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 ¼ teaspoon 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.
Notes
Make sure and drain all the veggies well so the mayo doesn't thin out.