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E(u(·, t)) is a decreasing function of t, in fact dt E(u(·, t)) = − S 1 ut (s, t) 2 ds. Since this quantity is also bounded from below, because nonnegative, we can ﬁnd a sequence tn → ∞ for which u(·, tn ) will converge to a curve with uiss + Γijk ujs uks = 0, that is, a geodesic. 5. A convexity argument shows that this convergence not only takes place for some sequence tn → ∞, but generally for t → ∞. 3. Therefore, we shall not discuss this here any further. 6) ∂s2 ∂t = 2gij uiss ujss + 2gij (uisss − uist )ujs + 4gij,k uks ujs uiss − gij,k ukt uis ujs + gij,kl uks uls uis ujs .
Let now M be a diﬀerentiable submanifold of the Riemannian manifold N ; dim M = m, dim N = n. We saw already that M then also carries a Riemannian metric. , . , as usual, is the scalar product given by the Riemannian metric. The spaces Tx⊥ M are the ﬁbers of a vector bundle T ⊥ M over M, and T M and T ⊥ M are both subbundles of T N|M , the restriction of T N to M (in a more complicated manner: T N|M = i∗ T N, where i : M → N is the diﬀerentiable embedding of M as a submanifold of N ). In order to see this, one may choose the ﬁrst m basis vectors v1 , .
13) Since E is nonnegative and the integrand also satisﬁes pointwise estimates by step 2, we obtain the conclusion of step 4. 13) ∂ d2 E(u(·, t)) = − 2 dt ∂t =− =− = S1 S1 S1 S1 gij uit ujt 2gij uitt ujt 2gij uisst ujt 2gij uist ujst ≥ 0. 14) Thus, the energy E(u(·, t)) is a convex function of t, and since we already know that d d dt E(u(·, tn )) → 0 for some sequence tn → ∞, we conclude that dt E(u(·, t)) → 0 for t → ∞. Thus, again invoking our pointwise estimates, ut (s, t) → 0 for t → ∞. This implies that u(s) = limt→∞ u(s, t) exists and is geodesic.