Yes, as the title says, .Net DateTime Minimum does not equal Sql Servers equivalent. For whatever reason you come across this gem of an issue you have a couple of options to get around it.
1. Use a .net Nullable datetime instead. i.e. Declare your "DateTime" as "DateTime?". Putting question marks after a lot of .Net variable types in C# turns them into nullable types.
2. If you really do need the min date, use SqlDateTime.Min.Value. "Value" gives you access to a DateTime type containing SQLs min date.