This entry devotes the existence of global super-solutions to the Lichnerowicz equations in the study of the Einstein equations in general relativity. This terminology plays an important role in the study of non-CMC case of the Einstein equations in vacuum case.

This terminology first introduced in 2009 by a paper due to M. Holst, G. Nagy and G. Tsogtgerel published in *Comm. Math. Phys.* [here]. In that paper, the solvability comes from the existence of both global sub- and global super-solutions together with some new fixed-point arguments which we have already discussed before [here]. Maxwell recently got a significant result by relaxing the existence of global sub-solutions [here] so that the existence of global super-solutions is enough to guarantee the solvability of the Einstein equations in non-CMC case.

In the vacuum case, the classification depends on the sign of the Yamabe invariants.

In the conformal method, in three dimensions the study of the Einstein equations becomes the study of existence of solution to a coupled system:

and

.

The first equation is usually called the Lichnerowicz equation

where is a symmetric -tensor.

Definition. We say is asuper-solutionof the Lichnerowicz equation if.

Now we have

Definition. We say is aglobal super-solutionof the Lichnerowicz equation if wheneverthen

where is a solution of second equation obtained from .

We are now in a position to derive main results.

**The case of non-negative Yamabe invariants**** (far-from-CMC)**. This result was due to M. Holst et al.

Theorem. Suppose with and , and . If is sufficiently small, then there exists a global super-solution.

*Proof*. Pick such that the scalar curvature of is strictly positive. We claim that if is sufficiently small, and if is additionally sufficiently small, then is a global super-solution.

Suppose , and let be the corresponding solution of the second equation. Note that

by the Cauchy-Schwarz inequality.

It is well-known that there exists a constant such that

.

Hence

Now pick so small that

is positive. It then follows that is a global super-solution so long as is so small that the RHS of the above inequality remains positive.

**The case of negative Yamabe invariants (near-by-CMC)**. This is my own result. I found this about several weeks ago.

Theorem. Suppose with and , and . If is sufficiently small and is bounded away from zero, say , then there exists a global super-solution.

*Proof*. In order to deal with this case, we need a refined pointwise estimate for , precisely, we have

.

This result can be found [here]. We now pick such that the scalar curvature of is constant which also satisfies

.

This is nothing but the Yamabe problem. We claim that if is additionally sufficiently small, then is a global super-solution.

Indeed,

Note that

.

So if is sufficiently small, we have

.

It then follows that is a global super-solution so long as is so small that the RHS of the above inequality remains positive.

## Leave a Reply