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Harmonic maps on spaces with conical singularities. (English) Zbl 0758.53023
A compact metric space \(X^{m+1}\) is said to be a space with conical singularities [J. Cheeger, Proc. Natl. Acad. Sci. USA 76, 2103-2106 (1979; Zbl 0411.58003)] if it is a smooth Riemannian manifold except at a finite set \(\Sigma\) of points at which a deleted neighbourhood is isometric to \(M^ m\times (0,\infty)\) equipped with the metric \(r^ 2g_ M+dr^ 2\) for some Riemannian manifold \((M^ m,g_ M)\). The authors extend the celebrated existence theorem of J. Eells and J. H. Sampson [Am. J. Math. 86, 109-160 (1964, Zbl 0122.401)] that any continuous harmonic map from a compact Riemannian manifold \(X\) to a compact Riemannian manifold \(N\) with non-positive sectional curvature is homotopic to a harmonic map to the case where \(X\) is a space with conical singularities, the harmonic map obtained being continuous on \(X\) and smooth on \(X\setminus\Sigma\). The proof is an adaptation of the heat flow method of Eells and Sampson in which fundamental solutions to the heat equation near singularities are blended with that on \(X\setminus\Sigma\). A nice application is given showing the existence of harmonic maps from singular algebraic curves, these having conformally conical singularities.
Reviewer: J.C.Wood (Leeds)

MSC:
53C20 Global Riemannian geometry, including pinching
58E20 Harmonic maps, etc.
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