In this post, we proved the following result (appeared in a paper by Y.Y. Li published in J. Eur. Math. Soc. (2004))
Lemma 1. For and , let be a function defined on and valued in satisfying
Then is constant or .
Later, we considered the equality case in this post and proved the following result:
Lemma 2. Let , and . Suppose that for every there exists such that
Then for some , and
In this post, we consider the third result which can be stated as follows:
Lemma 3. For and , let be a function defined on and valued in satisfying
Then , restricted to , is constant or .
In the statement of Lemma 3 above, by we mean those points such that the last coordinate . Then by we mean the boundary of . Hence, we may identify .
In addition, to make the difference possible for each and , we may identify by a new point sitting in . Hence, by definition, we simply set
where, and from now on, we denote for each point . We are now in a position to prove Lemma 3, see a paper by Dou and Zhu.
Proof of Lemma 3. For each we set with such that as . Then we choose in such a way that
Note that this is possible since we can always choose such that . We also set
Then we immediately obtain
Therefore, by our assumption
Our lemma follows since are arbitrary.