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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 $$\|I_t-\tilde{\Psi }_t\|_{C^{k, \alpha }(M)}=O(t^{l+\frac{1}{2}-\frac{k+\alpha }{2}})$$.

##### MSC:
 53C40 Global submanifolds 57R40 Embeddings in differential topology 35K08 Heat kernel
##### Keywords:
isometric embedding; heat kernel
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