AWS and TLS Client Authentication

Does anyone know how I can enable TLS Client Authentication on an application running inside an AWS Ubuntu machine.

To be specific, I have an Ubuntu machine on AWS running Linux Container (LXC) and LXD (a framework on top of LXC that provides REST APIs to access Linux Containers, among other things). I generated certificate and key on the Ubuntu host using LXC command line utility. I then tested whether the certificate works locally by running curl command providing the --cert and --key options to it, and everything works fine. I then copied the Certificate over to my local machines (Mac OS X) keyChain and tried accessing the Ubuntu Server (which btw has an open security, allows traffic from everywhere on any port.) It gives me the error : "This server could not prove that it is X.X.X.X . Its security certificate is from ip-X.X.X.X".

I noticed that the certificate has the DNS name value as the private IP address given to the machine by AWS instead of public IP address.

Does any one know how I can access my TLS enabled application inside an AWS Ubuntu machine from outside, public network?

Please let me know if things are not clear and I would be happy to provide more details.