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Isometric embeddings via heat kernel. (English) Zbl 1318.53057
Fix two constants \(\rho >0\) and \(0<\alpha <1\). The main result of the paper is Theorem 1. Let \((M, g)\) be a smooth \(n\)-dimensional compact Riemannian manifold without boundary. Then:
1) For any integer \(l\geq 1\) there exists a canonical family of almost isometric embeddings \(\tilde{\Psi }_t:M\rightarrow l^2\) such that \(\tilde{\Psi }^*_tg_{\mathrm{can}}=g+O(t^l)\) as \(t\rightarrow 0_+\), where the above convergence is in \(C^r\)-norm for any \(r\geq 0\).
2) For any integer \(k\geq 2\) satisfying \(k+\alpha <l+\frac{1}{2}\) there exists a constant \(t_0>0\) depending on \(k, \alpha , l, \rho \) and \(g\) such that for \(0<t\leq t_0\) we can truncate \(\tilde{\Psi }_t\) to \(\mathbb{R}^{q(t)}\subset l^2\) and perturb it to a unique \(C^{k, \alpha }\) isometric embedding \(I_t:M\rightarrow \mathbb{R}^{q(t)}\), where dimension \(q(t)\geq t^{-\frac{n}{2}-\rho }\) or \(q(t)=\infty \), and \(\|I_t-\tilde{\Psi }_t\|_{C^{k, \alpha }(M)}=O(t^{l+\frac{1}{2}-\frac{k+\alpha }{2}})\).

53C40 Global submanifolds
57R40 Embeddings in differential topology
35K08 Heat kernel
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