1. Scrub sweet potato and pierce with a fork. Cook in a microwave oven at full power (100%) for 4 minutes. Turn over and cook until tender when pressed, 2 to 4 minutes more. (Or bake in a 400° oven until tender when pressed, about 45 minutes.)
2. Slit potato, then let stand until cool enough to handle. Scoop flesh into a blender. Add coconut milk and whirl until smoothly purèed.
3. Meanwhile, in a 10- to 12-inch frying pan over medium heat, frequently stir oil, onion, and garlic until onion is limp, 8 to 10 minutes.
4. To onion, add curry paste, coriander, cumin, broth, and sweet potato mixture, mixing well. Bring to a boil over high heat, stirring, then reduce heat and simmer, covered, to blend flavors, about 20 minutes.
5. Stir crab and red pepper into pan. Ladle soup into bowls. Season to taste with salt.