Prove
Lemma 2 ([Hopf-weight distribution]). For m=2s-1 and 0≤b≤l , the eigenvalue…Lemma 1. The volume form of G(a,q,J) is a^m\,dvol_ g_0 . Its Ricci endomorphism…Proposition
Put r=m-1 . The beta density, equivalently polar coordinates for independent standard normal variables, gives the representation rY_m d=(Z^2)/(Z^2/r+W_r/r), where Z is standard normal and W_r is an independent sum of r squared standard normal variables. Since 𝔼[W_r/r]=1 and Var(W_r/r)=2/r , we obtain rY_m⇒Z^2 . The limiting distribution is continuous and strictly increasing on (0,∞) , so its upper quantile y_∞(v) satisfies r y_m(v)→y_∞(v) for every 0<v<1 . Fatou's lemma therefore gives _ m→∞ (m-1) J_m ≥∫_0^1 y_∞(v)(1/v)\,dv =𝔼[R^2(1)/(Q(R))], where R=|Z| and Q(x)=ℙ(R>x) . The last equality follows since Q(R) is uniform on (0,1) . This expectation is finite: integration of the half-normal density on [x,x+1] gives Q(x)≥√2/π\,e^ -(x+1)^2/2 . For x>0 , integration of the same density gives the classical Mills upper bound (see ) Q(x)=√2π∫_x^∞e^ -t^2/2 \,dt ≤√2π\, e^ -x^2/2 x . Consequently, writing A=𝔼[R^2 R] , we have 𝔼[R^21 Q(R) ] ≥32+A+12π2. Integration by parts against the half-normal density yields 𝔼[R^4 R]=3A+1 . Under the probability measure dν=R^4\,dℙ/3 , Jensen's inequality for the logarithm applied to 1/R now gives A+13=𝔼_ν[ R] ≥-𝔼_ν[1/R] =3 𝔼[R^3] =3 2√2/π . Combining this with proves 𝔼[R^21 Q(R) ] ≥76+(3π)/(4)>2. Here the final strict inequality can be checked without decimal approximations. Indeed, x>x-x^3/3 for x>0 gives π4=12+13 >(505)/(648)>79, so 3π/4>7/3 . Strict convexity of t↦1/t gives the midpoint bounds 73 =∫_1^ 5/3 (dt)/(t)+∫_ 5/3 ^ 7/3 (dt)/(t) >(2/3)/(4/3)+(2/3)/(2)=56. Equations and imply.