Isometric immersions with controlled curvatures (Q6601488)

The author presents new results related to the existence and approximation of\Nimmersions. Let \(X\) be a smooth manifold, \(Y=\left( Y,h\right) \) be a smooth\NRiemannian manifold, and \(f_{0}:X\rightarrow Y\) be a \(C^{\infty}\)-immersion.\NLet \(\mathcal{G}^{+}\left( X\right) \) denote the space of smooth\Nsemi-positive definite quadratic forms on \(X\). Let \(\boldsymbol{I}:C^{\infty\N}\left( X,Y\right) \rightarrow\mathcal{G}^{+}\left( X\right) \) denote the\Ncanonical mapping \(f\in C^{\infty}\left( X,Y\right) \longmapsto f^{\ast\N}\left( h\right) \). The author defines the immersion \(f_{0}\) to be\N\(\mathfrak{II}\) if there is an open neighborhood \(\mathcal{F}_{0}\subseteq\NC^{\infty}\left( X,Y\right) \) of \(f_{0}\) such that for every \(f\in\N\mathcal{F}_{0}\), the differential of \(\boldsymbol{I}\) at \(f\), \(\mathcal{L}\N_{f}:T_{f}\left( C^{\infty}\left( X,Y\right) \right) \rightarrow\NT_{\boldsymbol{I}\left( f\right) }\left( \mathcal{G}\right) \), is right\Ninvertible. The author presents an example (1.B) of a free isometric immersion\Nof a torus \(\mathbb{T}^{n}\) with a flat Riemannian metric into \(\mathbb{R}\N^{\frac{n\left( n+1\right) }{2}+n+2}\), and an \(\mathfrak{II}\) isometric\Nimmersion of \(\mathbb{T}^{n}\) into \(\mathbb{R}^{\frac{n\left( n+1\right) }{2}+n+1}\). Let \(f:X\rightarrow Y\) be a smooth\Nimmersion. Let \(m\leq n=\dim\left( X\right) \). Then \(f\) is said to be\N\(m\)-free if the restrictions of \(f\) to all \(m\)-dimensional submanifolds in \(X\)\Nare free. A smooth immersion \(f:X\rightarrow Y\) is called flat \(\mathfrak{II}\N^{\left[ m\right] }\), \(m\leq n=\dim\left( X\right) \), if the induced\NRiemannian metric in \(X\) is flat, and if the restrictions of \(f\) to all flat\N\(m\)-dimensional submanifolds are \(\mathfrak{II}\). For example,\ \(m\)-free\Nisometric immersions of the flat torus \(\mathbb{T}^{n}\) are flat\N\(\mathfrak{II}^{\left[ m\right] }\)-immersions. Recall that for an immersion\N\(f:X\rightarrow Y\), \(\operatorname*{curv}\left( f\right)\N=\operatorname*{curv}\left( f\left( X\right) \right) \) is the supremum of\Ncurvatures in \(Y\) of geodesics in \(\left( X,f^{\ast}h\right) \).\N\NLet \(F:\mathbb{T}^{n}\hookrightarrow B^{N}\left( 1\right) \), where\N\(B^{N}\left( 1\right) \) is the open unit ball in \(\mathbb{R}^{N}\), be a flat \(\mathfrak{II}^{\left[ m\right] }\)-immersion. Let\N\(X=\left( X,g\right) \) be an \(m\)-dimensional compact Riemannian manifold\N(possibly with boundary) such that there is a smooth \((1+\varepsilon\N)\)-bi-Lipschitz immersion \(\Phi_{\varepsilon}:X\rightarrow\mathbb{T}^{n}\). The\Ncompression lemma (1E) states that if \(0<\varepsilon\leq\varepsilon_{0}\left(\Nm\right) \), then there exist \(C^{\infty}\)-isometric immersions \(f_{j}^{\circ\N}:X\hookrightarrow B^{N}\left( \frac{1}{j}\right) ,j=1,2,...\), with\Ncontrolled curvatures of \(f_{j}\):\N\[\N\operatorname*{curv}\left( f_{j}\right) \leq j\operatorname*{curv}\left(\NF\left( \mathbb{T}^{n}\right) \right) +O\left( 1\right) \text{.}\N\]\NAs a corollary, the author derives the local compression corollary (1.F) from\Nthe above compression lemma. Specifically, if the absolute values of all\Nsectional curvatures of \(X=\left( X,g\right) \) are bounded by \(1\), and at a\Npoint \(x_{0}\in X\), the injectivity radius of \(\left( X,g\right) \) at\N\(x_{0}\) is not less than \(1\), then there exists a positive constant \(\rho\N=\rho_{m}\), such that the open ball \(B\left( \rho\right) =B_{x_{0}}\left(\N\rho\right) \subseteq X\) admits a smooth isometric immersion \(f\) to the\NEuclidean space \(\mathbb{R}^{\frac{m\left( m+1\right) }{2}+m+1}\). In addition, there exist immersions\N\(f_{\delta}:B\left( \rho\right) \rightarrow\mathbb{R}^{\frac{m\left( m+1\right) }{2}+m+1}\) for all \(\delta\in(0,1]\) such that\N\(\operatorname*{diam}\left( f_{\delta}\left( B\left( \rho\right) \right)\N\right) \leq\delta\) and \(\operatorname*{curv}\left( f_{\delta}\left(\NB\left( \rho\right) \right) \right) \leq\frac{C_{m}}{\delta}\). For the\Nproof, one should note that the curvature and injectivity radius condition\Nimply that \(B\left( \rho\right) \) is \(\left( 1+3\rho\right) \)-bi-Lipschitz\Nto the \(\rho\)-ball in the flat torus. Section 1 is concluded with the global\Napproximation corollary (1.H), which is a corollary of the approximation lemma\N(1.G). Let \(X=\left( X,g\right) \) and \(Y=\left( Y,h\right) \) be two smooth\NRiemannian manifolds of dimensions \(m\) and \(N\), respectively. Let\N\(f_{0}:X\rightarrow Y\) be a smooth strictly short mapping, i.e., \(g-f^{\ast\N}\left( h\right) \) being positive definite. If \(\left( X,g\right) \) is\Ncompact, \(X\) admits a smooth immersion to \(\mathbb{R}^{n}\) and \(N\geq\frac{n\left( n+1\right) }{2}+n+1\), then there exists a\N\(\delta_{j}\)-approximation of \(f_{0}\), \(0<\delta_{j}\leq1/j,j\in\mathbb{N}\), by isometric \(C^{\infty}\)-immersions \(f_{j}:X\rightarrow Y\) with the\Nfollowing curvatures control: \(\operatorname*{curv}\left( f_{j}\left(\NX\right) \right) \leq j\cdot C_{m}+o\left( j\right) \), \(j\in\mathbb{N}\). The author discusses estimates for \(N\) for hypersurfaces (i.e., \(n=m+1\))\Nand presents the flat torus approximation theorem (1.I) where \(N\geq\N\frac{m\left( m+1\right) }{2}+m+1\).\N\NIn Section 2, the author presents results related to free immersions of\Ntori into the unit balls and introduces another version of the global\Napproximation theorem involving \(m=\dim\left( X\right) \) and \(n\), the\Ndimension of the Euclidean space into which \(X\) can be immersed (see, 2C).\NSection 2 is concluded with a statement about immersions with prescribed\Ncurvatures (2D). Section 3 concludes with an interesting discussion of\Nrelated conjectures.