If a mead is actively fermenting, the answer is that there isn't a problem of "too much" headspace. During fermentation, you're more likely to suffer from too little.
Once a mead finishes fermentation, however (the specific gravity is near or below 1.000, or it stabilizes and does not change for several days), you will want to minimize headspace or control the gas inside the headspace.
If your palette is very sensitive to oxidation, you may never want your meads to fall below the carboy necks. For most people, as long as it remains sealed, a few inches of head space isn't going to be a major issue.
You can fill the headspace with inert gas. CO2 works just fine if you already have the set up for kegging. If not, you can buy “wine preserver spray.” It’s a little pricy, but so are the ingredients for 5 gallons of mead. Whether you're using CO2 or a preserver spray, you need to actually flush the headspace. CO2 easily diffuses with oxygen and ambient air, and does not form a long-lasting blanket of inert gas over the surface of a liquid.
In either case, you need to make sure your airlock stays intact as much as possible. If you go this route, give it another flush every time you have to open the airlock for a sample or test.
You can top up a mead. The simplest solution is to keep a few bottles of a neutral traditional mead on-hand for to use for this purpose. You can also boil water, allow it mostly cool, and add it to the carboy, though this dilutes the mead. The best solution is to make enough mead in the fermenter that you have plenty to completely fill the aging vessel, then rack carefully, losing as little as possible.
To fill the extra space, you can fill extra space with sanitized glass marbles.
This is not a recommended method, but as you may see references to it on reddit and other sites, be aware that it has significant drawbacks:
Glass marbles can easily break glass carboys.
The marbles create a lot of surface area in contact with the mead - surface area that may or may not have been perfectly sanitized.
This creates a lot of clean-up - it can be painstaking to separate marbles, fruit solids, and lees.