Will metabolism slow down on a very low calorie diet?

Yes: If your body does not get enough calories, it sends stress signals to the brain. These stress hormones try and preserve the body by decreasing energy used and in doing so slows the body's metabolism.

Most likely: On a very low calorie diet the metabolism will most likely slow down unless you're getting most of your calories from protein or very complex carbohydrates. In general weight-loss causes the metabolism to go down because the body has to run a smaller weight. If you're trying to increase your metabolism go on a little car ketogenic diet for most of your calories come from lean protein and some bastable's and a little bit of fruits as well as healthy fats like planned or fish-based fat.