To manage pigmentation and dark spots under your eyes, use a gentle, hydrating eye cream containing ingredients like hyaluronic acid, vitamin C, or niacinamide. Avoid rubbing or scratching the area, and limit exposure to potential irritants. Regular application of sunscreen can prevent the pigmentation from worsening, and maintaining adequate sleep and hydration may also help. If the problem persists, consult a dermatologist for proper evaluation and targeted treatment.