dg.differential geometry – What is the geometric meaning of a Riemannian metric larger than the other on a smooth variety?

Gromov conjectured in 1985 and LLarull proved in 1998 that: If g> g_0 on a sphere, then there exists a point p in a sphere with Sc (p) <Sc_0 (p). Here, g, g_0 are Riemannian metrics and g_0 is the standard metric on the sphere. Sc is the scalar curvature. His proof is by contradiction and uses a non-disappearing index on the sphere.

Is the theorem also valid for sectional curvature? Or if a Riemannian metric greater than the other on the sphere, then there must be a point such that the scalar curvature at this point is smaller than the other?

In general, if the largest metric, the smallest curvature (section curvature, Ricci or scalar) at least in the punctual setting?