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 ${\displaystyle \mathbb {C} }$ .

The Loewner matrix (in linear algebra) is a square matrix or, more specifically, a linear operator (of real ${\displaystyle C^{1}}$  functions) associated with 2 input parameters consisting of (1) a real continuously differentiable function on a subinterval of the real numbers and (2) an ${\displaystyle n}$ -dimensional vector with elements chosen from the subinterval; the 2 input parameters are assigned an output parameter consisting of an ${\displaystyle n\times n}$  matrix.^[3]

Let ${\displaystyle f}$  be a real-valued function that is continuously differentiable on the open interval ${\displaystyle (a,b)}$ .

For any ${\displaystyle s,t\in (a,b)}$  define the divided difference of ${\displaystyle f}$  at ${\displaystyle s,t}$  as

${\displaystyle f^{[1]}(s,t)={\begin{cases}\displaystyle {\frac {f(s)-f(t)}{s-t}},&{\text{if }}s\neq t\\f'(s),&{\text{if }}s=t\end{cases}}}$ .

Given ${\displaystyle t_{1},\ldots ,t_{n}\in (a,b)}$ , the Loewner matrix ${\displaystyle L_{f}(t_{1},\ldots ,t_{n})}$  associated with ${\displaystyle f}$  for ${\displaystyle (t_{1},\ldots ,t_{n})}$  is defined as the ${\displaystyle n\times n}$  matrix whose ${\displaystyle (i,j)}$ -entry is ${\displaystyle f^{[1]}(t_{i},t_{j})}$ .

In his fundamental 1934 paper, Loewner proved that for each positive integer ${\displaystyle n}$ , ${\displaystyle f}$  is ${\displaystyle n}$ -monotone on ${\displaystyle (a,b)}$  if and only if ${\displaystyle L_{f}(t_{1},\ldots ,t_{n})}$  is positive semidefinite for any choice of ${\displaystyle t_{1},\ldots ,t_{n}\in (a,b)}$ .^[3][4][5] Most significantly, using this equivalence, he proved that ${\displaystyle f}$  is ${\displaystyle n}$ -monotone on ${\displaystyle (a,b)}$  for all ${\displaystyle n}$  if and only if ${\displaystyle f}$  is real analytic with an analytic continuation to the upper half plane that has a positive imaginary part on the upper plane. See Operator monotone function.

"During [Loewner's] 1955 visit to Berkeley he gave a course on continuous groups, and his lectures were reproduced in the form of duplicated notes. Loewner planned to write a detailed book on continuous groups based on these lecture notes, but the project was still in the formative stage at the time of his death." Harley Flanders and Murray H. Protter "decided to revise and correct the original lecture notes and make them available in permanent form."^[6] Charles Loewner: Theory of Continuous Groups (1971) was published by The MIT Press,^[7] and re-issued in 2008.^[8]

In Loewner's terminology, if ${\displaystyle x\in S}$  and a group action is performed on ${\displaystyle S}$ , then ${\displaystyle x}$  is called a quantity (page 10). The distinction is made between an abstract group ${\displaystyle {\mathfrak {g}},}$  and a realization of ${\displaystyle {\mathfrak {g}},}$  in terms of linear transformations that yield a group representation. These linear transformations are Jacobians denoted ${\displaystyle J({\overset {u}{v}})}$  (page 41). The term invariant density is used for the Haar measure, which Loewner attributes to Adolph Hurwitz (page 46). Loewner proves that compact groups have equal left and right invariant densities (page 48).

A reviewer said, "The reader is helped by illuminating examples and comments on relations with analysis and geometry."^[9]

• Löwner-John ellipsoid
• Schramm–Loewner evolution
• Loop-erased random walk
• Systoles of surfaces

• Berger, Marcel: À l'ombre de Loewner. (French) Ann. Sci. École Norm. Sup. (4) 5 (1972), 241–260.
• Loewner, Charles; Nirenberg, Louis: Partial differential equations invariant under conformal or projective transformations. Contributions to analysis (a collection of papers dedicated to Lipman Bers), pp. 245–272. Academic Press, New York, 1974.

• Stanford memorial resolution
• O'Connor, John J.; Robertson, Edmund F., "Charles Loewner", MacTutor History of Mathematics Archive, University of St Andrews