Convergence of Riemannian manifolds and Laplace operators. I
Let \(\mathcal F=\{M\}\) be a family of compact, connected Riemannian manifolds. Let \({\Delta}_M\), \(P_{M,t}=e^{-t {\Delta}_M}\) and \(p_M(t, x, y)\) respectively denote the Laplace operator, the heat semigroup and the heat kernel of \(M\). Assume there exist positive constants \(\nu\) and \(C\) such that for any \(M\in {\mathcal F}\), \[ p_M(t, x, x){\leq} Ct^{-\frac {\nu}{2}}, \quad t{\in}(0, 1], \quad x{\in}M. \] In an earlier paper the present author and \textit{H. Kumura} [Tôhoku Math. J., II. Ser. 48, 71--120 (1996; Zbl 0883.58100)] proved i) The family \({\mathcal F}\) is precompact with respect to both the spectral distance SD and the Gromov-Hausdorff distance HD. ii) Let \(\{M_n\}\) be an SD-Cauchy sequence in \({\mathcal F}\), then there exists a compact, connected Hausdorff space \(X\), a nonnegative Radon measure \({\mu}_X\) on \(X\) and a Dirichlet form \((\varepsilon_X, D_{[{\varepsilon}_X]})\) defined on \(L^2(X, {\mu}_X)\), such that the heat semigroup \(P_{X, t}\) has a continuous heat kernel \(p_X(t, x, x')\). There exists a pair of \({\varepsilon}_n\)- spectral approximating maps \(f_n: (M_n, p_{M_n})\rightarrow (X, p_X)\) and \(h_n: (X, p_X)\rightarrow (M_n, p_{M_n})\) with \({\lim}_{n{\rightarrow} {\infty}} {\varepsilon}_n=0\). Furthermore, these maps \(f_n\) and \(h_n\) are also a pair of \({\varepsilon}_n\)-Hausdorff approximating maps with respect to the distances \(D^{\text{spe}}_{X_n}\) on \(M_n\) and \(D^{\text{spe}}_X\) on \(X\). Moreover, the sequence of image measures \(f_{n*}{\mu}_{M_n}\) converges to \({\mu}_X\) with respect to the weak* topology as \(n{\rightarrow}{\infty}.\) iii) \({\forall}i=0, 1, 2, \dots,\) the \(i\)-th eigenvalue \({\lambda}_i(M_n)\) converges to the \(i\)-th eigenvalue \({\lambda}_i(X)\) of the generator \({\mathcal L}_X\) of the Dirichlet form \({\varepsilon}_X\), and if \(u\) is an eigenfunction of \(M_n\) with eigenvalue \({\lambda}_i(M_n)\) and unit \(L^2\)-norm, then there exists an eigenfunction \(v\) of \({\mathcal L}_X\) with eigenvalue \({\lambda}_i(X)\) and unit \(L^2\)-norm such that \[ \sup_{a\in M_n}\| u(a)-v(f_n(a)) \| < \varepsilon_n(i); \quad \sup_{x\in X}\| u(h_n(x))- v(x)\| <{\varepsilon}_n(i) \] with \({\varepsilon}_n(i)\rightarrow 0\) as \(n\rightarrow \infty.\) The present paper proves that: (1) There exist a subsequence \(\{M_m\}\), a sequence of positive numbers \(\{\varepsilon_m\}\) tending to zero, and a continuous pseudo-distance \(\delta\) on \(X\) such that the maps \(f_m: M_m \rightarrow X\) and \(h_m: X \rightarrow M_m\) are a pair of \(\varepsilon_m\)-Hausdorff approximating maps of \((M_m, d_{M_m})\) and \((X, \delta)\). (2) The heat kernel \(p_X\) has the following off-diagonal upper bound \[ p_X(t, x, x')\leq \frac{C(\alpha)} {t^{\frac{\nu}{2}}} \exp\left(-\frac{\delta(x, x')^2} {(4+\alpha)t} \right), \quad t{\in}(0, 1], \quad x, x'\in X, \] where \(\alpha\) is any positive constant and \(C(\alpha)\) is a positive constant depending on C and \({\alpha}.\) (3) Let \(\delta\) be one of the continuous pseudo-distances on \(X\) as in i), and \(C^{0,1}(X, \delta)\) be the space of functions on \(X\) which are Lipschitz continuous with respect to \(\delta\). Then \(C^{0,1}(X, \delta)\subset d[\varepsilon_X]\cap C(X)\) and for \(u\in C^{0,1}(X,\delta)\) and \(v\in D[\varepsilon_X]\), \(\varepsilon_X(u, v)=0\) if \(\text{supp} (u){\cap} \text{supp}(v)=\emptyset\). Moreover, the energy measure of \(u{\in} C^{0,1}(X, \delta)\) is absolutely continuous with respect to the measure \({\mu}_X\) and the Radon-Nikodym derivative \({\Gamma}(u, v)=\frac{d\mu_{\langle u, v\rangle}}{d{\mu}_X}\) is bounded from above by the square of the local dilatation of \(u\), \[ {\Gamma}(u, v)(x)\leq \text{dil} _{\delta}u(x)^2, \] where \(\text{dil}_{\delta}u(x)= \lim_{r\rightarrow 0}\text{dil}_{\delta}(u| _{B_{\delta}(x, r)}),\) and \(\text{dil}_{\delta}(u|_ A)= \inf\{\lambda\mid \| u(x)-u(y)\| \leq \lambda\delta(x, y)\), \(\forall x, y\in A\}\).
scientific article; zbMATH DE number 1794829