Store it flat under a book.
Yes, that simple. The curling at the edges happens when you store the mousepad rolled up or leaned against something. Cloth pads with a rubber or foam base want to stay flat. If you force them into a tube or let them sit on edge, the material memory kicks in and those edges curl.
The fix: when you’re not using it, put it flat on a desk or shelf, then place something flat and heavy on top — a large book, a monitor stand, even a cutting board. Give it a day or two and it should flatten back out. Prevent it in the first place by never rolling it tightly for storage. If you must roll it (for travel), roll it loosely and unroll it as soon as you can.
Humidity can also make the edges lift. If your room is damp, that rubber base expands unevenly. A dehumidifier or just keeping the pad in a dry spot helps.
Future you will thank you for the two seconds it takes to set a book on it.
