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Additional resources for A Comprehensive Introduction to Differential Geometry, Vol. 4, 3rd Edition

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Sign control of interior terms In applications to global problems, the brute force strategy is generally too crude, so we discuss now the sign strategy. If the multiplier X happens to be a Killing field, (X) π ≡ 0 and the interior term is identically 0. For instance, this is the case for the multiplier X = ∂t for the flat metric. Leaving aside this trivial and miraculous case, we prove now the following theorem. Theorem For any C 2 function R, the following identity holds D R|∇φ|2 dV = 1 2 D − 12 φ 2 ( R)dV − Rφ( φ)dV D φ 2 N, ∇R dv + ∂D Rφ N, ∇φ dv.

The boundary terms will have to be controlled separately, using the standard energy inequality (corresponding to X = −∂t ). Note that in this example λ = 2/r, and (1/r), which is zero for r > 0, is singular at the origin. T As a result, the new interior term D φ 2 ( λ)dV is 0 φ 2 (0, t)dt. 2( φ)(Xφ) = ∂t [· · · ] + ∂i [· · · ] + The preceding examples suggest the following definition. Definition A positive field X for the metric g is a field such that, for some R, I = Qαβ (X) π αβ + R|∇φ|2 is a positive quadratic form in ∇φ.

48 The good components Recall the formula for the components of k, kij = − 12 g 0α (∂i gαj + ∂j gαi − ∂α gij ). Finally, we define the energy at time t to be E(t) = [(T φ)2 + (N φ)2 + | ∇ φ|2 ]dv, 1 2 t recalling the notation | ∇ φ|2 = e1 (φ)2 + e2 (φ)2 . Theorem Assume that the components of k satisfy, for some (i) t − r (ii) t − r 1+ 1+ > 0, 2 2 [k1N + k2N + (k11 + k22 )2 ] ∈ L1t L∞ x , [|T c/c| + |k1N | + |k2N | + |k11 | + |k12 | + |k22 |] ∈ L∞ x,t . Then, for some constant C = C and all T ≥ 0, t −r E(T ) + −1− 0≤t≤T ≤ CE(0) + C [e4 (φ)2 + | ∇ φ|2 ]dV T | φ||T φ|dV + C 0≤t≤T A(t)E(t)dt.

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