When using the ASP.NET Core OpenID Connect middleware, after a user has signed in you can access the name of the user by using the User.Identity.Name property. What this does under the cover is to look for the http://schemas.xmlsoap.org/ws/2005/05/identity/claims/name claim type and then return that value. In the case of Auth0, we actually return a user’s name in the name claim of an id_token and this does not get automatically mapped to the http://schemas.
I am busy working on some more samples for ASP.NET Core to demonstrate various techniques people can use Auth0 to authenticate their users. In most of our samples we use the standard OpenID Connect middleware, and one of the things I wanted to do was to pass extra parameters when the request is made to the Authorization endpoint. At Auth0 we allow users to authenticate with multiple social and Enterprise providers.