Diffusion determines the manifold (Q2898921)

The main theorem of the paper states the following. Let \(M_j\), \(j\in\{1,2\}\), be connected Riemannian manifolds which are regular in capacity (this holds in particular if they are complete). Let \(p\in[1,\infty)\), let \(\Delta_j\) be the Dirichlet Laplace-Beltrami operator on \(M_j\), and let \(S^{(j)}\) be the associated semigroup on \(L_p(M_j)\). Then the following conditions are equivalent: (I) \(M_1\) is isometric to \(M_2\); and (II) there is a lattice homomorphism with dense image, \(U:L_p(M_1)\to L_p(M_2)\), such that \(US^{(1)}_t=S^{(2)}_tU\) for all \(t>0\). Moreover any \(U\) satisfying (II) is an order isomorphism, and there exists \(c>0\) and an isometry \(\tau:M_2\to M_1\) such that \(U\phi=c\,\phi\circ\tau\) for all \(\phi\in L_p(M_1)\). Thus any Riemannian manifold is determined by its heat diffusion if it is regular in capacity.NEWLINENEWLINENow let us explain the concepts used in the statement. For a connected Riemannian manifold \(M\), let \(\tilde{M}\) denote its metric completion, and let \(\partial M=\tilde{M}\setminus M\). It is said that \(M\) is regular in capacity if any function in \(C_0(\tilde M)\cap H_0^1(M)\) vanishes on \(\partial M\), where \(C_0(\tilde M)\) is the closure of \(C_c(\tilde M)\) with respect to the supremum norm, \(H_0^1(M)\) is the closure of \(C^\infty_c(M)\) in \(H^1(M)\), and the intersection means the set of functions in \(C_0(\tilde M)\) whose restrictions to \(M\) are in \(H_0^1(M)\). This condition is clearly satisfied if \(M\) is complete. The condition on \(U:L_p(M_1)\to L_p(M_2)\) to be a lattice homomorphism means that \(U(\phi\wedge\psi)=U\phi\wedge U\psi\), where \((\phi\wedge\psi)(x)=\min\{\phi(x),\psi(x)\}\) a.e. Finally, \(U\) is called an order isomorphism when it is bijective and satisfies \(U\phi\geq0\) if and only if \(\phi\geq0\).