While not all researchers prioritize social impact, it is undeniably a crucial aspect that adds significance to their work. The objective of this paper is to explore potential gender differences in the social attention paid to researchers and to examine their association with specific fields of study. To achieve this goal, the paper analyzes four dimensions of social influence and examines three measures of social attention to researchers. The dimensions are media influence (mentions in mainstream news), political influence (mentions in public policy reports), social media influence (mentions in Twitter), and educational influence (mentions in Wikipedia). The measures of social attention to researchers are: proportion of publications with social mentions (social attention orientation), mentions per publication (level of social attention), and mentions per mentioned publication (intensity of social attention). By analyzing the rankings of authors -- for the four dimensions with the three measures in the 22 research fields of the Web of Science database -- and by using Spearman correlation coefficients, we conclude that: 1) significant differences are observed between fields; 2) the dimensions capture different and independent aspects of the social impact. Finally, we use non-parametric means comparison tests to detect gender bias in social attention. We conclude that for most fields and dimensions with enough non-zero altmetrics data, gender differences in social attention are not predominant, but are still present and vary across fields.