Chapter 1 DIFFERENTIAL GEOMETRY OF REAL MANIFOLDS

292 CHAPTER 1. DIFFERENTIAL GEOMETRY ...
This formula enables one to relate integration on M to that relative to r ∈ R. In fact, if
A(r) denotes the volume of Sr (relative to ω1 ∧ · · · ∧ ωm−1) and U b a(M) denotes the portion
of M defined by the inequalities a ≤ φ(x) ≤ b, then
U b a (M)
ds 2 (dφ, dφ)dv =
b
a
A(r)dr. (1.3.28)
It is customary to refer to (1.3.27) or its integrated form (1.3.28) as the co-area formula.
Proposition 1.3.4 Let M be a compact Riemannian manifold, then λ1 ≥ 1
4 h2 .
Proof - For a C 2 function φ on M let U+(r) = {x ∈ M | φ(x) ≥ r} and U−(r) = {x ∈
M | φ(x) ≤ r}. If r is a regular value then Sr = U+(r)∩U−(r) is a hypersurface decomposing
M into two pieces. Let φ be an eigenfunction for eigenvalue λ1, then
λ1 =
< dφ, dφ >
< φ, φ >
≥ [
M |φ(x)| ds 2 (dφ(x), dφ(x))dv] 2
< φ, φ > 2
where ≥ follows from the Cauchy-Schwartz inequality. Since dφ2 = 2φdφ we obtain
λ1 ≥ 1 [
4
ds2 (dφ2 , dφ2 2 )dv] M
< φ, φ > 2 . (1.3.29)
Assume 0 is a regular value for φ, A(r) be the area of the submanifold of U+(0) defined by
φ 2 = r and V (r) denote the volume of the portion of U+(r). Then
ds2 (dφ2 , dφ2 ) =
∞
A(r)dr
U+(0)
◦
∞
(by definition of h) ≥ h V (r)dr
◦
∞
(integration by parts) = −h rV
◦
′ (r)dr
(−V ′
= ω1 ∧ · · · ∧ ωm−1) = h
φ
U+(0)
2 ω1 ∧ · · · ∧ ωm
= h < φ, φ > .
We obtain a similar inequality by looking at U−(0). The assumption that 0 is a regular value
is inessential, since by looking at U±(ɛ) the same inequalities can be proven. ♣
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