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Heat kernel asymptotics on sub-Riemannian manifolds with symmetries and applications to the bi-Heisenberg group. (English. French summary) Zbl 1437.58018

The authors study small-time asymptotics for the heat kernel of the sub-Laplacian on a sub-Riemannian manifold whose exponential map satisfies certain symmetry conditions. They prove a general result (Theorem 3.2) which is then applied to obtain explicit small-time asymptotics, especially on the cut locus, for the bi-Heisenberg group.
Let \(M\) be a complete sub-Riemannian manifold of dimension \(n\) equipped with a smooth measure \(\mu\), and let \(d\) be the sub-Riemannian distance. Let \(\Delta = \operatorname{div}_\mu \nabla\) be the sub-Laplacian, where \(\nabla\) is the horizontal sub-gradient, and let \(p_t(x,y)\) denote the heat kernel of \(\Delta\) at points \(x,y \in M\). The analysis of \(p_t(x,y)\) typically depends on whether \(y\) is on the cut locus from \(x\); that is, whether minimizing geodesics from \(x\) to \(y\) continue to be minimizing when extended past \(y\). The key hypothesis of the main technical result (Theorem 3.2) is, very roughly, that the minimizing geodesics from \(x\) to \(y\) should form a nice \(r\)-dimensional manifold for some \(r\); the conclusion is then that \(p_t(x,y)\) has small-time asymptotics of the form \[ p_t(x,y) = \frac{C+O(t)}{t^{(n+r)/2}} e^{-d^2(x,y)/4t} \] as \(t \to 0\). The proof is based on a general asymptotic formula from D. Barilari et al. [J. Differ. Geom. 92, No. 3, 373–416 (2012; Zbl 1270.53066)], following techniques developed by S. A. Molchanov [Russ. Math. Surv. 30, No. 1, 1–63 (1975; Zbl 0315.53026)].
Section 4 of the paper applies this result to the bi-Heisenberg group. Strictly speaking, this is not a single group, but rather the two-parameter family of \(5\)-dimensional nilpotent Lie groups whose Lie algebra is spanned by vectors \(X_1, X_2, Y_1, Y_2, Z\) satisfying the bracket relations \[ [X_1, Y_1] = \alpha_1 Z, \quad [X_2, Y_2] = \alpha_2 Z \] for parameters \(\alpha_1 \ge \alpha_2 \ge 0\), with all other brackets vanishing. (The cases \(\alpha_1 = \alpha_2\) and \(\alpha_1 > \alpha_2 = 0\) turn out to be special.) It carries a natural left-invariant sub-Riemannian geometry in which the left-invariant vector fields \(\{X_1, X_2, Y_1, Y_2\}\) are an orthonormal frame for the horizontal distribution. The measure \(\mu\) is taken to be Haar measure, so that the sub-Laplacian \(\Delta\) is also left-invariant.
In Theorem 1.1, the authors explicitly compute the cut locus in this geometry. This is a key ingredient in Theorem 1.2, which obtains small-time asymptotics of the form \(p_t(x,y) = (C+O(t)) t^{-k}\) [G. Ben Arous, Ann. Sci. Éc. Norm. Supér. (4) 21, No. 3, 307–331 (1988; Zbl 0699.35047); Ann. Inst. Fourier 39, No. 1, 73–99 (1989; Zbl 0659.35024)] to show that the leading exponent \(k\) is given by \(k=3\) on the diagonal (\(x=y\)) and by \(k=5/2\) when \(y \ne x\) is not on the cut locus from \(x\). In the remaining case, when \(y \ne x\) is on the cut locus from \(x\), Theorem 3.2 is used to show that \(k=4\) if \(\alpha_1 = \alpha_2\) and \(k=3\) otherwise.
In Section 5, the authors compare the present results with those of R. Beals et al. [Adv. Math. 121, No. 2, 288–345 (1996; Zbl 0858.43009)], where there appears an integral formula for \(p_t\). They show how this formula can also be used to obtain asymptotics for \(p_t(x,y)\) for a restricted set of \((x,y)\) (omitting parts of the cut locus), which are consistent with those of Theorem 1.2.

MSC:

58J37 Perturbations of PDEs on manifolds; asymptotics
58J35 Heat and other parabolic equation methods for PDEs on manifolds
53C17 Sub-Riemannian geometry
53C22 Geodesics in global differential geometry
35H10 Hypoelliptic equations
35K08 Heat kernel
35B40 Asymptotic behavior of solutions to PDEs
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References:

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