ERRATUM TO “THE HEAT KERNEL WEIGHTED

HODGE LAPLACIAN ON NONCOMPACT MANIFOLDS.”

Ed Bueler

6 July 2000

Theorem 3.4 in [B] requires an additional hypothesis. Add the first sentence: “Suppose that either ker △_{μ}^{p} ≅ L_{μ}^{2}H^{p} for all p or ker △_{μ^{−}}^{p} ≅ L_{μ^{−}}^{2}H^{p} for all p (cf. theorems 5.9 and 5.10).”

The last three sentences of the proof of theorem 3.4 are incorrect and should be replaced by: “Let [ω′] ∈ L_{μ}^{2}H be nonzero. Let ω ∈ [ω′] be the harmonic (△_{μ}ω = 0) representative, so ω ≠ 0 and

d (⋆_{μ}ω) = ± ⋆ ⋆d ⋆ e^{2h}ω = ±e^{2h} (e^{−2h}δe^{2h}) ω = ±e^{2h}δ_{μ}ω = 0

(recall δ = ± ⋆ d⋆ and ⋆⋆ = ±1). Since ∫_{M} ω ∧ ⋆_{μ}ω = ∥ω∥_{μ}^{2} > 0, the pairing is nondegenerate. □”

Finally, the sentence following theorem 3.4 should read: “We have shown exactly that [⋆_{μ}ω] is the Poincaré dual of [ω′] if ω ∈ [ω′] is the harmonic representative.”

Thanks to Denis Bell for finding this error.

References

[B] Edward L. Bueler, The heat kernel weighted Hodge Laplacian on noncompact manifolds, Trans. Amer. Math. Soc. 351 no. 2 (1999), 683–713.

Department of Mathematical Sciences, University of Alaska, Fairbanks, AK.

E-mail address: ffelb@uaf.edu