The heat kernel weighted Hodge Laplacian

on noncompact manifolds

Author:
Edward L. Bueler

Journal:
Trans. Amer. Math. Soc. **351** (1999), 683-713

MSC (1991):
Primary 58A14, 35J10, 58G11

DOI:
https://doi.org/10.1090/S0002-9947-99-02021-8

MathSciNet review:
1443866

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Abstract | References | Similar Articles | Additional Information

Abstract: On a compact orientable Riemannian manifold, the Hodge Laplacian has compact resolvent, therefore a spectral gap, and the dimension of the space of harmonic -forms is a topological invariant. By contrast, on complete noncompact Riemannian manifolds, is known to have various pathologies, among them the absence of a spectral gap and either ``too large'' or ``too small'' a space . In this article we use a heat kernel measure to determine the space of square-integrable forms and to construct the appropriate Laplacian . We recover in the noncompact case certain results of Hodge's theory of in the compact case. If the Ricci curvature of a noncompact connected Riemannian manifold is bounded below, then this ``heat kernel weighted Laplacian'' acts on functions on in precisely the manner we would wish, that is, it has a spectral gap and a one-dimensional kernel. We prove that the kernel of on -forms is zero-dimensional on , as we expect from topology, if the Ricci curvature is nonnegative. On Euclidean space, there is a complete Hodge theory for . Weighted Laplacians also have a duality analogous to Poincaré duality on noncompact manifolds. Finally, we show that heat kernel-like measures give desirable spectral properties (compact resolvent) in certain general cases. In particular, we use measures with Gaussian decay to justify the statement that every topologically tame manifold has a strong Hodge decomposition.

**[A]**Michael T. Anderson,*harmonic forms on complete Riemannian manifolds*, in ``Geometry and Analysis on Manifolds: Proceedings Katata-Kyoto 1987,'' (Toshikazu Sunada, ed.) Springer, Berlin (1988), 1-19. MR**89j:58004****[AV]**A. Andreotti and E. Vesentini,*Carleman estimates for the Laplace-Beltrami equation on complex manifolds*, Publications mathematiques de l'Institut der Hautes Etudes Scientifiques**no. 25**(1965). MR**30:5333****[AM]**A. Arai & I. Mitoma,*De Rham-Hodge-Kodaira decomposition in -dimensions*, Math. Ann.**291**(1991), 51-73. MR**92k:58007****[Bi]**Jean-Michel Bismut,*The Witten complex and the degenerate Morse inequalities*, J. Differential Geom.**23**(1986), 207-240. MR**87m:58169****[Cha1]**Issac Chavel,*Eigenvalues in Riemannian Geometry*, Academic Press, Orlando, 1984. MR**86g:58140****[Cha2]**Issac Chavel,*Riemannian Geometry-A Modern Introduction*, Cambridge University Press, Cambridge, U. K., 1993. MR**95j:53001****[CG]**J. Cheeger & D. Gromoll,*On the structure of complete manifolds of nonnegative curvature*, Ann. of Math.**96**(1972), 413-443. MR**46:8121****[Che]**Paul R. Chernoff,*Essential self-adjointness of powers of generators of hyperbolic equations*, J. Funct. Anal.**12**(1973), 401-414. MR**51:6119****[CFKS]**Hans L. Cycon, Werner Kirsch, Richard G. Froese, & Barry Simon,*Schrödinger Operators with Application to Quantum Mechanics and Global Geometry*, Springer, Berlin, 1987. MR**88g:35003****[Da1]**E. B. Davies,*Heat Kernels and Spectral Theory*, Cambridge University Press, Cambridge, U. K., 1989. MR**90e:35123****[Da2]**E. B. Davies,*Pointwise bounds on the space and time derivatives of heat kernels*, J. Operator Theory**21**(1989), 367-378. MR**90k:58214****[Dod]**Jozef Dodziuk,*-harmonic forms on rotationally symmetric Riemannian manifolds*, Proc. Amer. Math. Soc.**77 no. 3**(1979), 395-400. MR**81e:58004****[Don]**Harold Donnelly,*The differential form spectrum of hyperbolic space*, Manuscripta Math.**33**(1981), 365-385. MR**82f:58085****[DonL]**Harold Donnelly & Peter Li,*Lower Bounds for the Eigenvalues of Riemannian Manifolds*, Michigan Math. J.**29**(1982), 149-161. MR**83g:58069****[DonX]**Harold Donnelly & Frederico Xavier,*On the differential form spectrum of negatively curved Riemannian manifolds*, Amer. J. Math.**106**(1984), 169-185. MR**85i:58115****[DriH]**Bruce Driver & Yaozhong Hu,*On heat kernel logarithmic Sobolev inequalities*, in ``Proceedings of the Fifth Gregynog Symposium" (I. M. Davies et al. eds.) Stochastic Analysis and Applications, World Scientific, NJ (1996), 189-200. MR**98h:58183****[DriL]**Bruce Driver & Terry Lohrenz,*Logarithmic Sobolev inequalities for pinned loop groups*, J. Funct. Anal.**40**(1996), 381-448. MR**97h:58176****[ER]**K. D. Elworthy & S. Rosenberg,*The Witten Laplacian on negatively curved simply connected manifolds*, Tokyo J. Math.**16**(1992), 513-524. MR**94j:58171****[EF1]**Jose F. Escobar & Alexandre Freire,*The spectrum of the Laplacian of manifolds of positive curvature*, Duke Math. J.**65, no. 1**(1992), 1-21. MR**93d:58174****[EF2]**Jose F. Escobar & Alexandre Freire,*The differential form spectrum of manifolds of positive curvature*, Duke Math. J.**69, no. 1**(1993), 1-41. MR**94b:58097****[Ga]**Matthew P. Gaffney,*Hilbert space methods in the theory of harmonic integrals*, Trans. Amer. Math. Soc.**78**(1955), 426-444. MR**16:957a****[GW]**R. E. Greene & H. Wu,*Function Theory on Manifolds Which Possess a Pole*, Lecture Notes in Mathematics 699, Springer, Berlin, 1979. MR**81a:53002****[Gri]**A. A. Grigor'yan,*Heat kernel of a noncompact Riemannian manifold*, Stochastic analysis (Ithaca, NY, 1993), Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI**57**(1995), 239-263. MR**96f:58155****[Gro]**Leonard Gross,*Analysis on Loop Groups*, in ``Stochastic Analysis and Applications in Physics'' (A.I. Cardoso, et al., eds.) Kluwer Academic Publishers, Dordrecht (1994). MR**96j:58023****[Hs]**Elton P. Hsu,*Logarithmic Sobolev Inequalities on Path Space*, C. R. Acad. Sci. Paris Sér. I Math.**320**(1995), 1009-1012. MR**96e:58167****[K1]**S. Kusuoka,*Analysis on Wiener spaces I. Nonlinear maps*, J. Funct. Anal.**98**(1991), 122-168. MR**93a:58176****[K2]**S. Kusuoka,*Analysis on Wiener spaces II. Differential forms*, J. Funct. Anal.**103**(1992), 229-274. MR**93c:58230****[K3]**S. Kusuoka,*De Rham cohomology of Wiener-Riemannian manifolds*, in ``Proceedings of the International Congress of Mathematicians (Kyoto, 1990)," Math. Soc. Japan, Tokyo (1991), 1075-1082. MR**93e:58198****[L]**John Lott,*The Zero-in-the-Spectrum Question*, Enseign. Math. (2)**42**(1996), 341-376. MR**97h:58145****[MS]**P. Malliavin & D. Stroock,*Short time behavior of the heat kernel and its logarithmic derivatives*, J. Differential Geomerty**44**(1996), 550-570. MR**98c:58164****[P]**M. Ann Piech,*The Exterior Algebra of Wiemann Manifolds*, J. Funct. Anal.**28**(1978), 279-308. MR**58:2851****[RaS]**D. B. Ray & I. M. Singer,*-Torsion and the Laplacian on Riemannian Manifolds*, Adv. Math.**7**(1971), 145-210. MR**45:4447****[ReS]**Michael Reed & Barry Simon,*Methods of Modern Mathematical Physics*, Volume II: Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975. MR**58:12429b****[deR]**Georges de Rham,*Differentiable Manifolds*, Springer, Berlin, 1984. MR**85m:58005****[Sh]**Ichiro Shigekawa,*De Rham-Hodge-Kodaira's decomposition on an abstract Wiener space*, J. Math. Kyoto Univ.**26**(1986), 191-202. MR**88h:58009****[St]**Daniel W. Stroock,*An Estimate on the Hessian of the Heat Kernel*, in ``Itô's Stochastic Calculus and Probability Theory'' (N. Ikeda et al. eds.), Springer, Tokyo (1996). MR**97m:58212****[Wa]**Frank W. Warner,*Foundations of Differentiable Manifolds and Lie Groups*, Springer, New York, 1983. MR**84k:58001****[Wi]**Edward Witten,*Supersymmetry and Morse theory*, J. Differential Geom.**17**(1982), 661-692. MR**84b:58111**

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Additional Information

**Edward L. Bueler**

Affiliation:
Department of Mathematics Sciences, University of Alaska, Fairbanks, Alaska 99775

DOI:
https://doi.org/10.1090/S0002-9947-99-02021-8

Keywords:
Hodge theory,
heat kernels,
weighted cohomology,
Schr\"{o}dinger operators

Received by editor(s):
July 29, 1996

Received by editor(s) in revised form:
February 5, 1997

Article copyright:
© Copyright 1999
American Mathematical Society