Isolated singularities of extrema of geometric variational problems.

*(English)*Zbl 0583.49028
Harmonic mappings and minimal immersions, Lect. 1st 1984 Sess. C.I.M.E., Montecatini/Italy 1984, Lect. Notes Math. 1161, 206-277 (1985).

[For the entire collection see Zbl 0566.00013.]

The author studies isolated singularities of extrema of functions of geometric type. The basic problem is set up on a cylindrical domain \(\Sigma\) \(\times (0,\infty)\), where \(\Sigma\) is a compact manifold. The functional considered is of the form \[ (*)\quad F(u)=\int^{\infty}_{0}\int_{\Sigma}e^{-mt}(F(\omega,u,\nabla u,\partial u/\partial t)+E(\omega,u,\nabla u,\partial u/\partial t))d\omega dt, \] where m is a constant, \(\nabla\) denotes the gradient on \(\Sigma\), where E has exponential decay with respect to t as \(t\uparrow \infty\), and u may be vector-valued. By an appropriate, and simple, change of variables, the asymptotic behavior as \(t\uparrow \infty\) of extrema of (*) can be related to the asymptotic behavior of solutions to various geometric problems on approach to a given point, which may be a singular point.

The paper is divided into two parts. The goal in the first part is an existence result, obtained essentially by an implicit function theorem method. This result is used to construct harmonic maps and minimal submanifolds with singularities of specified type at a given point. In the second part of the paper, results are obtained showing when a given extremal of a functional of the form (*) has an asymptotic limit as \(t\uparrow \infty\). These results are used to give information concerning the asymptotic behavior of minimal submanifolds and energy minimizing maps near isolated singular points. For example, it is shown that if M is a minimal submanifold with an isolated singularity at q and C is a multiplicity one regular tangent cone for M at q, then C is the unique tangent cone for M at q.

The author studies isolated singularities of extrema of functions of geometric type. The basic problem is set up on a cylindrical domain \(\Sigma\) \(\times (0,\infty)\), where \(\Sigma\) is a compact manifold. The functional considered is of the form \[ (*)\quad F(u)=\int^{\infty}_{0}\int_{\Sigma}e^{-mt}(F(\omega,u,\nabla u,\partial u/\partial t)+E(\omega,u,\nabla u,\partial u/\partial t))d\omega dt, \] where m is a constant, \(\nabla\) denotes the gradient on \(\Sigma\), where E has exponential decay with respect to t as \(t\uparrow \infty\), and u may be vector-valued. By an appropriate, and simple, change of variables, the asymptotic behavior as \(t\uparrow \infty\) of extrema of (*) can be related to the asymptotic behavior of solutions to various geometric problems on approach to a given point, which may be a singular point.

The paper is divided into two parts. The goal in the first part is an existence result, obtained essentially by an implicit function theorem method. This result is used to construct harmonic maps and minimal submanifolds with singularities of specified type at a given point. In the second part of the paper, results are obtained showing when a given extremal of a functional of the form (*) has an asymptotic limit as \(t\uparrow \infty\). These results are used to give information concerning the asymptotic behavior of minimal submanifolds and energy minimizing maps near isolated singular points. For example, it is shown that if M is a minimal submanifold with an isolated singularity at q and C is a multiplicity one regular tangent cone for M at q, then C is the unique tangent cone for M at q.

Reviewer: H.Parks

##### MSC:

49Q20 | Variational problems in a geometric measure-theoretic setting |

31B05 | Harmonic, subharmonic, superharmonic functions in higher dimensions |

53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |

58C15 | Implicit function theorems; global Newton methods on manifolds |

58C25 | Differentiable maps on manifolds |

58K99 | Theory of singularities and catastrophe theory |