# Charles Loewner

Charles Loewner
Charles Loewner in '63
Born 29 May 1893
Lány, Bohemia
Died 8 January 1968 (aged 74)
Stanford, California
Nationality American
Fields Mathematics
Institutions Stanford University
Syracuse University
University of Prague
Alma mater Karl-Ferdinands-Universität
Doctoral students Lipman Bers
Pao Ming Pu

Charles Loewner (29 May 1893 – 8 January 1968) was an American mathematician. His name was Karel Löwner in Czech and Karl Löwner in German.

Karl Loewner was born into a Jewish family in Lany, about 30 km from Prague, where his father Sigmund Löwner was a store owner.[1][2]

Loewner received his Ph.D. from the University of Prague in 1917 under supervision of Georg Pick. One of his central mathematical contributions is the proof of the Bieberbach conjecture in the first highly nontrivial case of the third coefficient. The technique he introduced, the Loewner differential equation, has had far-reaching implications in geometric function theory; it was used in the final solution of the Bieberbach conjecture by Louis de Branges in 1985. Loewner worked at the University of Berlin, University of Prague, University of Louisville, Brown University, Syracuse University and eventually at Stanford University. His students include Lipman Bers, Roger A. Horn, Adriano Garsia, and P. M. Pu.

## Loewner's torus inequality

In 1949 Loewner proved his torus inequality, to the effect that every metric on the 2-torus satisfies the optimal inequality

${\displaystyle \operatorname {sys} ^{2}\leq {\frac {2}{\sqrt {3}}}\operatorname {area} (\mathbb {T} ^{2}),}$

where sys is its systole. The boundary case of equality is attained if and only if the metric is flat and homothetic to the so-called equilateral torus, i.e. torus whose group of deck transformations is precisely the hexagonal lattice spanned by the cube roots of unity in ${\displaystyle \mathbb {C} }$.

## Book by Loewner

• Loewner, C.: Theory of continuous groups. Notes by H. Flanders and M. Protter. Mathematicians of Our Time 1, The MIT Press, Cambridge, Mass.—London, 1971.