By scrolling through your Leagues' Leaderboard you may see a PROMOTION ZONE on the top and a DEMOTION ZONE on the bottom.
Make sure you make enough scores during the league period, so that you are above the PROMOTION ZONE marker, to be promoted to a higher league.
If on the contrary, you did a poor score during the league period, and you are below the DEMOTION ZONE marker, then you will be demoted to a lower league.

Keep in mind that the League System is based on how fast other players progress on its Leaderboard,  by being one of the last players on the League Leaderboard, you might drop up to a maximum of 4 Rank Levels if many other players have achieved to obtain a higher amount of points than you have.

If you want to maintain your League Status, make sure to get enough points each week to maintain your current League Rank or drop only one League Rank.