A bit more than 20% of South Koreans have the surname “Kim”.
In Korea, the family name goes at the start (“Kim”), while the given name comes next (“Min-jae”).
Official Revised Romanization of Korean states that a hyphen should be used for separating different syllables (in this case “Min-jae”). Min-jae is one name that consists of two syllables (“민재”), so “-jae” should be written in lowercase, because it’s the second syllable of his first name.
A few facts in case some people didn’t know: