No aero bars are not installed on carbon handlebars, because carbon fiber is very sensitive and it does not tolerate the pushing force of the hands and elbows of the rider which are continuously pushing a downward force toward the carbon bars. Aerobars are usually installed on aluminum and steel handlebars due to their greater compression strength and tensile strength. mountain bikers and road bikers usually follow longer mileage trails and if you install aero bars on carbon bars then it will go against the smooth and comfortable ride. carbon bars would easily break or crack after installing aero bars, and if an accident or crash happened, the aero bars would hit the trail trees or trail rocks then which will result in the breaking of the carbon handlebars.
Aerobars are of aluminum metal and steel, these are hard and firm materials and would cause much more scratches and dents on the carbon handlebars, while holding the aero bars, the clamp bolts of the aero bars would become loose and start rotating within the tube of the handlebars which will cause damage to carbon handlebars and also weight of the aluminum metal and steel aero bars would negatively affect pressure on weightless carbon bars. carbon bars do not tolerate enough rotating torque and compression force as a result of any wrench tool or screwdriver, the normal and recommended rotating torque force is 5 newton meters. aluminum and steel aero bars would tolerate much more torque force but if you are clamping them on carbon bars then enough force will break or crack carbon bars. another safety measure is that you can replace aluminum or steel clamps and bolts with carbon clamps for clamping on the carbon handlebars, these carbon clamps would not exert enough force, and also aluminum clamps would rotate when you are placing hands or elbows above the aero bars, but carbon clamps do not allow the clamps of the aero bars to rotate and prevents dents and scratches.
There is also another common issue, the shape, and structure of the carbon handlebar tubes are flat, rounded, and oval-shaped, while the clamp or clips of the aero bars are of different shapes and designs, it is a common difference, and if the handlebar pipe is flat then it is not possible to install flat clips or flat clamps on these bars. weight of the clamps and clip-on are also not compatible with the carbon handlebars, you have to look for compatible weight clip bars. carbon handlebars are also manufactured with different thicknesses and strengths, it depends on the brand and quality of the carbon bars, if the carbon is enough durable to support the clamps and lips on the bar, then the handlebar is compatible with the aero bars. these bars are usually not recommended for carbon handlebars, the reason is that when going on longer rides there are many risks and dangers of falling and crashing because you are riding in a group, due to these issues mostly carbon handlebars are not manufactured and are non-compatible with the aero bars.
Carbon handlebars are usually not manufactured to lift heavyweights of different accessories for mountain and road ikes, trails contain bumps and jumps, and when the tire absorbs bumps of the trail, your hands would also feel vibrations and dampening which will exert and push downward force on the carbon handlebars and aero bars, and the industry and brands of carbon handlebars would not advise putting extra weight pressure on the handlebars.
The appropriate and correct way of installing aero bars on carbon handlebars is that you can rub the handlebar clamp place with sandpaper where you are going to clamp the aero bars, and also rub the internal surface and edges of the clip-on of the aero bars with sandpaper so that these edges would fit and sit accurately and tightly with the carbon handlebars without exerting of too much force. including the sandpaper you also have 2nd option of using carbon assembly paste or carbon preparation, for this purpose, but keep in mind never wrap tape or wrapping material on the clamping place of the carbon handlebars, because these wrapping tapes are not able to keep the aero bar clamps on their place and after a single ride the clamps would start rotating and affecting the carbon handlebars of your bike.
Aerobars are not compatible with carbon fiber handlebars, but in one case, depends on your body weight, if you are exerting a larger force by your elbows on the aero bars, it will affect the carbon bars, but if you are not using the aero bars so much and you are carefully placing both of your elbow and hands on the aero bars, then it would be possible to install these aero bars on the carbon bars, but the better solution would be that replace the carbon handlebars with the steel or aluminum bar for installation of the aero bars and decrease the risk of falling and crashing or sudden accidents.