Will my doctor be able to cure my rosacea?

No. He can control the appearance and discomfort but the condition can be chronic.
No, but can improve! Rosacea is a skin condition that is characterized by increased vascularity, sebaceous hyperplasia, acne cyst formation, and possibly skin thickening (rhinophyma). It can be improved by oral/topical antibiotics in some cases, and the redness can be treated by laser. Rhinophyma can be reduced via laser and surgery. But "cure" is not possible, only reduction of symptoms and signs of the condition!
Improve not cure. Symptoms of rosacea may be improved with education, dietarty restrictions, skin care and a medical regimen. However a cure is genrally not possible at this time.