Peel and dice potatoes, making sure all are in about 1 inch cubes. Place in a large pot and cover with water. Bring to a boil over medium-high heat and then reduce heat to medium low to maintain a rolling boil. Cook until potatoes are fork tender and you can easily pierce them with a fork when touched.
While the potatoes are cooking, lightly salt the cucumbers with 1/4 tsp salt and let sit for 10 minutes to release their juices. Place the cucumbers on a towel and gently squeeze out the excess water after 10 minutes.
Using a spider strainer, remove the potatoes from the water, drain and set aside in a bowl. Meanwhile, blanch the carrots for about 1 minute and remove.
Place the diced potatoes through a ricer or masher and transfer to a large bowl. Add mayonnaise, heavy cream, salt, sugar, black pepper, corn, cucumber, and carrot and stir to incorporate.
Taste and adjust seasoning as needed. Serve.