The stekloff problem for rotationally invariant metrics on the ball (2013)
Let (Br,g) be a ball of radius r and gt;0 in Rn (n≥ 2) endowed with a rotationally invariant metricds2+f2(s)dw2, where dw2 represents the standard metric on Sn-1, the (n-1)--dimensional unit sphere. Assume that Br has non--negative sectional curvature. In this paper we prove that ifh(r) and gt;0 is the mean curvature on ∂ Br and ν1 is the first eigenvalue of the Stekloff problem, thenν1 ≥ h(r). Equality (ν 1 = h(r)) holds only for the standard metric of Rn.