In this short note I will give a characterization when a Riemannian manifold with boundary is isometric to a hemisphere. This result is due to Huang-Wu.
Theorem 1 Let be the unit ball in . Suppose is function on such that , and let be it graph. Suppose has scalar curvature , then is isometric to the standard hemisphere .
Remark 1 This is a special case of the Min-Oo conjecture, which states that for a compact Riemannian manifold with boundary, if , is isometric to and is totally geodesic in , then is isometric to the hemisphere. This conjecture is true when (by Eichmair), and is true under different variation of the assumptions, e.g. strengthened to , the case which was proved by Hang-Wang. The conjecture turns out to be false for , as counterexamples were constructed by Brendle-Marques-Neves very recently, using deformation techniques.
We begin with two lemmas.
Proof: As the unit normal is , it is easy to see that
Thus by divergence theorem,
where is the ball of radius and is the normal of its boundary, taking , we are done.
Lemma 3 Let be the shape operator of any hypersurface , and let be its trace-free part (i.e. and ), then
Proof: By Gauss equation,
as . So by Pythagoras theorem,
Substitute it into the Gauss equation, we are done.
Note that if and only if the shape operator is a multiple of the identity, in other words, the hypersurface is umbilical at that point (meaning all directions are principal directions. )
We now prove the main theorem.
Proof: As , Lemma 3 implies . Combining this with Lemma 2, we conclude that . But then this implies , i.e. is umbilical of constant curvature , so it is contained in a unit sphere. Together with the boundary condition , we can get the result.