Extending the UserManager

Your View shouldn't need to call back-end services on its own, you should provide it all the information it requires either through the @Model or through ViewBag/ViewData/Session.
However, if you need to get the current user you could just use:

var user = await UserManager.GetUserAsync(User);
string userName = user.Name;

If you want to have your own UserManager, though, you'd have to do this:

public class MyManager : UserManager<ApplicationUser>
{
    public MyManager(IUserStore<ApplicationUser> store, IOptions<IdentityOptions> optionsAccessor, IPasswordHasher<ApplicationUser> passwordHasher, IEnumerable<IUserValidator<ApplicationUser>> userValidators, IEnumerable<IPasswordValidator<ApplicationUser>> passwordValidators, ILookupNormalizer keyNormalizer, IdentityErrorDescriber errors, IServiceProvider services, ILogger<UserManager<ApplicationUser>> logger) : base(store, optionsAccessor, passwordHasher, userValidators, passwordValidators, keyNormalizer, errors, services, logger)
    {

    }

    public async Task<string> GetNameAsync(ClaimsPrincipal principal)
    {
        var user = await GetUserAsync(principal);
        return user.Name;
    }
}

And add it to the services:

services.AddIdentity<ApplicationUser, ApplicationRole>()
    .AddEntityFrameworkStores<SomeContext>()
    .AddUserManager<MyManager>()
    .AddDefaultTokenProviders();

Then you'd need to replace the references to UserManager<ApplicationUser> for MyManager.

Thanks to @Camilo-Terevinto I was able to figure out a solution. In my _Layout.cshtml

<span class="m-topbar__username rust-text">
   @{ var u = await UserManager.GetUserAsync(User); }
   @u.Name
 </span>