Nope, it is a rumor. It would make your condition worse if you do so. The skin under your eyes is the thinnest part in your body. The sunlight would increase the level of melanin in your skin to make the skin darker, especially you exposure to bright sunlight. In order to reduce the dark circles under the eyes, you should intake supplement of iron in daily diet, because a deficiency of iron could cause dark circles under the eyes. And you had better wear sunglasses in the sun, which could prevent your dark circles under the eyes to become darker. What is more, you can place tomato slices over your eye areas due to it can lighten the skin.
Yes, the suitable amount of sunlight will make your body add some Ca which will stimulate the blood circulation around the eyes. That's why you see the effect of reducing dark circles. However, too much amount of sun lights will harm your eyes which will make your eyes look red because of the uv and other bad lights.
Looks like you are not aware of the causes of dark circles. As a matter of fact, it is resulted from a lot of reasons but not a lack of exposure to sun lights. Perhaps you have got a lack of sleep recently, or have overused your eyes and irregular life routines. Therefore, you need to foster a healthy life mode, take more exercises and get adequate sleep before it is too late.