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Analysis of the cut locus via the heat kernel. (English) Zbl 1081.58013

Grigor’yan, Alexander (ed.) et al., Surveys in differential geometry. Eigenvalues of Laplacians and other geometric operators. Somerville, MA: International Press (ISBN 1-57146-115-9/hbk). Surveys in Differential Geometry 9, 337-349 (2004).
The very interesting survey under review presents some recent results connecting the behavior of the heat kernel to properties of the cut locus through the study of the Hessian of the logarithm of the heat kernel. It is shown, in particular, that the cut locus of a point is the set of all points at which this Hessian diverges faster than \(1/t\) as \(t\searrow 0.\) Moreover, this rate of divergence is related to the conjugacy and other structural properties.
For the entire collection see [Zbl 1050.53002].

MSC:

58J35 Heat and other parabolic equation methods for PDEs on manifolds
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