Bauer studied at the University of Erlangen-Nuremberg and received his PhD there in 1953 under the supervision of Otto Haupt and finished his habilitation in 1956, both for work with Otto Haupt. After a short time from 1961 to 1965 as professor at the University of Hamburg he stayed his whole career at the University of Erlangen-Nuremberg. His research focus was the Potential theory, Probability theory and Functional analysis
Bauer received the Chauvenet Prize in 1980 and became a member of the German Academy of Sciences Leopoldina in 1986. Bauer died in Erlangen.