpolation
• Kobayashi hyperbolicity
• Bloch's principle

• K. Broder (2022), The Schwarz Lemma: An Odyssey, Rocky Mountain Journal of Mathematics, 52, 4, pp. 1141--1155.
• S. Dineen (1989).The Schwarz Lemma. Oxford.ISBN 0-19-853571-6.
• J. Jost,Compact Riemann Surfaces (2002), Springer-Verlag, New York.ISBN 3-540-43299-X(See Section 2.3)
• S.-T. Yau (1978), A general Schwarz lemma for Kähler manifolds, American Journal of Mathematics, 100:1,197--203.

This article incorporates material from Schwarz lemma onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.

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