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[P=NP]

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Proof. If h satisfies the ordinary heat equation ht = △h with respect to the evolving metric gij(t), then we have d dt R hu = 0 and d dt R hv ≥ 0. Thus we only need to check that for everywhere positive h the limit of R hv as t → T is nonpositive. But it is easy to see, that this limit is in fact zero. 9.4 Corollary. Under assumptions of the previous corollary, for any smooth curve γ(t) in M holds − d dt f(γ(t), t) ≤ 1 2 (R(γ(t), t) + |γ˙ (t)|2) − 1 2(T − t) f(γ(t), t) (9.2) Proof. From the evolution equation ft = −△f + |∇f|2 − R + n 2(T−t) and v ≤ 0 we get ft+1 2R−1 2 |∇f|2− f 2(T−t) ≥ 0. On the other hand,− d dtf(γ(t), t) = −ft− < ∇f, γ˙ (t) >≤ −ft + 1 2 |∇f|2 + 1 2 |γ˙ |2. Summing these two inequalities, we get (9.2). 9.5 Corollary. If under assumptions of the previous corollary, p is the point where the limit δ-function is concentrated, then f(q, t) ≤ l(q, T − t), where l is the reduced distance, defined in 7.1, using p and τ (t) = T − t.

References

Proof P Equals NP is contained in this paper.

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