×

Diffusion determines the manifold. (English) Zbl 1260.58016

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.
Now 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 \(\phi\geq0\).

MSC:

58J35 Heat and other parabolic equation methods for PDEs on manifolds
58J53 Isospectrality
58J65 Diffusion processes and stochastic analysis on manifolds
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Arendt W., Math. 550 pp 97– (2002)
[2] DOI: 10.3934/dcds.2008.21.21 · Zbl 1155.35011 · doi:10.3934/dcds.2008.21.21
[3] DOI: 10.1023/A:1024181608863 · Zbl 1028.31004 · doi:10.1023/A:1024181608863
[4] DOI: 10.1007/s11118-006-9018-0 · Zbl 1198.35108 · doi:10.1007/s11118-006-9018-0
[5] Gordon C. S., Math. Res. Lett. 1 pp 539– (1994) · Zbl 0840.58046 · doi:10.4310/MRL.1994.v1.n5.a2
[6] DOI: 10.1007/BF01231320 · Zbl 0778.58068 · doi:10.1007/BF01231320
[7] DOI: 10.2307/2313748 · Zbl 0139.05603 · doi:10.2307/2313748
[8] DOI: 10.1073/pnas.51.4.542 · Zbl 0124.31202 · doi:10.1073/pnas.51.4.542
[9] DOI: 10.1007/BF02392720 · Zbl 0912.58041 · doi:10.1007/BF02392720
[10] DOI: 10.1006/jfan.1999.3557 · doi:10.1006/jfan.1999.3557
[11] DOI: 10.1002/cpa.20060 · Zbl 1078.53028 · doi:10.1002/cpa.20060
[12] DOI: 10.1155/S1073792892000047 · Zbl 0769.58054 · doi:10.1155/S1073792892000047
[13] DOI: 10.1007/BF01053457 · Zbl 0840.31006 · doi:10.1007/BF01053457
[14] Urakawa H., Ann. Sci. E’c. Norm. Sup. 15 (4) pp 441– (1982)
[15] DOI: 10.1002/cpa.3160200404 · doi:10.1002/cpa.3160200404
[16] S. R., Comm. Pure Appl. Math. 20 pp 431– (1967)
[17] DOI: 10.1007/PL00001633 · Zbl 0961.58012 · doi:10.1007/PL00001633
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.